Relevant bibliographies by topics / Polynomial collocation method

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Polynomial collocation method'

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Published: 7 June 2025
Last updated: 25 June 2025

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Journal articles on the topic "Polynomial collocation method"

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C, Kayelvizhi, and Emimal Kanaga Pushpam A. "Application of Polynomial Collocation Method Based on Successive Integration Technique for Solving Delay Differential Equation in Parkinson's Disease." Indian Journal of Science and Technology 17, no. 2 (2024): 112–19. https://doi.org/10.17485/IJST/v17i2.2573.

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Abstract <strong>Background/Objective:</strong>&nbsp;Parkinson&rsquo;s disease (PD) is a neurological disorder that is often prevalent in elderly people. This is induced by the reduction or loss of dopamine secretion. The main objective of this work is to apply the polynomial collocation method using successive integration technique for solving delay differential equations (DDEs) arising in PD models.&nbsp;<strong>Methods:</strong>&nbsp;The polynomial collocation method based on successive integration techniques is proposed to obtain approximate solutions of the PD models. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered.&nbsp;<strong>Findings:</strong>&nbsp;Numerical examples of two PD models have been considered to demonstrate the efficiency of the proposed method. Numerical simulations of the proposed method are well comparable to the simulation by step method using Picard approximation.&nbsp;<strong>Novelty:</strong>&nbsp;The numerical simulation demonstrates the reliability and efficiency of the proposed polynomial collocation method. The proposed method is very effective, simple, and suitable for solving the nonlinear DDEs model of PD and similar real-world problems exist in different fields of science and engineering. <strong>Keywords:</strong> Polynomial collocation method, Successive Integration Technique, Delay differential equation, Parkinson's disease, Simulation
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Molla, Hasib Uddin, and Goutam Saha. "Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 11–25. http://dx.doi.org/10.3329/ganit.v38i0.39782.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method.&#x0D; GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25
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Mirkov, Nikola, Nicola Fabiano, Dušan Nikezić, Vuk Stojiljković, and Milica Ilić. "Symmetries of Bernstein Polynomial Differentiation Matrices and Applications to Initial Value Problems." Symmetry 17, no. 1 (2024): 47. https://doi.org/10.3390/sym17010047.

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In this study, we discuss the symmetries underlying Bernstein polynomial differentiation matrices, as they are used in the collocation method approach to approximate solutions of initial and boundary value problems. The symmetries are brought into connection with those of the Chebyshev pseudospectral method (Chebyshev polynomial differentiation matrices). The treatment discussed here enables a faster and more accurate generation of differentiation matrices. The results are applied in numerical solutions of several initial value problems for the partial differential equation of convection–diffusion reaction type. The method described herein demonstrates the combination of advanced numerical techniques like polynomial interpolation, stability-preserving timestepping, and transformation methods to solve a challenging nonlinear PDE efficiently. The use of Bernstein polynomials offers a high degree of accuracy for spatial discretization, and the CGL nodes improve the stability of the polynomial approximation. This analysis shows that exploiting symmetry in the differentiation matrices, combined with the wise choice of collocation nodes (CGL), leads to both accurate and efficient numerical methods for solving PDEs and accuracy that approach pseudospectral methods that use well-known orthogonal polynomials such as Chebyshev polynomials.
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Bin Jebreen, Haifa, and Carlo Cattani. "Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials." Symmetry 15, no. 11 (2023): 2076. http://dx.doi.org/10.3390/sym15112076.

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To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy.
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Imani, Ahmad, Azim Aminataei, and Ali Imani. "Collocation Method via Jacobi Polynomials for Solving Nonlinear Ordinary Differential Equations." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/673085.

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We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials.
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Pushpam, A. E. K., and C. Kayelvizhi. "Polynomial Collocation Methods Based on Successive Integration Technique for Solving Neutral Delay Differential Equations." Journal of Scientific Research 16, no. 2 (2024): 343–54. http://dx.doi.org/10.3329/jsr.v16i2.65152.

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This paper presents a new approach to using polynomials such as Hermite, Bernoulli, Chebyshev, Fibonacci, and Bessel to solve neutral delay differential equations. The proposed method is based on the truncated polynomial expansion of the function together with collocation points and successive integration techniques. This method reduces the given equation to a system of nonlinear equations with unknown polynomial coefficients which can be easily calculated. The convergence of the proposed method is discussed with several mild conditions. Numerical examples are considered to demonstrate the efficiency of the method. The numerical results reveal that the proposed new approach gives better results than the conventional operational matrix approach of the polynomial collocation method. It demonstrates the reliability and efficiency of this method for solving linear and nonlinear neutral delay differential equations.
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Wu, Nan-Jing, and Ting-Kuei Tsay. "A robust local polynomial collocation method." International Journal for Numerical Methods in Engineering 93, no. 4 (2012): 355–75. http://dx.doi.org/10.1002/nme.4380.

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C. Kayelvizhi and A. Emimal Kanaga Pushpam. "Subdomain collocation method based on successive integration technique for solving delay differential equations." International Journal of Science and Research Archive 11, no. 2 (2024): 382–90. http://dx.doi.org/10.30574/ijsra.2024.11.2.0429.

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The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.
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Leyk, Zbigniew. "A C0-Collocation-like method for elliptic equations on rectangular regions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 38, no. 3 (1997): 368–87. http://dx.doi.org/10.1017/s0334270000000734.

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AbstractWe describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.
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Arif, M. Ziaul, Ahmad Kamsyakawuni, and Ikhsanul Halikin. "Modified Chebyshev Collocation Method for Solving Differential Equations." CAUCHY 3, no. 4 (2015): 208. http://dx.doi.org/10.18860/ca.v3i4.2923.

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This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.
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Dissertations / Theses on the topic "Polynomial collocation method"

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Allen, James Brandon. "Estimating Uncertainties in the Joint Reaction Forces of Construction Machinery." Thesis, Virginia Tech, 2009. http://hdl.handle.net/10919/33046.

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In this study we investigate the propagation of uncertainties in the input forces through a mechanical system. The system of interest was a wheel loader, but the methodology developed can be applied to any multibody systems. The modeling technique implemented focused on efficiently modeling stochastic systems for which the equations of motion are not available. The analysis targeted the reaction forces in joints of interest. <p> The modeling approach developed in this thesis builds a foundation for determining the uncertainties in a Caterpillar 980G II wheel loader. The study begins with constructing a simple multibody deterministic system. This simple mechanism is modeled using differential algebraic equations in Matlab. Next, the model is compared with the CAD model constructed in ProMechanica. The stochastic model of the simple mechanism is then developed using a Monte Carlo approach and a Linear/Quadratic transformation method. The Collocation Method was developed for the simple case study for both Matlab and ProMechanica models. <p> Thus, after the Collocation Method was validated on the simple case study, the method was applied to the full 980G II wheel loader in the CAD model in ProMechanica. <p> This study developed and implemented an efficient computational method to propagate computational method to propagate uncertainties through â black-boxâ models of mechanical systems. The method was also proved to be reliable and easier to implement than traditional methods.<br>Master of Science
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Parts, Inga. "Piecewise polynomial collocation methods for solving weakly singular integro-differential equations /." Online version, 2005. http://dspace.utlib.ee/dspace/bitstream/10062/851/5/parts.pdf.

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Kaiser, Robert. "Polynomiale Kollokations-Quadraturverfahren für singuläre Integralgleichungen mit festen Singularitäten." Doctoral thesis, Universitätsbibliothek Chemnitz, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-229930.

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Viele Probleme der Riss- und Bruchmechanik sowie der mathematischen Physik lassen sich auf Lösungen von singulären Integralgleichungen über einem Intervall zurückführen. Diese Gleichungen setzen sich im Wesentlichen aus dem Cauchy'schen singulären Integraloperator und zusätzlichen Integraloperatoren mit festen Singularitäten in den jeweiligen Kernen zusammen. Zur numerischen Lösung solcher Gleichungen werden polynomiale Kollokations-Quadraturverfahren betrachet. Als Ansatzfunktionen und Kollokationspunkte werden dabei gewichtete Polynome und Tschebyscheff-Knoten gewählt. Die Gewichte sind so gewählt, dass diese das asymptotische Verhalten der Lösung in den Randpunkten widerspiegeln. Mit Hilfe von C*-Algebra Techniken, werden in dieser Arbeit notwendige und hinreichende Bedingungen für die Stabilität der Kollokations-Quadraturverfahren angegeben. Die theoretischen Resultate werden dabei durch numerische Berechnungen anhand des Problems der angerissenen Halbebene und des angerissenen Loches überprüft.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.

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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Rosa, Miriam Aparecida. "Método de colocação polinomial para equações integro-diferenciais singulares: convergência." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-26092014-104429/.

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Esta tese analisa o método de colocação polinomial, para uma classe de equações integro-diferenciais singulares em espaços ponderados de funções contínuas e condições de fronteira não nulas. A convergência do método numérico em espaços com norma uniforme ponderada, é demonstrada, e taxas de convergências são determinadas, usando a suavidade dos dados das funções envolvidas no problema. Exemplos numéricos confirmam as estimativas<br>This thesis analyses the polynomial collocation method, for a class of singular integro-differential equations in weighted spaces of continuous functions, and non-homogeneous boundary conditions. Convergence of the numerical method, in weighted uniform norm spaces, is demonstrated and convergence rates are determined using the smoothness of the data functions involved in problem. Numerical examples confirm the estimates
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Kokron, Carlos J. "Modeling large temperature swings in heat regenerators using orthogonal collocation." Thesis, 1991. http://hdl.handle.net/1957/36585.

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This thesis examines the transient performance of packed bed heat regenerators when very large temperature differences are involved. The effects of gas temperature on the key gas physical properties of velocity, density and heat capacity were studied via simulation. Three models were developed and compared. The first model (HRKDV) considers heat balances for both solid and gas phases, the second (HRVDV) considers mass balances in addition to the heat balances set up in the first model and the third one (HRASO) considers that the only significant rate of accumulation term is that of the energy of the solid phase. The governing partial differential equations were solved by the method of lines with the spatial discretization accomplished by the method of orthogonal collocation. The findings of this work reveal that whereas the effects of large temperature changes on the gas velocity and density are completely negligible, the effects of temperature on the gas heat capacity must be considered "continuously" when large temperature swings occur. Considering the heat capacity as a constant, even at an average value, leads to significant errors in temperature profiles.<br>Graduation date: 1992
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Matandos, Marcio. "Use of orthogonal collocation in the dynamic simulation of staged separation processes." Thesis, 1991. http://hdl.handle.net/1957/36669.

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Two basic approaches to reduce computational requirements for solving distillation problems have been studied: simplifications of the model based on physical approximations and order reduction techniques based on numerical approximations. Several problems have been studied using full and reduced-order techniques along with the following distillation models: Constant Molar Overflow, Constant Molar Holdup and Time-Dependent Molar Holdup. Steady-state results show excellent agreement in the profiles obtained using orthogonal collocation and demonstrate that with an order reduction of up to 54%, reduced-order models yield better results than physically simpler models. Step responses demonstrate that with a reduction in computing time of the order of 60% the method still provides better dynamic simulations than those obtained using physical simplifications. Frequency response data obtained from pulse tests has been used to verify that reduced-order solutions preserve the dynamic characteristics of the original full-order system while physical simplifications do not. The orthogonal collocation technique is also applied to a coupled columns scheme with good results.<br>Graduation date: 1992
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Gehre, Nico. "Lösungsoperatoren für Delaysysteme und Nutzung zur Stabilitätsanalyse." 2017. https://monarch.qucosa.de/id/qucosa%3A21053.

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In diese Dissertation werden lineare retardierte Differentialgleichungen (DDEs) und deren Lösungsoperatoren untersucht. Wir stellen eine neue Methode vor, mit der die Lösungsoperatoren für autonome und nicht-autonome DDEs bestimmt werden. Die neue Methode basiert auf dem Pfadintegralformalismus, der aus der Quantenmechanik und von der Analyse stochastischer Differentialgleichungen bekannt ist. Es zeigt sich, dass die Lösung eines Delaysystems zum Zeitpunkt t durch die Integration aller möglicher Pfade von der Anfangsbedingung bis zur Zeit t gebildet werden kann. Die Pfade bestehen dabei aus verschiedenen Schritten unterschiedlicher Längen und Gewichte. Für skalare autonome DDEs können analytische Ausdrücke des Lösungsoperators in der Literatur gefunden werden, allerdings existieren keine für nicht-autonome oder höherdimensionale DDEs. Mithilfe der neuen Methode werden wir die Lösungsoperatoren der genannten DDEs aufstellen und zusätzlich auf Delaysysteme mit mehreren Delaytermen erweitern. Dabei bestätigen wir unsere Ergebnisse sowohl analytisch wie auch numerisch. Die gewonnenen Lösungsoperatoren verwenden wir anschließend zur Stabilitätsanalyse periodischer Delaysysteme. Es werden zwei neue Verfahren präsentiert, die mithilfe des Lösungsoperators den transformierten Monodromieoperator des Delaysystems nähern und daraus die Stabilität bestimmen können. Beide neue Verfahren sind spektrale Methoden für autonome sowie nicht-autonome Delaysysteme und haben keine Einschränkungen wie bei der bekannten Chebyshev-Kollokationsmethode oder der Chebyshev-Polynomentwicklung. Die beiden bisherigen Verfahren beschränken sich auf Delaysysteme mit rationalem Verhältnis zwischen Periode und Delay. Außerdem werden wir eine bereits bekannte Methode erweitern und zu einer spektralen Methode für periodische nicht-autonome Delaysysteme entwickeln. Wir bestätigen alle drei neue Verfahren numerisch. Damit werden in dieser Dissertation drei neue spektrale Verfahren zur Stabilitätsanalyse periodischer Delaysysteme vorgestellt.<br>In this thesis linear delay differential equations (DDEs) and its solutions operators are studied. We present a new method to calculate the solution operators for autonomous and non-autonomous DDEs. The new method is related to the path integral formalism, which is known from quantum mechanics and the analysis of stochastic differential equations. It will be shown that the solution of a time delay system at time t can be constructed by integrating over all paths from the initial condition to time t. The paths consist of several steps with different lengths and weights. Analytic expressions for the solution operator for scalar autonomous DDEs can be found in the literature but no results exist for non-autonomous or high dimensional DDEs. With the help of the new method we can calculate the solution operators for such DDEs and for time delay systems with several delay terms. We verify our results analytically and numerically. We use the obtained solution operators for the stability analysis of periodic time delay systems. Two new methods will be presented to approximate the transformed monodromy operator with the help of the solution operator and to get the stability. Both new methods are spectral methods for autonomous and non-autonomous delay systems and have no limitations like the known Chebyshev collocation method or Chebyshev polynomial expansion. Both previously known methods are limited to time delay systems with a rational relation between period and delay. Furthermore we will extend a known method to a spectral method for non-autonomous time delay systems. We verify all three new methods numerically. Hence, in this thesis three new spectral methods for the stability analysis of periodic time delay systems are presented.
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Books on the topic "Polynomial collocation method"

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Kokron, Carlos J. Modeling large temperature swings in heat regenerators using orthogonal collocation. 1991.

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Matandos, Marcio. Use of orthogonal collocation in the dynamic simulation of staged separation processes. 1991.

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Douglas, J. jr, and T. Dupont. Collocation Methods for Parabolic Equations in a Single Space Variable : (Based on C1-Piecewise-Polynomial Spaces). Springer London, Limited, 2006.

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Book chapters on the topic "Polynomial collocation method"

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Singh, Sudhir, and K. Murugesan. "Bernstein Polynomial Collocation Method for Acceleration Motion of a Vertically Falling Non-spherical Particle." In Advances in Fluid Dynamics. Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-4308-1_53.

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Zhao, Dangjun, Zhiwei Zhang, and Mingzhen Gui. "Birkhoff Pseudospectral Method and Convex Programming for Trajectory Optimization." In Autonomous Trajectory Planning and Guidance Control for Launch Vehicles. Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-99-0613-0_4.

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AbstractTrajectory optimization, an optimal control problem (OCP) in essence, is an important issue in many engineering applications including space missions, such as orbit insertion of launchers, orbit rescue, formation flying, etc. There exist two kinds of solving methods for OCP, i.e., indirect and direct methods. For some simple OCPs, using the indirect methods can result in analytic solutions, which are not easy to be obtained for complicated systems. Direct methods transcribe an OCPs into a finite-dimensional nonlinear programming (NLP) problem via discretizing the states and the controls at a set of mesh points, which should be carefully designed via compromising the computational burden and the solution accuracy. In general, the larger number of mesh points, the more accurate solution as well as the larger computational cost including CPU time and memory [1]. There are many numerical methods have been developed for the transcription of OCPs, and the most common method is by using Pseudospectral (PS) collocation scheme [2], which is an optimal choice of mesh points in the reason of well-established rules of approximation theory [3]. Actually, there have several mature optimal control toolkits based PS methods, such as DIDO [4], GPOPS [5]. The resulting NLP problem can be solved by the well-known algorithm packages, such as IPOPT [6] or SNOPT [7]. However, these algorithms cannot obtain a solution in polynomial-time, and the resulting solution is locally optimal. Moreover, a good initial guess solution should be provided for complicated problems.
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Junghanns, Peter, Giuseppe Mastroianni, and Incoronata Notarangelo. "Collocation and Collocation-Quadrature Methods for Strongly Singular Integral Equations." In Weighted Polynomial Approximation and Numerical Methods for Integral Equations. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-77497-4_7.

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Lacor, Chris, and Éric Savin. "General Introduction to Polynomial Chaos and Collocation Methods." In Uncertainty Management for Robust Industrial Design in Aeronautics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-77767-2_7.

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Monegato, G., and L. Scuderi. "Polynomial Collocation Methods for 1D Integral Equations with Nonsmooth Solutions." In Mathematical Aspects of Boundary Element Methods. Chapman and Hall/CRC, 2024. http://dx.doi.org/10.1201/9780429332449-19.

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Lacor, Chris, and Éric Savin. "Polynomial Chaos and Collocation Methods and Their Range of Applicability." In Uncertainty Management for Robust Industrial Design in Aeronautics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-77767-2_42.

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Pedas, A., E. Tamme, and M. Vikerpuur. "Piecewise Polynomial Collocation for a Class of Fractional Integro-Differential Equations." In Integral Methods in Science and Engineering. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-16727-5_39.

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Singh, Vinai K., and A. K. Singh. "RETRACTED CHAPTER: A Collocation Method for Integral Equations in Terms of Generalized Bernstein Polynomials." In Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-1454-3_23.

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Singh, Vinai K., and A. K. Singh. "Retraction Note to: A Collocation Method for Integral Equations in Terms of Generalized Bernstein Polynomials." In Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-1454-3_26.

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Bansu, Hitesh, and Svetozar Margenov. "Parametric Analysis of Space-Time Fractional Pennes Bioheat Model Using a Collocation Method Based on Radial Basis Functions and Chebyshev Polynomials." In Large-Scale Scientific Computations. Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-56208-2_12.

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Conference papers on the topic "Polynomial collocation method"

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Cheng, Haiyan, and Adrian Sandu. "Collocation least-squares polynomial chaos method." In the 2010 Spring Simulation Multiconference. ACM Press, 2010. http://dx.doi.org/10.1145/1878537.1878621.

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Çevik, Mehmet, and Mehmet Sezer. "Engineering Applications of Polynomial Matrix Method: A Review." In 7th International Students Science Congress. Izmir International guest Students Association, 2023. http://dx.doi.org/10.52460/issc.2023.032.

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The modeling of natural phenomena is based on ordinary and partial differential equations, which appear in all branches of science and engineering. For this reason, applied mathematicians and engineers have tried to develop new methods of solutions to these equations. One of the widely used numerical methods for the solution of ordinary and partial differential, integro-differential, and integral equations is the Polynomial Matrix Method (PMM). In this study, these methods are introduced first and then a brief history of the development of the method is given. Almost 30 polynomials used in this collocation approach are mentioned. Fundamental principle of the PMM is explained. Engineering applications such as in single and multi-degree of freedom systems, mechanical vibrations, heat equations, diffusion equation and others are reviewed.
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Çevik, Mehmet, and Mehmet Sezer. "Engineering Applications of Polynomial Matrix Method: A Review." In 7th International Students Science Congress. Izmir International guest Students Association, 2023. http://dx.doi.org/10.52460/issc.2023.032.

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The modeling of natural phenomena is based on ordinary and partial differential equations, which appear in all branches of science and engineering. For this reason, applied mathematicians and engineers have tried to develop new methods of solutions to these equations. One of the widely used numerical methods for the solution of ordinary and partial differential, integro-differential, and integral equations is the Polynomial Matrix Method (PMM). In this study, these methods are introduced first and then a brief history of the development of the method is given. Almost 30 polynomials used in this collocation approach are mentioned. Fundamental principle of the PMM is explained. Engineering applications such as in single and multi-degree of freedom systems, mechanical vibrations, heat equations, diffusion equation and others are reviewed.
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Morgado, M. L., M. Rebelo, and N. J. Ford. "A non-polynomial collocation method for fractional terminal value problems." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics. AIP, 2012. http://dx.doi.org/10.1063/1.4756111.

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Pasic, Hajrudin. "Efficient Solution of Stiff ODE by Implicit Collocation Method." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/mech-5989.

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Abstract When s-stage fully implicit collocation method is used to solve n–dimensional system of implicit ordinary differential equations (ODE) f(x, y, y’) = 0 the resulting algebraic system has a dimension sn. Its solution by Gauss elimination is expensive and requires s3n3 / 3 operations. In this paper we present an efficient algorithm which uncouples the algebraic system into a block-diagonal matrix with sn-dimensional blocks that may be solved in parallel. It applies to both explicit and implicit ODEs. The algorithm is formally different from the implicit Runge-Kutta (RK) method in that the solution for y is assumed to have a form of an algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points. Locations of the collocation points are found from the derived stability function such as to guarantee both good stability and precision properties. Such a procedure leads to an algebraic system which has a sparse block-form and which may be easily diagonalized, providing an efficient solution of the problem, which is about s2 times faster than the Gauss elimination applied to the original, nondiagonalized system. In addition, the block-diagonal form of the algebraic system allows parallel processing and further increase of efficiency.
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Türkhan, Havva, and Kübra Erdem Biçer. "Fubini Polynomial Solution of Linear Delay Fredholm Integro Differential Equations." In 8th International Students Science Congress. ULUSLARARASI ÖĞRENCİ DERNEKLERİ FEDERASYONU (UDEF), 2024. https://doi.org/10.52460/issc.2024.036.

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In this paper, a numerical matrix method is used to solve linear delay Fredholm integro-differential equations with variable coefficients under mixed conditions. The technique consists of collocation points and the Fubini polynomials. The residual error functions of numerical solutions of the method are also presented. Firstly, the approximate solutions are formed and secondly, an error problem is constituted in favor of the residual error function. The numerical solutions are computed for this error problem by using the present method. The approximate solutions of the original problem and the error problem are the corrected Fubini polynomial solutions. As the exact solutions to the problem are not known, absolute errors can be approximated by approximate solutions to the error problem. Numerical examples are given to demonstrate the validity and applicability of the technique. Additionally, the calculations are made with the MATLAB program.
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Blanchard, Emmanuel, Corina Sandu, and Adrian Sandu. "A Polynomial-Chaos-Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34600.

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In this study, a new computational approach for parameter identification is proposed based on the application of the polynomial chaos theory. The polynomial chaos method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. In the new approach presented in this paper, the maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework. The new parameter estimation method is illustrated on a four degree-of-freedom roll plane model of a vehicle in which the vertical stiffnesses of the tires are estimated from periodic observations of the displacements and velocities across the suspensions. The results obtained with this approach are close to the actual values of the parameters even when only measurements with low sampling rates are available. The accuracy of the estimations has been shown to be sensitive to the number of terms used in the polynomial expressions and to the number of collocation points, and thus it may become computationally expensive when a very high accuracy of the results is desired. However, the noise level in the measurements affects the accuracy of the estimations as well. Therefore, it is usually not necessary to use a large number of terms in the polynomial expressions and a very large number of collocation points since the addition of extra precision eventually affects the results less than the effect of the measurement noise. Possible applications of this theory to the field of vehicle dynamics simulations include the estimation of mass, inertia properties, as well as other parameters of interest.
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Soudbakhsh, Damoon, Azim Eskandarian, and David Chichka. "Vehicle Evasive Maneuver Trajectory Optimization Using Collocation Technique." In ASME 2010 Dynamic Systems and Control Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/dscc2010-4213.

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Vehicle evasive maneuvers or sudden lane changes pose stringent conditions on trajectory control, which require not only a desired path following but also a complete consideration of lateral forces and vehicle dynamic stability. Experienced drivers steer the vehicle on a desired path, as much as possible, without creating large lateral forces beyond the stability limits. Steering control systems have been developed to perform similar lane change or evasive maneuvers automatically but with limitations. A control method is developed to find desired trajectory automatically based on the defined design criteria using constrained optimization via collocation technique. The results are compared with two known suitable trajectories. The results show that the proposed control method produces peak lateral acceleration that are lower than the 5th order polynomial trajectory, and overall lateral accelerations that are lower than a comparable trapezoidal acceleration profile.
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Cao, Lu, Xiaobo Wu, Huixing Li, Hao Chen, Yanqiang Shi, and Hao Wu. "Polynomial Approximation of Transient Voltage Stability Region Boundary in the Parameter Space Based on Collocation Method." In 2022 IEEE 6th Conference on Energy Internet and Energy System Integration (EI2). IEEE, 2022. http://dx.doi.org/10.1109/ei256261.2022.10116599.

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Hwang, Jon Li, and Ting Nung Shiau. "Application of Generalized Polynomial Expansion Method to Nonlinear Rotor Bearing Systems." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0269.

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Abstract The Generalized Polynomial Expansion Method (GPEM) is utilized to model a large-order flexible-rotor system with nonlinear supports. With the application of GPEM, a set of nonlinear ordinary differential equations are obtained. A hybrid method which combines the merits of the Harmonic Balance Method (HBM) and the Trigonometric Collocation Method (TCM) is used to solve for the nonlinear response of the system. This hybrid method together with reduction techniques can efficiently solve for the motion of the system. The overall algorithm presented provides a very efficient technique for investigating the periodic response of large-order nonlinear rotor systems. Two examples are used to illustrate the merits of the method. One is the simple Jeffcott rotor which is used to check the accuracy of the present numerical algorithms. The other is a flexible shaft with multiple disks, supported by multiple bearings. It is used to show the advantages of the linkage between GPEM and the presented hybrid numerical algorithm. Some of the support bearings are modeled as a squeeze-film damper associated with a center spring. The center spring is considered to be linear and the squeeze-film damper is nonlinear. The nonlinear hydrodynamic forces are obtained using short-bearing theory. Based on the example results, the conclusions can be summarized as follows: (l) For a nonlinear flexible-rotor system, the number of equations needed for the system described by the Generalized Polynomial Expansion Method are always smaller than the number required by the finite element method before applying the condensation technique. This is also true for the linear case which has been discussed by Shiau and Hwang (1989, 1990). (2) A technique of component mode synthesis has been developed based on the modeling approach of the Generalized Polynomial Expansion Method. It is applied to decouple the nonlinear independent and dependent modal coordinates. With this technique, the number of non-zero elements in the generalized modal-forces vector is equal to the number of nonlinear, dependent modal coordinates. The results indicate that the use of GPEM together with component mode synthesis not only retains its merits in solving a linear rotor-dynamic problem, but also provides attainable solutions for the nonlinear rotor-dynamic problem. (3) The hybrid numerical algorithm developed in the present study combines both the advantages of the harmonic-balance method and the collocation method. The use of this hybrid method with condensation technique can significantly save computing time. Furthermore, it can be used to predict the periodic response, including the sub-harmonic response and the super-harmonic response, of a nonlinear system. (4) The time required for the initial formulation processing with this method may be a little more than that required by the finite-element approach. However, it is very small compared to the overall time savings that are accrued.
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