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Journal articles on the topic 'Differential equations. Bipartite graphs'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

Yoshimura, H. "A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations." Mechanical Sciences 4, no. 1 (June 6, 2013): 243–50. http://dx.doi.org/10.5194/ms-4-243-2013.

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Abstract. In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

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2

Jimenez, Belmonte, Garrido, Ruz, and Vazquez. "Software Tool for Acausal Physical Modelling and Simulation." Symmetry 11, no. 10 (September 24, 2019): 1199. http://dx.doi.org/10.3390/sym11101199.

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Modelling and simulation are key tools for analysis and design of systems and processes from almost any scientific or engineering discipline. Models of complex systems are typically built on acausal Differential-Algebraic Equations (DAE) and discrete events using Object-Oriented Modelling (OOM) languages, and some of their key concepts can be explained as symmetries. To obtain a computer executable version from the original model, several algorithms, based on bipartite symmetric graphs, must be applied for automatic equation generation, removing alias equations, computational causality assignment, equation sorting, discrete-event processing or index reduction. In this paper, an open source tool according to OOM paradigm and developed in MATLAB is introduced. It implements such algorithms adding an educational perspective about how they work, since the step by step results obtained after processing the model equations can be shown. The tool also allows to create models using its own OOM language and to simulate the final executable equation set. It was used by students in a modelling and simulation course of the Automatic Control and Industrial Electronics Engineering degree, showing a significant improvement in their understanding and learning of the abovementioned topics after their assessment.

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3

Amin, Md Ruhul, and Marc R. Roussel. "Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration." Canadian Journal of Chemistry 92, no. 2 (February 2014): 85–93. http://dx.doi.org/10.1139/cjc-2013-0315.

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We develop and analyze a mathematical model based on a previously enunciated hypothesis regarding the origin of rapid, irregular oscillations observed in photosynthetic variables when a leaf is transferred to a low-CO2 atmosphere. This model takes the form of a set of differential equations with two delays. We review graph-theoretical methods of analysis based on the bipartite graph representation of mass-action models, including models with delays. We illustrate the use of these methods by showing that our model is capable of delay-induced oscillations. We present some numerical examples confirming this possibility, including the possibility of complex transient oscillations. We then use the structure of the identified oscillophore, the part of the reaction network responsible for the oscillations, along with our knowledge of the plausible range of values for one of the delays, to rule out this hypothetical mechanism.

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4

Vol’pert, A. I. "Differential Equations on Graphs." Mathematical Modelling of Natural Phenomena 10, no. 5 (2015): 6–15. http://dx.doi.org/10.1051/mmnp/201510502.

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5

Bothe, Dieter. "Multivalued differential equations on graphs." Nonlinear Analysis: Theory, Methods & Applications 18, no. 3 (February 1992): 245–52. http://dx.doi.org/10.1016/0362-546x(92)90062-j.

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6

Malinowski, Marek T. "Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition." Mathematical Problems in Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/3830529.

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We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.

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7

Fukuizumi, Reika, Jeremy Marzuola, Dmitry Pelinovsky, and Guido Schneider. "Nonlinear Partial Differential Equations on Graphs." Oberwolfach Reports 14, no. 2 (April 27, 2018): 1805–68. http://dx.doi.org/10.4171/owr/2017/29.

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8

Pokornyi, Yu V., and A. V. Borovskikh. "Differential Equations on Networks (Geometric Graphs)." Journal of Mathematical Sciences 119, no. 6 (March 2004): 691–718. http://dx.doi.org/10.1023/b:joth.0000012752.77290.fa.

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9

Pröll, Sebastian, Jan Lunze, and Fabian Jarmolowitz. "From Structural Analysis to Observer–Based Residual Generation for Fault Detection." International Journal of Applied Mathematics and Computer Science 28, no. 2 (June 1, 2018): 233–45. http://dx.doi.org/10.2478/amcs-2018-0017.

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Abstract This paper combines methods for the structural analysis of bipartite graphs with observer-based residual generation. The analysis of bipartite structure graphs leads to over-determined subsets of equations within a system model, which make it possible to compute residuals for fault detection. In observer-based diagnosis, by contrast, an observability analysis finds observable subsystems, for which residuals can be generated by state observers. This paper reveals a fundamental relationship between these two graph-theoretic approaches to diagnosability analysis and shows that for linear systems the structurally over-determined set of model equations equals the output connected part of the system. Moreover, a condition is proved which allows us to verify structural observability of a system by means of the corresponding bipartite graph. An important consequence of this result is a comprehensive approach to fault detection systems, which starts with finding the over-determined part of a given system by means of a bipartite structure graph and continues with designing an observerbased residual generator for the fault-detectable subsystem found in the first step.

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10

Avdonin, Sergei, and Victor Mikhaylov. "Controllability of partial differential equations on graphs." Applicationes Mathematicae 35, no. 4 (2008): 379–93. http://dx.doi.org/10.4064/am35-4-1.

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11

Borovskikh, A. V., and K. P. Lazarev. "Fourth-Order Differential Equations on Geometric Graphs." Journal of Mathematical Sciences 119, no. 6 (March 2004): 719–38. http://dx.doi.org/10.1023/b:joth.0000012753.65477.23.

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12

Rücker, Gerta, Christoph Rücker, and Ivan Gutman. "On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs." Zeitschrift für Naturforschung A 57, no. 3-4 (April 1, 2002): 143–53. http://dx.doi.org/10.1515/zna-2002-3-406.

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Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s1, and the analogous quantity sn, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s1 is interpreted as a measure of mixedness of a graph, and sn, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, sn is maximal for star graphs, while the minimal value of sn is zero. Mixedness s1 is maximal for regular graphs. Minimal values of s1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal s1, while the trees with minimal s1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s1, determined within trees and 4-trees (alkanes), was found to be high.

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13

Mironov, Andrey, Aleksey Morozov, and Sergey Natanzon. "Infinite-dimensional topological field theories from Hurwitz numbers." Journal of Knot Theory and Its Ramifications 23, no. 06 (May 2014): 1450033. http://dx.doi.org/10.1142/s0218216514500333.

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Classical Hurwitz numbers of a fixed degree together with Hurwitz numbers of seamed surfaces give rise to a Klein topological field theory (see [A. Alexeevski and S. Natanzon, The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math. 72(4) (2008) 627–646]). We extend this construction to Hurwitz numbers of all degrees simultaneously. The corresponding infinite-dimensional Cardy–Frobenius algebra is computed in terms of Young diagrams and bipartite graphs. This algebra turns out to be isomorphic to the algebra of differential operators introduced in [A. Mironov, A. Morozov and S. Natanzon, Cardy–Frobenius extension of algebra of cut-and-join operators, J. Geom. Phys. 73 (2012) 243–251, arXiv:1210.6955; A Hurwitz theory avatar of open-closed string, Eur. Phys. J. C 73(2) (2013) 1–10, arXiv:1208.5057], which serves a model for open-closed string theory. We prove that the operators corresponding to Young diagrams and bipartite graphs give rise to relations between Hurwitz numbers.

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14

Bhamidi, Shankar, Amarjit Budhiraja, and Sanchayan Sen. "Critical random graphs and the differential equations technique." Indian Journal of Pure and Applied Mathematics 48, no. 4 (December 2017): 633–69. http://dx.doi.org/10.1007/s13226-017-0249-0.

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15

Wormald, Nicholas C. "Differential Equations for Random Processes and Random Graphs." Annals of Applied Probability 5, no. 4 (November 1995): 1217–35. http://dx.doi.org/10.1214/aoap/1177004612.

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16

Pushpam, P. Roushini Leely, and D. Yokesh. "Differentials in certain classes of graphs." Tamkang Journal of Mathematics 41, no. 2 (June 30, 2010): 129–38. http://dx.doi.org/10.5556/j.tkjm.41.2010.664.

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Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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17

Granda, J. J. "Computer generation of physical system differential equations using bond graphs." Journal of the Franklin Institute 319, no. 1-2 (January 1985): 243–55. http://dx.doi.org/10.1016/0016-0032(85)90078-x.

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18

Freidlin, Mark, and Shuenn-Jyi Sheu. "Diffusion processes on graphs: stochastic differential equations, large deviation principle." Probability Theory and Related Fields 116, no. 2 (February 2000): 181–220. http://dx.doi.org/10.1007/pl00008726.

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19

ARTAMKIN, I. V. "COLORED GRAPHS, BURGERS EQUATION AND HESSIAN CONJECTURE." International Journal of Mathematics 18, no. 07 (August 2007): 797–808. http://dx.doi.org/10.1142/s0129167x0700431x.

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We prove that generating series for colored modular graphs satisfy some systems of partial differential equations generalizing Burgers or heat equations. The solution is obtained by genus expansion of the generating function. The initial term of this expansion is the corresponding generating function for trees. For this term the system of differential equations is equivalent to the inversion problem for the gradient mapping defined by the initial condition. This enables to state the Jacobian conjecture in the language of generating functions. The use of generating functions provides rather short and natural proofs of resent results of Zhao and of the well-known Bass–Connell–Wright tree inversion formula.

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20

Pivovarchik, Vyacheslav N., and Harald Woracek. "Sums of Nevanlinna functions and differential equations on star-shaped graphs." Operators and Matrices, no. 4 (2009): 451–501. http://dx.doi.org/10.7153/oam-03-26.

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21

Finta, István, Sándor Szénási, and Lóránt Farkas. "Input Pattern Classification Based on the Markov Property of the IMBT with Related Equations and Contingency Tables." Entropy 22, no. 2 (February 21, 2020): 245. http://dx.doi.org/10.3390/e22020245.

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In this contribution, we provide a detailed analysis of the search operation for the Interval Merging Binary Tree (IMBT), an efficient data structure proposed earlier to handle typical anomalies in the transmission of data packets. A framework is provided to decide under which conditions IMBT outperforms other data structures typically used in the field, as a function of the statistical characteristics of the commonly occurring anomalies in the arrival of data packets. We use in the modeling Bernstein theorem, Markov property, Fibonacci sequences, bipartite multi-graphs, and contingency tables.

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22

Sun, Yong-Li, Wen-Xiu Ma, Jian-Ping Yu, Bo Ren, and Chaudry Masood Khaliqu. "Lump and interaction solutions of nonlinear partial differential equations." Modern Physics Letters B 33, no. 11 (April 18, 2019): 1950133. http://dx.doi.org/10.1142/s0217984919501331.

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In this paper, lump solutions of nonlinear partial differential equations, the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional KP equation and the Jimbo–Miwa equation, are studied by using the Hirota bilinear method and carrying out symbolic computations in Maple. Moreover, the interaction solutions, i.e. collisions between lump waves and kink waves, are investigated. A group of graphs are plotted to illustrate the dynamics of the obtained results.

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23

Malik, M. Y., A. Hussain, T. Salahuddin, M. Awais, and S. Bilal. "Numerical Solution of Sisko Fluid Over a Stretching Cylinder and Heat Transfer with Variable Thermal Conductivity." Journal of Mechanics 32, no. 5 (April 11, 2016): 593–601. http://dx.doi.org/10.1017/jmech.2016.8.

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AbstractPresent paper addresses the numerical study of Sisko fluid model over stretching cylinder with variable thermal conductivity. The governing equations are simplified by incorporating the boundary layer approximations. After employing suitable similarity transformations partial differential equations are reduced to ordinary differential equations. To obtain numerical solution shooting method in conjunction with Runge-Kutta-Fehlberg method is used. For the analysis of model, variations due to different physical parameters involved in momentum and heat equations are reflected through graphs. Also, the effects of physical parameters on skin-friction coefficient and Nusselt number are represented through graphs as well as tables.

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24

O'Regan, Donal. "A note on multivalued differential equations on proximate retracts." Journal of Applied Mathematics and Stochastic Analysis 12, no. 2 (January 1, 1999): 169–78. http://dx.doi.org/10.1155/s1048953399000179.

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This paper discusses viable solutions for differential inclusions in Banach spaces. Existence will be established in two steps. In step 1, a nonlinear alternative of Leray-Schauder type [8] for maps with closed graphs will be used to establish a variety of existence principles for the Cauchy differential inclusion. Step 2 involves using the results in step 1 together with some tricks involving the Bouligand cone (and sometimes the Urysohn function) so that new existence criteria can be established for multivalued differential equations on proximate retracts.

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25

de Snoo, Henk, and Henrik Winkler. "Canonical systems of differential equations with self-adjoint interface conditions on graphs." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 2 (April 2005): 297–315. http://dx.doi.org/10.1017/s0308210500003899.

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For n canonical systems of differential equations, the corresponding n copies of their domain (0, ∞) are thought of as a graph with vertex 0. An interface condition at 0 is given by a so-called Nevanlinna pair. Explicit formulae are deduced for the spectral representation of the corresponding underlying self-adjoint relation and the generalized Fourier transformation. Furthermore, results on compressions of the Fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

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26

Barvinok, Alexander, and Guus Regts. "Weighted counting of solutions to sparse systems of equations." Combinatorics, Probability and Computing 28, no. 5 (April 15, 2019): 696–719. http://dx.doi.org/10.1017/s0963548319000105.

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AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

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27

Yatim, S. A. M., Z. B. Ibrahim, K. I. Othman, and M. B. Suleiman. "A Numerical Algorithm for Solving Stiff Ordinary Differential Equations." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/989381.

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An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

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28

VAN GENNIP, YVES, and CAROLA-BIBIANE SCHÖNLIEB. "Introduction: Big data and partial differential equations." European Journal of Applied Mathematics 28, no. 6 (November 7, 2017): 877–85. http://dx.doi.org/10.1017/s0956792517000304.

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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

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29

Yurko, Vjacheslav Anatoljevich. "Recovering differential pencils on graphs with a cycle from spectra." Tamkang Journal of Mathematics 45, no. 2 (June 30, 2014): 195–206. http://dx.doi.org/10.5556/j.tkjm.45.2014.1492.

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We study boundary value problems on compact graphs with a cycle for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate inverse spectral problems of recovering coefficients of the differential equation from spectra. For these inverse problems we prove uniqueness theorems and provide procedures for constructing their solutions.

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Houwe, A., M. Inc, S. Y. Doka, B. Acay, and L. V. C. Hoan. "The discrete tanh method for solving the nonlinear differential-difference equations." International Journal of Modern Physics B 34, no. 19 (July 28, 2020): 2050177. http://dx.doi.org/10.1142/s0217979220501775.

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We investigated analytical solutions for the nonlinear differential-difference equations (DDEs) having fractional-order derivative. We employed the discrete tanh method in computations. Performance of trigonometric functions, dark one solitons and rational solutions are discussed in detail. The results with reliable parameters are illustrated via 2-D and 3-D graphs.

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31

Liu, Jian-Gen, Xiao-Jun Yang, and Yi-Ying Feng. "Analytical solutions of some integral fractional differential–difference equations." Modern Physics Letters B 34, no. 01 (December 9, 2019): 2050009. http://dx.doi.org/10.1142/s0217984920500098.

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The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as two couple space-time FDDEs. Furthermore, some properties of the analytical solutions are illustrated by graphs.

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32

Bonaccorsi, Stefano, Francesca Cottini, and Delio Mugnolo. "Random Evolution Equations: Well-Posedness, Asymptotics, and Applications to Graphs." Applied Mathematics & Optimization 84, no. 3 (March 11, 2021): 2849–87. http://dx.doi.org/10.1007/s00245-020-09732-w.

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AbstractWe study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs.

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WILDS, ROY, and LEON GLASS. "AN ATLAS OF ROBUST, STABLE, HIGH-DIMENSIONAL LIMIT CYCLES." International Journal of Bifurcation and Chaos 19, no. 12 (December 2009): 4055–96. http://dx.doi.org/10.1142/s0218127409025225.

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We present a method for constructing dynamical systems with robust, stable limit cycles in arbitrary dimensions. Our approach is based on a correspondence between dynamics in a class of differential equations and directed graphs on the n-dimensional hypercube (n-cube). When the directed graph contains a certain type of cycle, called a cyclic attractor, then a stable limit cycle solution of the differential equations exists. A novel method for constructing regulatory systems that we call minimal regulatory networks from directed graphs facilitates investigation of limit cycles in arbitrarily high dimensions. We identify two families of cyclic attractors that are present for all dimensions n ≥ 3: cyclic negative feedback and sequential disinhibition. For each, we obtain explicit representations for the differential equations in arbitrary dimension. We also provide a complete listing of minimal regulatory networks, a representative differential equation, and a bifurcation analysis for each cyclic attractor in dimensions 3–5. This work joins discrete concepts of symmetry and classification with analysis of differential equations useful for understanding dynamics in complex biological control networks.

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López-Saldívar, Julio, Margarita Man’ko, and Vladimir Man’ko. "Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States." Entropy 22, no. 5 (May 23, 2020): 586. http://dx.doi.org/10.3390/e22050586.

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In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian H ^ . We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

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35

Aoki, Takashi, and Takayuki Iizuka. "Classification of Stokes Graphs of Second Order Fuchsian Differential Equations of Genus Two." Publications of the Research Institute for Mathematical Sciences 43, no. 1 (2007): 241–76. http://dx.doi.org/10.2977/prims/1199403816.

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36

Yurko, Vjacheslav. "Inverse problems for Bessel-type differential equations on noncompact graphs using spectral data." Inverse Problems 27, no. 4 (March 8, 2011): 045002. http://dx.doi.org/10.1088/0266-5611/27/4/045002.

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37

Voevodin, A. F. "Solution method for boundary value problems for ordinary differential equations on complexes (Graphs)." Differential Equations 48, no. 7 (July 2012): 929–39. http://dx.doi.org/10.1134/s001226611207004x.

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38

Kumar, Avadhesh, Ramesh K. Vats, Ankit Kumar, and Dimplekumar N. Chalishajar. "Numerical approach to the controllability of fractional order impulsive differential equations." Demonstratio Mathematica 53, no. 1 (September 19, 2020): 193–207. http://dx.doi.org/10.1515/dema-2020-0015.

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AbstractIn this manuscript, a numerical approach for the stronger concept of exact controllability (total controllability) is provided. The proposed control problem is a nonlinear fractional differential equation of order \alpha \in (1,2] with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.

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39

Debela, Habtamu Garoma, Solomon Bati Kejela, and Ayana Deressa Negassa. "Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations." International Journal of Differential Equations 2020 (June 17, 2020): 1–13. http://dx.doi.org/10.1155/2020/5768323.

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This paper presents a numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms. A fourth-order exponentially fitted numerical scheme on uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method.

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Hussain, Azad, Sobia Akbar, Lubna Sarwar, and Sohail Nadeem. "Probe of Radiant Flow on Temperature-Dependent Viscosity Models of Differential Type MHD Fluid." Mathematical Problems in Engineering 2020 (November 6, 2020): 1–16. http://dx.doi.org/10.1155/2020/2927013.

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This paper numerically investigates the combined effects of the radiation and MHD on the flow of a viscoelastic Walters’ B liquid fluid model past a porous plate with temperature-dependent variable viscosity. To study the effects of variable viscosity on the fluid model, the equations of continuity, momentum with magnetohydrodynamic term, and energy with radiation term have been expanded. To understand the phenomenon, Reynold’s model and Vogel’s model of variable viscosity are also incorporated. The dimensionless governing equations are two-dimensional coupled and highly nonlinear partial differential equations. The highly nonlinear PDEs are transferred into ODEs with the assistance of suitable transformations which are solved with the help of numerical techniques, namely, shooting technique coupled with Runge–Kutta method and BVP4c solution method for the numerical solutions of governing nonlinear problems. Viscosity is considered as a function of temperature. Skin friction coefficient and Nusselt number are investigated through tables and graphs in the present probe. The behavior of emerging parameters on the velocity and temperature profiles is studied with the help of graphs. For Reynold’s model, we have shrinking stream lines and increasing three-dimensional graphs. γ and Pr are reduced for both models.

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Tan, Jinglu, Kim A. Stelson, and Kevin A. Janni. "Compressible-Flow Modeling With Pseudo Bond Graphs." Journal of Dynamic Systems, Measurement, and Control 116, no. 2 (June 1, 1994): 272–80. http://dx.doi.org/10.1115/1.2899220.

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A finite lump pseudo bond graph scheme for isothermal and isentropic pipe flows of ideal gas is developed on the basis of the governing differential equations and an assumption of staggered spatial profiles for the variables. Element constitutive relationships are formally derived. A new CONVECTANCE element is defined to model convective acceleration. The bond graph is simple and convenient to use.

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Yurko, Vjacheslav. "Inverse problems for higher order differential systems with regular singularities on star-type graphs." Tamkang Journal of Mathematics 46, no. 3 (September 30, 2015): 257–68. http://dx.doi.org/10.5556/j.tkjm.46.2015.1754.

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We study an inverse spectral problem for arbitrary order ordinary differential equations on compact star-type graphs when differential equations have regular singularities at boundary vertices. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

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Hayat, Tasawar, Ambreen Safdar, Muhammad Awais, and Awatif A. Hendi. "Unsteady Three-Dimensional Flow in a Second-Grade Fluid Over a Stretching Surface." Zeitschrift für Naturforschung A 66, no. 10-11 (November 1, 2011): 635–42. http://dx.doi.org/10.5560/zna.2011-0032.

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The three-dimensional unsteady flow induced in a second-grade fluid over a stretching surface has been investigated. Nonlinear partial differential equations are reduced into a system of ordinary differential equations by using the similarity transformations. The homotopy analysis method (HAM) has been implemented for the series solutions. Graphs are displayed for the effects of different parameters on the velocity field.

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Aziz, Taha, and F. M. Mahomed. "A Note on the Solutions of Some Nonlinear Equations Arising in Third-Grade Fluid Flows: An Exact Approach." Scientific World Journal 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/109128.

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In this communication, we utilize some basic symmetry reductions to transform the governing nonlinear partial differential equations arising in the study of third-grade fluid flows into ordinary differential equations. We obtain some simple closed-form steady-state solutions of these reduced equations. Our solutions are valid for the whole domain [0,∞) and also satisfy the physical boundary conditions. We also present the numerical solutions for some of the underlying equations. The graphs corresponding to the essential physical parameters of the flow are presented and discussed.

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Subbotin, A. I. "Generalized solutions of partial differential equations of the first order. The invariance of graphs relative to differential inclusions." Journal of Mathematical Sciences 78, no. 5 (February 1996): 594–611. http://dx.doi.org/10.1007/bf02363859.

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Xia, Yarong, Ruoxia Yao, Xiangpeng Xin, and Yan Li. "Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations." Symmetry 13, no. 7 (July 15, 2021): 1268. http://dx.doi.org/10.3390/sym13071268.

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In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

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Abdel-Rahman, Reda G., and Ahmed M. Megahed. "Lie Group Analysis for a Mixed Convective Flow and Heat Mass Transfer Over a Permeable Stretching Surface with Soret and Dufour Effects." Journal of Mechanics 30, no. 1 (November 14, 2013): 67–75. http://dx.doi.org/10.1017/jmech.2013.72.

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ABSTRACTThe Lie group transformation method is applied for solving the problem of mixed convection flow with mass transfer over a permeable stretching surface with Soret and Dufour effects. The application of Lie group method reduces the number of independent variables by one and consequently the system of governing partial differential equations reduces to a system of ordinary differential equations with appropriate boundary conditions. Further, the reduced non-linear ordinary differential equations are solved numerically by using the shooting method. The effects of various parameters governing the flow and heat transfer are shown through graphs and discussed. Our aim is to detect new similarity variables which transform our system of partial differential equations to a system of ordinary differential equations. In this work a special attention is given to investigate the effect of the Soret and Dufour numbers on the velocity, temperature and concentration fields above the sheet.

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Xu, Youjun, and Shu Zhou. "Optimal Control of Pseudoparabolic Variational Inequalities Involving State Constraint." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/641736.

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We establish the necessary condition of optimality for optimal control problem governed by some pseudoparabolic differential equations involving monotone graphs. Some approximating control process and examples are given.

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Motsa, S. S., Y. Khan, and S. Shateyi. "A New Numerical Solution of Maxwell Fluid over a Shrinking Sheet in the Region of a Stagnation Point." Mathematical Problems in Engineering 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/290615.

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The mathematical model for the incompressible two-dimensional stagnation flow of a Maxwell fluid towards a shrinking sheet is proposed. The developed equations are used to discuss the problem of being two dimensional in the region of stagnation point over a shrinking sheet. The nonlinear partial differential equations are transformed to ordinary differential equations by first-taking boundary-layer approximations into account and then using the similarity transformations. The obtained equations are then solved by using a successive linearisation method. The influence of the pertinent fluid parameters on the velocity is discussed through the help of graphs.

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Bakushev, Sergey V. "Differential equations of equilibrium of continuous medium for plane one-dimensional deformation at closing equations approximation by biquadratic functions." Structural Mechanics of Engineering Constructions and Buildings 16, no. 6 (December 15, 2020): 481–92. http://dx.doi.org/10.22363/1815-5235-2020-16-6-481-492.

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Problems of differential equations construction of equilibrium of a geometrically and physically nonlinear continuous medium under conditions of one-dimensional plane deformation are considered, when the diagrams of volumetric and shear deformation are approximated by quadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations, the secant modulus of volumetric expansion - contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity, the secant modulus of volumetric expansion - compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, six main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation, each approximated by two parabolas. The differential equations of equilibrium in displacements constructed in the article can be applied in determining the stressed and deformed state of a continuous medium under conditions of one-dimensional plane deformation, the closing equations of physical relations for which, constructed on the basis of experimental data, are approximated by biquadratic functions.

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