Numerical Solutions of Differential Equations Using Modified B-spline Differential Quadrature Method

1 - INTRODUCTION

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.

In the present paper, the initial value problem for the first order ordinary differential equation with involution is studied. We obtain equivalent initial value problem for the second order ordinary differential equations to the initial value problem for first order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the first order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for first order ordinary nonlinear differential equation with involution is proved.

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This paper studies a two-dimensional wave equation in the presence of power and derivative nonlinearity subject to suitable initial conditions. It aims to construct the exact-analytical solution of a two-dimensional nonlinear wave equation using two different semi-analytical methods and to compare the obtained results. First, a well-known Adomian decomposition method (ADM) based on operators is employed. Many researchers use the ADM to investigate several scientific applications, and this method straightforwardly attacks the problem without using linearization, discretization, perturbation, or any other restrictive assumption that may change the physical behavior of the problem. Second, the variational iteration method (VIM) also provides rapid convergent successive approximations of the closed-form solutions if it exists; otherwise, it provides an approximation of a high degree accuracy level even in case of few obtained iterations. The obtained results are drawn graphically and presented in the tables. Both methods provide almost the same solutions in each nonlinearity case, and VIM has computational advantages over ADM in computation size. Further, ADM and VIM provide a series of solutions that converge in a very small time domain, which limits these methods. Keywords: Adomian decomposition method, variational iteration method, approximations, partial differential equations, two-dimensional wave equation. https://doi.org/10.55463/issn.1674-2974.50.2.3

У статті розглянуто метод отримання одновимірних інтегральних динамічних моделей систем із розподіленими параметрами в інтегральній формі на основі застосування диференціальних рівнянь з дробовими похідними, які отримуються шляхом перетворень ірраціональних передатних функцій.

A first, we used our knowledge of Fourier series to solve several interesting boundary value problems by the method of separation of variables. The success of our method depended to a large extent on the fact that the domains under consideration were easily described in Cartesian coordinates. In this paper/research we address problems where the domains are easly described in polar and cylindrical coordinates. Spesifically we consider boundary value problems for the wave, heat, Laplace and Poisson equation over disks or cylinders. Upon restating these problems in suitable coordinat systems and separating variables, we will encounter new ordinary differential equations, Bessel’s equation, whose solutions are called Bessel function in ways analogous to Fourier series expansions. The vibrations of the membrane are governed by the two-dimensional wave equation, which will be expressed in polar coordinantes, because these are the coordinates best suited to this problem. Finally, we will solve the two dimensional wave equation in polar coordinates (general case).

Structure of Lie point and variational symmetry algebras for a class of odes

Bernstein basis functions based algorithm for solving system of third order initial value problems

We consider systems of ordinary differential equations (ODEs) of the form \documentclass[12pt]{minimal}\begin{document}${\cal B}{\mathbf {K}}=0$\end{document}BK=0, where \documentclass[12pt]{minimal}\begin{document}$\cal B$\end{document}B is a Hamiltonian operator of a completely integrable partial differential equation hierarchy, and K = (K, L)T. Such systems, while of quite low order and linear in the components of K, may represent higher-order nonlinear systems if we make a choice of K in terms of the coefficient functions of \documentclass[12pt]{minimal}\begin{document}$\cal B$\end{document}B. Indeed, our original motivation for the study of such systems was their appearance in the study of Painlevé hierarchies, where the question of the reduction of order is of great importance. However, here we do not consider such particular cases; instead we study such systems for arbitrary K, where they may represent both integrable and nonintegrable systems of ordinary differential equations. We consider the application of the Prelle-Singer (PS) method—a method used to find first integrals—to such systems in order to reduce their order. We consider the cases of coupled second order ODEs and coupled third order ODEs, as well as the special case of a scalar third order ODE; for the case of coupled third order ODEs, the development of the PS method presented here is new. We apply the PS method to examples of such systems, based on dispersive water wave, Ito and Korteweg-de Vries Hamiltonian structures, and show that first integrals can be obtained. It is important to remember that the equations in question may represent sequences of systems of increasing order. We thus see that the PS method is a further technique which we expect to be useful in our future work.

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. 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Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Our main purpose in this project is to help reader find a clear and glaring relationship between linear algebra and differential equations, such that the applications of the former may solve the system of the latter using exponential of a matrix.Applications to linear differential equations on account of eigen values and eigenvectors, diagonalization of n-square matrix using computation of an exponential of a matrix using results and ideas from elementary studies form the core study of our project.

In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.

Ordinary differential equations of fractional order have been presented as a tool of vital importance in modeling the anomalous dynamics of various problems from the exact sciences and engineering, however, is still under discussion a clear and coherent theory analogous to the classical theory of ordinary differential equations. The nonlocal character of their fractional operators provides additional information that allows the development of a comprehensive analysis of the mathematical models. Within these fractional order differential equations, the qualitative approach is an open topic of study at present, in which stability analysis plays a preponderant role. This article presents a description of recent results on the stability of nonlinear fractional order ordinary differential equations and some analytical methods used. First of all, this article presents and describes the fundamental concepts of the study of stability of systems of ordinary differential equations of fractional order, both linear and nonlinear. The results of this research provide fundamental tools in the study of the stability of systems of nonlinear fractional order differential equations that can be applied to various models of applied sciences and engineering. Keywords: fractional analysis, fractional differential equations, stability of fractional equations, Mittag-Leffler stability, Lyapunov functions. https://doi.org/10.55463/issn.1674-2974.50.7.7