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Kisela, Tomáš. "Basics of Qualitative Theory of Linear Fractional Difference Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.
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Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
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Clinger, Richard A. "Stability Analysis of Systems of Difference Equations." VCU Scholars Compass, 2007. http://hdl.handle.net/10156/1318.
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Sevinik, Adiguzel Rezan. "On The Q-analysis Of Q-hypergeometric Difference Equation." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612758/index.pdf.
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In this thesis, a fairly detailed survey on the q-classical orthogonal polynomials of the Hahn
class is presented. Such polynomials appear to be the bounded solutions of the so called qhypergeometric
difference equation having polynomial coefficients of degree at most two. The
central idea behind our study is to discuss in a unified sense the orthogonality of all possible
polynomial solutions of the q-hypergeometric difference equation by means of a qualitative
analysis of the relevant q-Pearson equation. To be more specific, a geometrical approach has
been used by taking into account every posssible rational form of the polynomial coefficients,
together with various relative positions of their zeros, in the q-Pearson equation to describe a
desired q-weight function on a suitable orthogonality interval. Therefore, our method differs
from the standard ones which are based on the Favard theorem and the three-term recurrence
relation.
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Budke, Albrecht [Verfasser], Rüdiger [Akademischer Betreuer] Seydel, and Pascal [Akademischer Betreuer] Heider. "Finite Difference Methods for the Non-Linear Black-Scholes-Barenblatt Equation / Albrecht Budke. Gutachter: Rüdiger Seydel ; Pascal Heider." Köln : Universitäts- und Stadtbibliothek Köln, 2013. http://d-nb.info/1062696697/34.
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Morávková, Blanka. "Reprezentace řešení lineárních diskrétních systémů se zpožděním." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2014. http://www.nusl.cz/ntk/nusl-233649.
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Disertační práce se zabývá lineárními diskrétními systémy s konstantními maticemi a s jedním nebo dvěma zpožděními. Hlavním cílem je odvodit vzorce analyticky popisující řešení počátečních úloh. K tomu jsou definovány speciální maticové funkce zvané diskrétní maticové zpožděné exponenciály a je dokázána jejich základní vlastnost. Tyto speciální maticové funkce jsou základem analytických vzorců reprezentujících řešení počáteční úlohy. Nejprve je uvažována počáteční úloha s impulsy, které působí na řešení v některých předepsaných bodech, a jsou odvozeny vzorce popisující řešení této úlohy. V další části disertační práce jsou definovány dvě různé diskrétní maticové zpožděné exponenciály pro dvě zpoždění a jsou dokázány jejich základní vlastnosti. Tyto diskrétní maticové zpožděné exponenciály nám dávají možnost najít reprezentaci řešení lineárních systémů se dvěma zpožděními. Tato řešení jsou konstruována v poslední kapitole disertační práce, kde je řešení tohoto problému dáno pomocí dvou různých vzorců.
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Eslick, John. "A Dynamical Study of the Evolution of Pressure Waves Propagating through a Semi-Infinite Region of Homogeneous Gas Combustion Subject to a Time-Harmonic Signal at the Boundary." ScholarWorks@UNO, 2011. http://scholarworks.uno.edu/td/1367.
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In this dissertation, the evolution of a pressure wave driven by a harmonic signal on the boundary during gas combustion is studied. The problem is modeled by a nonlinear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for any transient effects to have dissipated. The zeroth, first and second-order perturbation solutions are obtained and their moduli are plotted against frequency. It is seen that the first and second-order corrections have unique maxima that shift to the right as the frequency decreases and to the left as the frequency increases. Dispersion relations are determined and their limiting behavior investigated in the low and high frequency regimes. It is seen that for low frequencies, the medium assumes a diffusive-like nature. However, for high frequencies the medium behaves similarly to one exhibiting relaxation. The phase speed is determined and its limiting behavior examined. For low frequencies, the phase speed is approximately equal to sqrt[ω/(n+1)] and for high frequencies, it behaves as 1/(n+1), where n is the mode number. Additionally, a maximum allowable value of the perturbation parameter, ε = 0.8, is determined that ensures boundedness of the solution. The location of the peak of the first-order correction, xmax, as a function of frequency is determined and is seen to approach the limiting value of 0.828/sqrt(ω) as the frequency tends to zero and the constant value of 2 ln 2 as the frequency tends to infinity. Analytic expressions are obtained for the approximate general perturbation solution in the low and high-frequency regimes and are plotted together with the perturbation solution in the corresponding frequency regimes, where the agreement is seen to be excellent. Finally, the solution obtained from the perturbation method is compared with the long-time solution obtained by the finite-difference scheme; again, ensuring that the transient effects have dissipated. Since the finite-difference scheme requires a right boundary, its location is chosen so that the wave dissipates in amplitude enough so that any reflections from the boundary will be negligible. The perturbation solution and the finite-difference solution are found to be in excellent agreement. Thus, the validity of the perturbation method is established.
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Bou, Saba David. "Analyse et commande modulaires de réseaux de lois de bilan en dimension infinie." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSEI084/document.
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Les réseaux de lois de bilan sont définis par l'interconnexion, via des conditions aux bords, de modules élémentaires individuellement caractérisés par la conservation de certaines quantités. Des applications industrielles se trouvent dans les réseaux de lignes de transmission électriques (réseaux HVDC), hydrauliques et pneumatiques (réseaux de distribution du gaz, de l'eau et du fuel). La thèse se concentre sur l'analyse modulaire et la commande au bord d'une ligne élémentaire représentée par un système de lois de bilan en dimension infinie, où la dynamique de la ligne est prise en considération au moyen d'équations aux dérivées partielles hyperboliques linéaires du premier ordre et couplées deux à deux. Cette dynamique permet de modéliser d'une manière rigoureuse les phénomènes de transport et les vitesses finies de propagation, aspects normalement négligés dans le régime transitoire. Les développements de ces travaux sont des outils d'analyse qui testent la stabilité du système, et de commande au bord pour la stabilisation autour d'un point d'équilibre. Dans la partie analyse, nous considérons un système de lois de bilan avec des couplages statiques aux bords et anti-diagonaux à l’intérieur du domaine. Nous proposons des conditions suffisantes de stabilité, tant explicites en termes des coefficients du système, que numériques par la construction d'un algorithme. La méthode se base sur la reformulation du problème en une analyse, dans le domaine fréquentiel, d'un système à retard obtenu en appliquant une transformation backstepping au système de départ. Dans le travail de stabilisation, un couplage avec des dynamiques décrites par des équations différentielles ordinaires (EDO) aux deux bords des EDP est considéré. Nous développons une transformation backstepping (bornée et inversible) et une loi de commande qui, à la fois stabilise les EDP à l'intérieur du domaine et la dynamique des EDO, et élimine les couplages qui peuvent potentiellement mener à l’instabilité. L'efficacité de la loi de commande est illustrée par une simulation numériqueNetworks of balance laws are defined by the interconnection, via boundary conditions, of elementary modules individually characterized by the conservation of physical quantities. Industrial applications of such networks can be found in electric (HVDC networks), hydraulic and pneumatic (gas, water and oil distribution) transmission lines. The thesis is focused on modular analysis and boundary control of an elementary line represented by a system of balance laws in infinite dimension, where the dynamics of the line is taken into consideration by means of first order two by two coupled linear hyperbolic partial differential equations. This representation allows to rigorously model the transport phenomena and finite propagation speed, aspects usually neglected in transient regime. The developments of this work are analysis tools that test the stability, as well as boundary control for the stabilization around an equilibrium point. In the analysis section, we consider a system of balance laws with static boundary conditions and anti-diagonal in-domain couplings. We propose sufficient stability conditions, explicit in terms of the system coefficients, and numerical by constructing an algorithm. The method is based on reformulating the analysis problem as an analysis of a delay system in the frequency domain, obtained by applying a backstepping transform to the original system. In the stabilization work, couplings with dynamic boundary conditions, described by ordinary differential equations (ODE), at both boundaries of the PDEs are considered. We develop a backstepping (bounded and invertible) transform and a control law that at the same time, stabilizes the PDEs inside the domain and the ODE dynamics, and eliminates the couplings that are a potential source of instability. The effectiveness of the control law is illustrated by a numerical simulation
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Clark, Rebecca G. "A Study of the Effect of Harvesting on a Discrete System with Two Competing Species." VCU Scholars Compass, 2016. http://scholarscompass.vcu.edu/etd/4497.
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This is a study of the effect of harvesting on a system with two competing species. The system is a Ricker-type model that extends the work done by Luis, Elaydi, and Oliveira to include the effect of harvesting on the system. We look at the uniform bound of the system as well as the isoclines and perform a stability analysis of the equilibrium points. We also look at the effects of harvesting on the stability of the system by looking at the bifurcation of the system with respect to harvesting.
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Goedhart, Eva Govinda. "Explicit bounds for linear difference equations /." Electronic thesis, 2005. http://etd.wfu.edu/theses/available/etd-05102005-222845/.
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Šafařík, Jan. "Slabě zpožděné systémy lineárních diskrétních rovnic v R^3." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2018. http://www.nusl.cz/ntk/nusl-378908.
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Dizertační práce se zabývá konstrukcí obecného řešení slabě zpožděných systémů lineárních diskrétních rovnic v ${\mathbb R}^3$ tvaru \begin{equation*} x(k+1)=Ax(k)+Bx(k-m), \end{equation*} kde $m>0$ je kladné celé číslo, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ a $B=(b_{ij})$ jsou konstantní $3\times 3$ matice. Charakteristické rovnice těchto systémů jsou identické s charakteristickými rovnicemi systému, který neobsahuje zpožděné členy. Jsou získána kriteria garantující, že daný systém je slabě zpožděný a následně jsou tato kritéria specifikována pro všechny možné případy Jordanova tvaru matice $A$. Systém je vyřešen pomocí metody, která ho transformuje na systém vyšší dimenze, ale bez zpoždění \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} kde ${\mathrm{dim}}\ y = 3(m+1)$. Pomocí metod lineární algebry je možné najít Jordanovy formy matice $\mathcal{A}$ v závislosti na vlastních číslech matic $A$ and $B$. Tudíž lze nalézt obecné řešení nového systému a v důsledku toho pak odvodit obecné řešení počátečního systému.
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Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations." [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.
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Xue, Fei. "Asymptotic solutions of almost diagonal differential and difference systems." Morgantown, W. Va. : [West Virginia University Libraries], 2006. https://eidr.wvu.edu/etd/documentdata.eTD?documentid=4556.
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Halfarová, Hana. "Slabě zpožděné lineární rovinné systémy diskrétních rovnic." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2014. http://www.nusl.cz/ntk/nusl-233648.
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Dizertační práce se zabývá slabě zpožděnými lineárními rovinnými systémemy s konstantními koeficienty. Charakteristická rovnice těchto systémů je identická s charakteristickou rovnicí systému, který neobsahuje zpožděné členy. V takovém případě se počáteční dimenze prostoru řešení mění po několika krocích na menší. V jistém smyslu je tato situace analogická podobnému jevu v teorii lineárních diferenciálních systémů s konstantními koeficienty a speciálním zpožděním, kdy původně nekonečně rozměrný prostor řešení (na počátečním intervalu) přejde po několika krocích do konečného prostoru řešení. V práci je pro každý možný případ kombinace kořenů charakteristické rovnice konstruováno obecné řešení daného systému a jsou formulovány výsledky o dimenzi prostoru řešení. Také je zkoumána stabilita řešení.
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Smith, Dale T. "Expotential decay of resolvents of banded matrices and asymptotics of solutions of linear difference equations." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/29218.
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Pellegrini, Joseph Charles. "Neural network emulation of temporal second order linear difference equations." Thesis, Massachusetts Institute of Technology, 1991. http://hdl.handle.net/1721.1/42505.
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Xu, Yuhua. "Disconjugacy and Oscillation Theory of Linear Differential and Difference Equations." DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7130.
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This dissertation is both a literature survey and a presentation of new and independent results.
The survey gives an overview of disconjugacy and oscillation theory for linear differential and difference equations with an emphasis on comparing the theory of difference equations to the theory of differential equations. higher order scalar equations. Second order scalar equations, matrix equations (systems) and Hamiltonian systems are discussed. A chapter on three-term recurrences of systems is also included. Both similarities and differences between differential and difference equations are described.
The new and independent results are for Hamiltonian systems of difference equations. Those results include the representation of any solution in terms of an isotropic solution, necessary conditions for disconjugacy, the development of appropriate Riccati equations and the existence of principal solutions.
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El-Nakla, Jehad A. H. "Finite difference methods for solving mildly nonlinear elliptic partial differential equations." Thesis, Loughborough University, 1987. https://dspace.lboro.ac.uk/2134/10417.
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This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.
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Boquet, Grant Michael. "Geometric Properties of Over-Determined Systems of Linear Partial Difference Equations." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/26352.
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We relate linear constant coefficient systems of partial difference equations (a discretization of a system of linear partial differential equations) satisfying some collection of scalar polynomial equations to systems defined over the coordinate ring of an algebraic variety. This motivates the extension of behavioral systems theory (a generalization of classical systems theory where inputs and outputs are lumped together) to the setting where the ring of operators is an affine domain and the signal space is restricted to signals which satisfy the same scalar polynomial equations. By recognizing the role of the kernel representationâ s Gröbner basis in the Cauchy problem, we extend notions of controllability from the classical behavioral setting to accommodate this generalization. We then address the question as to when an autonomous behavior admits a LivÅ¡ic-system state-space representation, where the state update equations are overdetermined leading to the requirement that the input and output signals satisfy their own compatibility difference equations. This leads to a frequency domain setting involving input and output holomorphic vector bundles and a transfer function given by a meromorphic bundle map. An analogue of the Hankel realization theorem developed by J. Ball and V. Vinnikov then leads to a LivÅ¡ic-system state-space representation for an autonomous behavior satisfying some natural additional conditions.Ph. D.
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Er, Aynur. "Stability of Linear Difference Systems in Discrete and Fractional Calculus." TopSCHOLAR®, 2017. http://digitalcommons.wku.edu/theses/1946.
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The main purpose of this thesis is to define the stability of a system of linear difference equations of the form,
∇y(t) = Ay(t),
and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem.
This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example.
In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.
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Jesus, Hugo Naves. "Método compacto de diferenças finitas para resolver equações de Schrödinger não lineares com dispersão de quarta ordem." Universidade Federal de Goiás, 2016. http://repositorio.bc.ufg.br/tede/handle/tede/6495.
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Finite difference schemes belong to a class of numerical methods used to approximate derivatives. They are widely used to find approximations to differential equations. There are a lot of numerical methods, whose deductions are made through expansions in Taylor Series. Depending on the manner in which expansion is made, it can be combined with other expansions to obtain derivatives with better numerical approximations. Usually when we get numerical derivative with better approaches, it is necessary to increase the amount of points used in the grid. An alternative to this problem are compact methods, which achieve better approximations for the same derivative but without increasing the number of mesh points. This work is an attempt to develop the Compact-SSFD method for the Schrödinger Equation Nonlinear Fourth Order. SSFD methods are used to separate the parts of a differential equation so that each part can be solved separately. For example in the case of non-linear differential equations it is often used to separate the linear parts of nonlinear parts. In Compact-SSFD methods nonlinear parts are resolved exactly as the linear are resolved using compact methods. Our work is inspired in the Dehghan and Taleei’s work where was used the Compact-SSFD method for solving numerically the equation Nonlinear Schrödinger. Before we try to develop our method, the results of the authors was correctly reproduced. But when we try to deduce a method analogous to the differential equation we wanted to solve, which also involves derived from fourth order, we realized that a Compact type method does not get as trivially as in the case of used to approach second-order derivatives.
Métodos de diferenças finitas pertencem a uma classe de métodos numéricos usados para se aproximar derivadas. Eles são amplamente usados para encontrar-se soluções numéricas para equações diferenciais. Há uma grande quantidade de métodos numéricos, cuja as deduções são feitas através de expansões em séries de Taylor. Dependendo da forma em que uma expansão é feita, ela pode ser combinada com outras expansões para obter-se derivadas numéricas com melhores aproximações. Geralmente quando obtemos derivadas numéricas com aproximações melhores, é necessário aumentar-se a quantidade de pontos usados no domínio discretizado. Uma alternativa a este problema são os chamados métodos compact, que obtêm melhores aproximações para a mesma derivada mas sem precisar aumentar a quantidade de pontos da malha. Este trabalho é uma tentativa de desenvolver-se um método Compact-SSFD para a Equação de Schrödinger Não Linear de Quarta Ordem. Métodos SSFD são usados para separar-se as partes de uma equação diferencial tal que cada parte possa ser resolvida separadamente. Por exemplo no caso de equações diferenciais não lineares ele é bastante usado para separar-se as partes lineares das partes não lineares. Nos métodos Compact-SSFD as partes não lineares são resolvidas exatamente enquanto as lineares são resolvidas usando-se métodos compact. Nos baseamos no trabalho de Dehghan e Taleei onde foi usado o Método Compact-SSFD para resolver-se numericamente a Equação de Schrödinger Não Linear. Antes de tentarmos desenvolver nosso método, reproduzimos corretamente os resultados dos autores. Mas ao tentarmos deduzir um método análogo para a equação diferencial que queríamos resolver, que envolve também derivadas de quarta ordem, percebemos que um método do tipo Compact não se obtêm tão trivialmente como no caso dos usados para aproximar-se derivadas de segunda ordem.
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Allen, Benjamin T. Gravagne Ian A. "Experimental investigation of a time scales linear feedback control theorem." Waco, Tex. : Baylor University, 2007. http://hdl.handle.net/2104/5116.
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Junior, Walter Fernandes da Silva. "Equações de diferenças lineares de ordem superior e aplicações." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-26102016-165330/.
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As equações de diferenças desempenham papel fundamental na modelagem de problemas em que o tempo é medido em intervalos discretos, por exemplo, horas, dia, mês, ano. Elas têm aplicações em Matemática, Física, Engenharia, Economia, Biologia e Sociologia. O objetivo desse trabalho é estudar as equações de diferenças lineares de ordem superior, focando aspectos teóricos, métodos de determinação das soluções destas equações e análise da estabilidade de soluções de equações de diferenças de 2a ordem com coeficientes constantes. Exemplos e aplicações ilustram a teoria desenvolvida. É apresentada uma proposta didática relacionada ao tema para ser trabalhada no ensino médio.The difference equations play a key role in shaping problems in which time is measured in discrete intervals, e.g., hour, day, month, year. They may be applied to Mathematics, Physics, Engineering, Economics, Biology and Sociology. The aim of this work is to study the higher-order linear difference equations, focusing on the theoretical aspects, on the methods used to determine the solutions of these equations and also on the analysis of the stability of 2nd-order difference equations with constants coefficients. Examples and applications depict the developed theory. In addition, a didactic proposal related to the topic to be worked on high school is presented.
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Pefferly, Robert J. "Finite difference approximations of second order quasi-linear elliptic and hyperbolic stochastic partial differential equations." Thesis, University of Edinburgh, 2001. http://hdl.handle.net/1842/11244.
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This thesis covers topics such as finite difference schemes, mean-square convergence, modelling, and numerical approximations of second order quasi-linear stochastic partial differential equations (SPDE) driven by white noise in less than three space dimensions. The motivation for discussing and expanding these topics lies in their implications in such physical phenomena as signal and information flow, gravitational and electromagnetic fields, large scale weather systems, and macro-computer networks. Chapter 2 delves into the hyperbolic SPDE in one space and one time dimension. This is an important equation to such fields as signal processing, communications, and information theory where singularities propagate throughout space as a function of time. Chapter 3 discusses some concepts and implications of elliptic SPDE's driven by additive noise. These systems are key for understanding steady state phenomena. Chapter 4 presents some numerical work regarding elliptic SPDE's driven by multiplicative and general noise. These SPDE's are open topics in the theoretical literature, hence numerical work provides significant insight into the nature of the process. Chapter 5 presents some numerical work regarding quasi-geostrophic geophysical fluid dynamics involving stochastic noise and demonstrates how these systems can be represented as a combination of elliptic and hyperbolic components.
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Lazaryan, Shushan, Nika LAzaryan, and Nika Lazaryan. "Discrete Nonlinear Planar Systems and Applications to Biological Population Models." VCU Scholars Compass, 2015. http://scholarscompass.vcu.edu/etd/4025.
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We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential.
We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system.
We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd.
The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates.
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Song, Yongcun. "An ADMM approach to the numerical solution of state constrained optimal control problems for systems modeled by linear parabolic equations." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/551.
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We address in this thesis the numerical solution of state constrained optimal control problems for systems modeled by linear parabolic equations. For the unconstrained or control-constrained optimal control problem, the first order optimality condition can be obtained in a general way and the associated Lagrange multiplier has low regularity, such as in the L²(Ω). However, for state-constrained optimal control problems, additional assumptions are required in general to guarantee the existence and regularity of Lagrange multipliers. The resulting optimality system leads to difficulties for both the numerical solution and the theoretical analysis. The approach discussed here combines the alternating direction of multipliers (ADMM) with a conjugate gradient (CG) algorithm, both operating in well-chosen Hilbert spaces. The ADMM approach allows the decoupling of the state constraints and the parabolic equation, in which we need solve an unconstrained parabolic optimal control problem and a projection onto the admissible set in each iteration. It has been shown that the CG method applied to the unconstrained optimal control problem modeled by linear parabolic equation is very efficient in the literature. To tackle the issue about the associated Lagrange multiplier, we prove the convergence of our proposed algorithm without assuming the existence and regularity of Lagrange multipliers. Furthermore, a worst case O(1/k) convergence rate in the ergodic sense is established. For numerical purposes, we employ the finite difference method combined with finite element method to implement the time-space discretization. After full discretization, the numerical results we obtain validate the methodology discussed in this thesis.
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Hardy, Benjamin Arik. "A New Method for the Rapid Calculation of Finely-Gridded Reservoir Simulation Pressures." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1123.pdf.
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Bonomo, Wescley. "Sistemas dinâmicos discretos: estabilidade, comportamento assintótico e sincronização." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-01072008-164134/.
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Este trabalho é em parte baseado no livro The Stability and Control of Discrete Processes de Joseph P. LaSalle. Nós estudamos equações como x(n+1) = T(x(n)), onde T : \' R POT. m\' \' SETA\' \'R POT. m\' é uma aplicação contínua, com o sistema dinâmico associado \'PI\' (n,x) := \' T POT. n\' (x). Nós fornecemos condições suficientes para a estabilidade de equilíbrios usando o método direto de Liapunov. Também consideramos sistemas discretos da forma x(n+1)=T(n, x(n),\'lâmbda\' ) dependendo de uma parâmetro \' lâmbda\' e apresentamos resultados obtendo estimativas de atratores. Finalmente, nós apresentamos algumas simulações de sistemas acoplados como uma aplicação em sistemas de comunicaçãoThis work is in part based on the book The Stability and Control of Discrete Processes of Joseph P. LaSalle. We studing equations as x(n+1) = T(x(n)), where T : \' R POT.m\' \' ARROW\' \' \' R POT.m\' is continuous transformation, with the associated dynamic system \'PI\' (n,x) := \' T POT.n\' (x). We provide suddicient conditions for stability of equilibria, using Liapunov direct method. We also consider nonautonomous discrete systems of the form x(n + 1) = T(n, x(n), \' lâmbda\') depending on the parameter \'lâmbda\' and present results obtaining uniform estimatives of attractors. We finally we present some simulations on synchronization of coupled systems as an application on communication systems
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Carcuz, Jerez Juan Ramon de Jesus. "An AVO method toward direct detection of lithologies combining P-P and P-S reflection data." Texas A&M University, 2003. http://hdl.handle.net/1969/38.
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Miyaoka, Tiago Yuzo 1990. "Impacto ambiental e populações que interagem : uma modelagem inovadora, aproximação e simulações computacionais." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307267.
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Orientador: João Frederico da Costa Azevedo MeyerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
Made available in DSpace on 2018-08-26T22:13:08Z (GMT). No. of bitstreams: 1 Miyaoka_TiagoYuzo_M.pdf: 9483350 bytes, checksum: 13a6ce526d2a0eca797c7b2c56f65600 (MD5) Previous issue date: 2015
Resumo: Este trabalho trata da modelagem matemática e da simulação computacional de um problema de dinâmica populacional, mais precisamente a interação de um poluente tóxico a duas espécies que competem entre si por espaço e alimento. A modelagem é feita a partir de dispersão e advecção populacional juntamente com o modelo clássico de Lotka-Volterra e reprodução do tipo de Verhulst, mas com um termo inovador para a interação entre poluente e população. Este termo inovador visa a melhoria do modelo a médio e longo prazos, pois tem comportamento assintótico em relação ao tempo. Temos assim um sistema de equações diferenciais parciais não-linear, cuja solução analítica é impossível de ser obtida. Recorremos então a métodos numéricos e simulações computacionais para obter soluções aproximadas. Para isso, utilizamos os métodos de Elementos Finitos (com elementos triangulares de primeira ordem) nas variáveis espaciais e de Diferenças Finitas (mais especificamente, o método de Crank-Nicolson) na temporal, além do método preditor-corretor de Douglas e Dupont para tratar não linearidades, detalhando o procedimento de se obter um software capaz de gerar cenários qualitativamente realistas (os parâmetros utilizados foram estimados). Com o software obtido apresentamos gráficos das soluções aproximadas em cenários hipotéticos distintos, de forma a poder analisar possíveis impactos ambientais causados pela poluição despejada no meio ambiente
Abstract: This work treats the mathematical modeling and computational simulation of a populational dynamics problem, more precisely the interaction of a toxic pollutant in two species which compete with each other for space and food. The modeling is done from populational dispersion and advection together with the classical model of Lotka-Volterra and Verhulst type reproduction, but with a innovative term for the interaction of pollutant and population. This innovative term aims the improvement of the model in the medium and long time, because it has asymptotic behaviour in relation to time. Therefore we have a system of non linear partial differential equations, whose analytical solution is impossible to be obtained. We then appeal to numerical methods and computational simulations to obtain approximated solutions. For this, we use the Finite Elements method (with first order triangular elements) in spatial variables and Finite Differences method (more specifically the Crank-Nicolson method), in addition to the Douglas and Dupont predictor-corrector method to treat non linearities, detailing the process of obtaining a software capable of generating qualitatively realistic scenarios (the parameters used were estimated). With the obtained software we present plots of approximate solutions in different hypothetical scenarios, in order to analyze possible enviromental impacts caused by pollution released into the environment
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada
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Bénézet, Cyril. "Study of numerical methods for partial hedging and switching problems with costs uncertainty." Thesis, Université de Paris (2019-....), 2019. http://www.theses.fr/2019UNIP7079.
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Nous apportons dans cette thèse quelques contributions à l’étude théorique et numérique de certains problèmes de contrôle stochastique, ainsi que leurs applications aux mathématiques financières et à la gestion des risques financiers. Ces applications portent sur des problématiques de valorisation et de couverture faibles de produits financiers, ainsi que sur des problématiques réglementaires. Nous proposons des méthodes numériques afin de calculer efficacement ces quantités pour lesquelles il n’existe pas de formule explicite. Enfin, nous étudions les équations différentielles stochastiques rétrogrades liées à de nouveaux problèmes de switching, avec incertitude sur les coûtsIn this thesis, we give some contributions to the theoretical and numerical study to some stochastic optimal control problems, and their applications to financial mathematics and risk management. These applications are related to weak pricing and hedging of financial products and to regulation issues. We develop numerical methods in order to compute efficiently these quantities, when no closed formulae are available. We also study backward stochastic differential equations linked to some new switching problems, with costs uncertainty
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Lai, Chien-Chou, and 賴建州. "Disconjugacy of Linear Difference Equation." Thesis, 1997. http://ndltd.ncl.edu.tw/handle/45186607864377540117.
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碩士淡江大學
數學學系
85
In this work,we adopt the Green's functions of ordinary difference equations to find the sufficient condition for any even order linear difference equationto be disconjugate.First, we discuss the properties of the Green's functions and the conditions of the nontrivial solutions of boundary-value problems. Applying these properties, we derive the sufficient conditions of the solutions of boundary-value problems to be disconjugate. From the literature, we see that to use Green's functions to research the disconjugacy is not easy. It will be good for working on the properties of disconjugacy by applying the well-known Green's functions. We hope that our work will be good for the future research on the sufficient conditions of the solutions of the boundary-value problems to be disconjugacy.
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Ciou, Bo-Siang, and 邱柏翔. "Stability for the finite difference solutions of the linear wave equation." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/27726201651261662305.
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碩士國立中正大學
應用數學研究所
100
We consider in this paper methods of stability analysis for the finite difference solutions of the linear wave equation utt = uxx (0 < x < 1; t > 0). First, traditional methods, such as separation of variables, Von Neumann analysis are studied. Recently, Furihata [4] and Matuso [6] proposed the so-called discrete variational derivative method, which is applicable to our problem and gives several stable schemes, On the other hand, Samarskii et al. [11] considered difference equations defined in a finite-dimensional Hilbert space and analyzed the stability of their difference solutions. However all the methods mentioned above could not solve the stability for the most primitive explicit scheme (2) where nonuniform time meshes are taken into consideration. Thus we also introduce Cho's result [2], in which the stability is proved by a modified energy.
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Bernardes, Adérito Giordan. "Transferência de calor com condições de Robin." Master's thesis, 2020. http://hdl.handle.net/10316/93583.
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Dissertação de Mestrado em Matemática apresentada à Faculdade de Ciências e TecnologiaNeste trabalho é estudado, do ponto de vista analítico e numérico, o problema de calor linear e não linear com condições de fronteira de Robin. Salientamos que na literatura são consideradas frequentemente condições de fronteira de Dirichlet e o estudo apresentado para o caso contínuo e discreto é baseado no método de energia. No caso discreto esta análise requer a definição de um quadro funcional adequado.No caso linear provamos a estabilidade do problema continuo e, para o problema semi-discreto e completamente discreto, são estabelecidos resultados de estabilidade e convergência.No caso não linear, para o modelo continuo, a estabilidade é concluída localmente para uma solução suave. No caso semi-discreto, a estabilidade local é estabelecida se a solução verifica uma condição de regularidade que pode ser vista como uma versão discreta da imposta no caso continuo. A validade desta condição é concluída a partir do resultado de convergência. Assim, a estabilidade local é estabelecida para uma aproximação semi-discreta que seja uma aproximação de segunda ordem para a solução contínua. A estabilidade da aproximação completamente discreta definida por um método de Euler implícito-explícito é também estudada. Observamos que tal propriedade é válida localmente para aproximações que verificam uma condição análoga à considerada no caso semi-discreto.O comportamento qualitativo dos sistemas estudados é também incluído neste trabalho. Salientamos que no caso linear é também ilustrado o resultado de convergência.
In this work, we study, from analytical and numerical point of view, a initial boundary value problem for a linear and nonlinear heat equation with Robin boundary conditions. We remark that in the literature are often considered these problems but with Dirichlet boundary conditions and the study presented for the continuous and discrete case is based on the energy method. In the discrete case, the analysis requires the definition of a convenient functional context.In the linear case, for the continuous model, we prove the stability and for the semi-discrete and discrete approximations, we establish stability and convergence results.For the nonlinear problem and for the continuous model, the local stability is established for smooth solutions. For the semi-discrete case, the local stability is proved for solutions satisfying a condition that can be seen as a discrete version of the one assumed for the continuous case. We prove that this condition can the concluded from the convergence analysis. The local stability is established for semi-discrete approximations that are second-order approximations for the continuous solutions.The stability of the discrete approximation defined by an implicit-explicit Euler method is also studied. We point out that this property can be deduced locally for discrete solutions that satisfy a condition analogous to the one considered in the semi-discrete case. The qualitative behaviour of the solutions of the problems studied in this thesis is also included. For the linear case, the convergence result is also illustrated.
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Lin, Yu-Jen, and 林育任. "Solvability of Singular Linear Difference Equations." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/09685972858398477823.
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碩士國立臺灣大學
數學研究所
100
In this thesis, we focus on the solvability of singular linear difference equations. We use the geometric viewpoint to survey the properties about the solutions of (E, A, B)-system. First, we consider the simple system—(E, A)-system. We try to use the geometric technique to solve the properties about the solutions of (E, A)-system. And we hope that we can solve it by the way which is independent of the choice of the basis. And (E, A)-system is a special case of the (E, A, B)-system. So, we can use the conclusions which we got before to solve the solution of the (E, A, B)-system. Finally, we have described the solution space of (E, A, B)-system.
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"Mode-dependent finite-difference discretization of linear homogeneous differential equations." Laboratory for Information and Decision Systems, Massachusetts Institute of Technology], 1987. http://hdl.handle.net/1721.1/2983.
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C.-C. Jay Kuo, Bernard C. Levy.Bibliography: p. 32-35.
Supported in part by the Army Research Office under grant no. DAAG-29-84-K-0005 Supported in part by the Advanced Research projects Agency monitored by ONR under contract N00014-81-K-0742 Supported in part by AFOSR contract F49620-84-K-0742
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Jordão, Daniela Sofia Domingues. "Coupling Hyperbolic and Parabolic IBVP: Applications to Drug Delivery." Doctoral thesis, 2020. http://hdl.handle.net/10316/94361.
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Tese no âmbito do Programa Interuniversitário de Doutoramento em Matemática, apresentada à Faculdade de Ciências e Tecnologia da
Universidade de CoimbraIn this thesis, we study a system of partial differential equations defined by a hyperbolic equation - a wave equation, and two parabolic equations - a quasilinear diffusion-reaction equation and a convection-diffusion-reaction equation. In this system, the reaction term of the first parabolic equation depends on the solution of the wave equation, the convective velocity of the second parabolic equation depends on the solution of the wave equation and its gradient, and the diffusion coefficient of the convection-diffusion-reaction equation depends on the solutions of the other two equations. This system arises in the mathematical modeling of several multiphysics processes, as for instance in ultrasound enhanced drug delivery. In this case, the propagation of the acoustic pressure wave, which is described by the hyperbolic equation, induces an increase in the temperature of the target tissue, an increase of the convective drug transport, and the increase of the temperature induces an increase of the diffusion drug transport. Here we propose an algorithm to solve this coupled problem defined in a two-dimensional spatial domain. Our numerical method can be seen, simultaneously, as a fully discrete in space, piecewise linear finite element method, where special quadrature rules are considered, and as a finite difference method defined in nonuniform rectangular grids. We provide the theoretical convergence support where we show that the numerical approximations for the solution of the hyperbolic equation are second order convergent with respect to a discrete $H^1$- norm. This result allows us to conclude that the numerical approximations for the gradient do not deteriorate the quality of the numerical approximations for the solution of the last parabolic equation. For the numerical approximations for the two parabolic equations, we also establish second order convergence but with respect to a discrete $L^2$- norm. These convergence results are proved assuming lower regularity conditions than those usually imposed. In the scope of the finite difference methods, our results can be seen as supraconvergence results because the method uses nonuniform rectangular grids where the correspondent truncation errors are only first order convergent with respect to the norm $\| . \|_\infty$. As the method can be constructed considering piecewise linear finite element method, in the language of the finite element methods our results can be seen as superconvergence results. In fact, it is well known that piecewise linear finite element methods for elliptic equations lead to first order convergent approximations with respect to the usual $H^1$- norm. Numerical results illustrating the theoretical support are also included, highlighting the sharpness of the smoothness assumption on the solutions of the multiphysics problem. It is reported in the literature the use of ultrasound to increase the drug transport and its absorption within the target tissue in different contexts, as for instance in cancer treatment. A simple version of the mathematical problem studied in this work is considered to illustrate the effectiveness of the use of ultrasound to enhance the drug transport.
Nesta tese estudamos um sistema de equações diferenciais de derivadas parciais definido por uma equação hiperbólica – uma equação de onda, e duas equações parabólicas – uma equação de difusão-reação quase linear e uma equação de convecção-difusão-reação. Neste sistema, o termo reativo da primeira equação parabólica depende da solução da equação da onda, e a velocidade convectiva da segunda equação parabólica depende da solução da primeira equação e do seu gradiente. O coeficiente de difusão da última equação depende também das soluções das duas primeiras equações. O problema matemático que motivou esta dissertação surge no contexto de diversos problemas físicos, como por exemplo, no contexto da libertação controlada de fármacos estimulada por ultrassons. Neste caso, a propagação da onda de pressão acústica descrita pela equação hiperbólica, induz um aumento da temperatura no tecido alvo, um aumento no transporte do fármaco, e o aumento da temperatura induz um aumento do transporte difusivo do fármaco. Neste trabalho, propomos um método numérico para o sistema diferencial definido num domínio espacial de duas dimensões. O nosso método pode ser visto, simultaneamente, como um método de elementos finitos segmentado linear discreto no espaço, e como um método de diferenças finitas definido em malhas retangulares não uniformes. Para este método provamos a segunda ordem de convergência, relativamente a uma norma que pode ser vista como uma versão discreta da norma usual de $H^1$, para a discretização da equação hiperbólica. Este resultado permite concluir que a aproximação para o gradiente não deteriora a qualidade da aproximação para a concentração. Estabelecemos que as aproximações para a temperatura e para a concentração também são de segunda ordem, mas relativamente a uma norma que pode ser vista como uma discretização da norma usual de $L^2$. Os resultados de convergência são demonstrados utilizando condições de regularidade mais fracas do que as usadas usualmente. No contexto dos métodos de diferenças finitas, uma vez que consideramos malhas não uniformes onde os erros de truncatura associados são de primeira ordem relativamente à norma $\| . \|_\infty$, os nossos resultados podem ser vistos como resultados de supraconvergência. Visto que o método proposto pode ser visto como um método de elementos finitos segmentado linear, no contexto dos métodos de elementos finitos os nossos resultados podem ser vistos como resultados de superconvergência. De facto, é bem conhecido que os métodos de elementos finitos segmentados lineares para equações elípticas levam a aproximações convergentes de primeira ordem, relativamente à norma usual de $H^1$. Os resultados teóricos obtidos são ilustrados numericamente. A precisão das condições de regularidade impostas às soluções do sistema diferencial contínuo é também analisada numericamente. Podemos encontrar na literatura que o uso de ultrassons leva a um aumento do transporte do fármaco e da sua absorção pelo tecido alvo em diferentes contextos, como por exemplo em tratamentos de cancro. Uma versão simples do sistema estudado neste trabalho é considerada para ilustrar a eficiência do uso dos ultrassons como estímulo ao transporte de fármacos.
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陳思嘉. "Stability criteria for a class of linear delay partial difference equations." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/97110468353432747490.
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碩士逢甲大學
應用數學系
88
This paper is concerned with two linear delay partial difference equations. Sufficient conditions for these equations to be stable and oscillatory are derived.Stable, oscillatory conditions and some examples for these equations are obtained.
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Chang, Chi-Hsin, and 張啟新. "A Study of Grey Modeling Using Linear Differential and Difference Equations." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/82506616738363432765.
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碩士國立臺灣科技大學
機械工程系
87
In this thesis, the comparison between using linear differential and linear difference equation for grey modeling has been presented. It is shown that the use of linear difference equation is more straightforward than that of linear differential equation for grey modeling. In order to reduce the modeling error, genetic algorithms are also used to search the coefficients of the grey model instead of using the tranditional least square approach. Experimental results have shown that the modeling error can be further reduced by using the genetic search of the coefficients of the grey model.
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Tasi, Yu-Dian, and 蔡育典. "Certain Positive Linear Operators Constructed by Some Special Functions and Some Difference Equations." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/86791135148341087296.
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碩士淡江大學
數學學系碩士班
99
For sequence , {c_(n)}, we consider the following difference equation. a_(n)=a_(n+1)-c_(n){[a_(n+1)]^2-S^2}. We will apply the method of backward induction to establish the existence, the uniqueness and behavior of the solution under certain conditions. We know that the difference equation has bounded monotone solution if the positive series sum_{n=1}^infinity c_(n) is convergent. However, the difference equation has no unbounded solution if the positive series sum_{n=1}^infinity c_(n) is divergent. Finally, we consider the existence, the uniqueness and behavior of the solution of the difference equation under sum_{n=1}^infinity c_(n) is not positive series.
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Ndou, Ndivhuwo. "Numerical Simulations of Stokes Flow by the Iterations of Boundary Conditions and Finite Difference Methods." Diss., 2018. http://hdl.handle.net/11602/1185.
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MSc (Applied Mathematics)Mathematics and Applied Mathematics Department
In this study the iteration of boundary conditions method (Chizhonkov and Kargin, 2006) is used together with the well known Finite difference numerical method to solve the Stokes problem over a rectangular domain as well as in irregular domain. The iteration of boundary conditions method has been applied to the Stokes problem in a rectangular domain, 2
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Wu, Yao-jhen, and 吳曜溱. "The Performance of the Seventh Graders’ formularizing on One-variable Linear Equation Presented by Different Representations." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/61077697249659270817.
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碩士國立臺南大學
數學教育學系碩士班
98
The purposes of this study were to examine the seventh graders’ problem solving performance on word problems of one-variable linear equation presented by different representations (word problems, word problems with graphic hint, word problems with keyword hints), and analyze the influences on students’ problem-solving performance among hints with different representations, linguistic intelligence, and mathematical achievement. The participants, who are 125 seventh graders’ students from east district of Tainan city, had to finish three different kinds of questionnaires designed by the author. Descriptive statistics and mix design two-way ANOVA were employed to analyze the collected data, and semi-structured interview was adopted to analyze the differences of the problem-solving strategies and the fondness of the hints’ representations among students’ different levels of linguistic intelligence. The followings are the results: 1.Students’ performance on the questionnaire of word problems：In terms of linguistic intelligence, the average of the questionnaire listed in descending order of scores are high linguistic intelligence, the medium ones and the low ones except the second、fourth、fifth question. In terms of mathematical achievement, the average of the questionnaire listed in descending order of scores is the high mathematical achievement, the medium mathematical achievement, and the low mathematical achievement. 2.Students’ performance on the questionnaire of word problems with graphic hints：In terms of linguistic intelligence, the average of the questionnaire listed in descending order of scores is the high linguistic intelligence, the medium linguistic intelligence, and the low linguistic intelligence except the first and third question. In terms of mathematical achievement, the average of the questionnaire listed in descending order of scores is the high mathematical achievement, the medium mathematical achievement, and the low mathematical achievement. 3.Students’ performance on the questionnaire of word problems with keyword hints：In terms of linguistic intelligence, the average of the questionnaire listed in descending order of scores is the high linguistic intelligence, the medium linguistic intelligence, and the low linguistic intelligence except the first and third question. In terms of mathematical achievement, the average of the questionnaire listed in descending order of scores is the high mathematical achievement, the medium mathematical achievement, and the low mathematical achievement except the fifth question. 4.On different levels of mathematical achievement, the average of the questionnaire with graphic hints is better than that of the questionnaire with the keyword hints, and the average of the questionnaire with keyword hints is better than that of the questionnaire with word problems. Besides, there is a significant difference between graphic hints and word problems in each group. But, the significant difference between keyword hints and word problems is only in the group of medium mathematical achievement and the high mathematical achievement. 5.On different levels of linguistic intelligence, the average of the questionnaire with graphic hints is better than that of the questionnaire with the keywords hint, and the average of the questionnaire with keyword hints is better than that of the questionnaire with word problems. There is a significant difference between the groups of the word problems and the groups of the keyword hints. And also, there is a significant difference between the groups of the word problems and the groups of the graphic hints. 6.Most of the solvers prefer the questionnaire with graphic hints and are used to employing graphs to understand the questions. We can also find the phenomenon in different levels of linguistic intelligence. Some students with low linguistic intelligence have difficulties in reading, and they can’t solve the questions just because of misunderstanding the words and phrases.
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Kuo, Chin-jung, and 郭錦蓉. "The Influence of the Seventh Graders’ Problem Solving Performance on Linear Equation Presented by Different Representations." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/48113021841875098262.
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碩士國立臺南大學
應用數學研究所碩士班
101
The purpose of this study was to understand the seventh graders’ problem-solving performance on linear equation by different representations (graphs (G), numbers(N), equations(E), and verbal(V)), investigate the relationship within the reciprocal concept of representation question, and analyze the differences of different gender and mathematical ability in the students’ overall problem-solving performance and three different dimensions (visual (visual), numbers (numeric), abstract (abstract)) problem-solving performance. The participants were 32 boys and 44 girls, coming from one junior high school in Tainan City. They completed the linear equation examination designed by the author. Descriptive statistics, Pearson-product moment Correlation, Independent-Sample T-Test, ANOVA were then employed to analyze the collected data, and Semi-structured interview was adopted to analyze the differences of the problem-solving strategies and the representation fondness among the students with different levels of mathematical ability. The research results are as follows: 1. The participants’ average correct rate on the questions of linear equation presented by different representations: higher correct rate on EN, VG, and GE, but lower correct rate on VE. Overall, students’ performance on graphs representation was the most centralized, but students’ performance on verbal representation was relatively scattered. 2. There was significantly positive correlation within the reciprocal concept of representation question (GN v.s. NG, GE v.s. EG, NE v.s. EN). 3. There was no significant difference between different genders in the overall problem-solving performance presented by different representations and three different dimensions (visual (visual), numbers (numeric), abstract (abstract)) problem-solving performance. 4. There were significant differences among students with different levels of mathematical ability in the overall problem-solving performance presented by different representations, where there were significant differences in the numeric and abstract dimension problem-solving performance, but no significant relation in the visual dimension problem-solving performance between high mathematical ability and low mathematical ability. 5. There were differences on the problem-solving strategies and the representation fondness among the students with different levels of mathematical ability. The fondness among students with high mathematical ability is most consistent, which graphs representation was most preferred and equation representation, numbers representation and verbal representation followed in sequence, while bigger fondness differences occurred among students with middle and low mathematical ability.
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Tsai, Fang-Yu, and 蔡芳榆. "Estimating Control Rates of Three Different Insecticides by Generalized Estimating Equation and Generalized Linear Mixed Model." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/05460098030021084419.
Full textAbstract:
碩士國立臺灣大學
農藝學研究所
93
The aim of this study is to estimate the control rate of three bait-formulated insecticides of red imported fire ants. Two field experiments were conducted, respectively, in Taoyuan and Chiayi county where the red ant infestation were spotted and the three different insecticides applied are Fipronil, Pyripronxyfen and Spinosyns. Repeated counts of ant mound number in each field plot of size 100 $m^2$ were recorded by the researchers in the local agricultural experimental station during the period of eight weeks. Two statistical procedures were employed to analyzed these two data sets and both are of generalized linear models. First one is a GEE model and the second one is a generalized mixed-effects model (GLMM). The former is relatively easy however the later demands more effort to determinea decent model. The estimates of control rate resulted from GEE and GLMM are quite similar though the standard errors are different substantially. We recommend that the SE''s due to GLMM be applied to construct relevant confidence intervals, since variance structure of GLMM does have a better description to the variation of data collected. One interesting result is that all three insecticides show remarkable consistancy in control rates in the two experiment sites.
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Tsai, Chi-lin, and 蔡其霖. "The Influence of the Eighth Graders’ Performance of the Problem Solving on the Linear Equation with Two Variables Concepts Presented by Different Representations." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/38363605822327957432.
Full textAbstract:
碩士國立臺南大學
應用數學研究所碩士班
100
The purpose of this study was to understand the current situation of the eighth graders’performance of the problem solving on different linear equation with two variables concepts and different representations, test if any significant correlation between Chinese comprehensive ability and the performance on the contextual representation questions, and investigate the relevance relationship among gender, different representations, different mathematical abilities and different linear equation with two variables concept problems.The researcher used questionnaires to collect research data. 139 participants,71 boys and 68 girls,coming from one of Tainan City school completed three types of instruments, including visual representation,phrase representation, contextual representation,and chinese comprehensive ability.Descriptive approach,Pearson correlation, Independent T-test,and Single-factor analysis method were employed to analyze the research data.The research results are in the following. 1.The participants’ average score on performance of the three different representations: contextual representation is higher than phrase and visual one. Phrase representation is lightly better than visual one. 2.Problem solving performance of students in different kinds of questions on the concept of the Linear Equations： (a) Two straight lines intersect on the concept of kinds of questions: the phrase representation better than contextual representation, and contextual representation better than visual representation. (b) Change of point in a straight line on the concept of kinds of questions: contextual representation better than phrase representation, and phrase representation better than visual representation. (c) Linear Equations application of the concept of questionsl：the phrase representation better than contextual representation, and contextual representation better than visual representation. 3.There was significantly correlation between chinese comprehensive ability and the performance on the contextual representation questions. Solving the first step is to understand the meaning of the questions. 4.About 50% of participants passed the linear equation with two variables’type questions,and reached a mature situation , but still there is room for improvement. 5.The girls of the eighth graders surpassed the boys in the every type regarding to representations,but there doesn''t exist significant difference. Gender was not significant correlation among visual representation, phrase representation and contextual representation. 6.Contextual representation is higher than phrase and visual one and phrase representation is lightly better than visual one for students of different mathematical abilities.
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Pan, Heng-tsu, and 潘亨足. "A Study of Problem-Solving Strategies in Linear Equations with One Unknown for Junior High School Students under the Different Understanding of the Equal sign." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/11521157682018591266.
Full textAbstract:
碩士國立中山大學
教育研究所
98
The purpose of this study is to investigate students’ understanding of the equal sign, problem-solving strategies of equations with one unknown, and the strategies of solving equations with one unknown under different understanding types of the equal sign. To achieve this purpose, the investigator did a survey and development instruments. The participants were 203 seventh-grade students in a convenient sample. Descriptive statistics were used to analyze data in frequency and percentages. The main results was that participants with a relational definition of the equal sign were the most (close to 50%), and an operational definition of the equal sign was approximately 1/4. There was a higher successful performance associated with a relational definition than an operational definition. The primary strategy of operations on the left-hand side of equal sign is the mathematical operations; the main strategy of an unknown quantity on the right-hand side of the equal sign was by going to the parenthesis-reverse and bringing different denominators into a common denominator; the principal strategies of one number on the right-hand side of the equal sign, equations with operations on the right side of the equal sign and equations with operations on both sides of the equal sign are cover-up and transposing. To use the strategies of trial and error substitution and undoing is minority in a linear equation with one unknown. The strategy of an operational definition participant in five equal sign topics is similar to the strategy of one with a relational definition. However, those with a relational definition apply multiple strategies and exhibited varying particular and algebraic property. On the other hand, participants with an operational definition used arithmetic strategies more frequently than participants with a relational definition. From the above results, the researcher suggested instruction to include strategies with algebraic property to help learners to develop stable understanding of the equal sign in Algebra. In addition, the recommendation is to have teachers to encourage students to apply multi-dimensional thinking and different strategies in algebraic problem-solving.