We discuss the cubic spline collocation method with two parameters for solving the initial value problems (IVPs) of fractional differential equations (FDEs). Some results of the local truncation error, the convergence, and the stability of this method for IVPs of FDEs are obtained. Some numerical examples verify our theoretical results. A ninth order block hybrid collocation method is proposed for solving general second order ordinary differential equations directly. The derivation involves interpolation and collocation of basic polynomial that generates the main and additional methods.
Ordinary Differential Equation (ODE), Initial Value Problem (IVP), Canonical Polynomial, Collocation 1. Introduction The subject of Ordinary Differential Equation (ODE) is an important aspect of mathematics. It is useful in mod-eling a wide variety of physical phenomena—chemical reactions, satellite orbit, electrical networks, and so on.
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. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations. Their use is also known as "numerical integration", although this term is sometimes taken to mean the computation of integrals. Many differential equations cannot be solved using symbolic computation. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms ...
Runge Kutta Collocation Method for the Solution of First Order Ordinary Di erential Equations A. O. Adesanya1 Department of Mathematics Modibbo Adama University of Technology Yola, Adamawa State, Nigeria A. U. Fotta Department of Mathematics Adamawa State Polytechnic Yola, Adamawa State, Nigeria R. O. Onsachi Department of Mathematics
been given such as, homotopy perturbation method , homotopy analysis method , collocation method [7,14] and others . Representation of a function in terms of a series expansion using orthogonal polynomials is a fundamental concept in approximation theory and form the basis of the solution of differential equations [15, 16]. Chebyshev
Some of the topics covered in this volume are: discrete variable methods, Runge-Kutta methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, differential-algebraic and delay-differential equations, and stochastic initial value problems. Collocation and Galerkin Time-Stepping Methods H.T. Huynh National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Abstract We study the numerical solutions of ordinary differential equations by one-step methods where the solution at t n is known and that at t n+1 is to be calculated. The approaches employed are ...
Efﬁcient solution of ordinary differential equations with high-dimensional parametrized uncertainty. Zhen Gao1 and Jan S. Hesthaven2, 1 Research Center for Applied Mathematics, Ocean University of China, Qingdao, 266071, PRC & Division of Applied Mathematics, Brown University, Providence, 02912, USA. Discrete methods were given which were used in block and implemented for solving the initial value problems, being continuous interpolant derived and collocated at grid points. Some numerical examples of ordinary differential equations were solved using the derived methods to show their validity and the accuracy.
Discrete methods were given which were used in block and implemented for solving the initial value problems, being continuous interpolant derived and collocated at grid points. Some numerical examples of ordinary differential equations were solved using the derived methods to show their validity and the accuracy. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature.
General Linear Methods for Ordinary Differential Equations is an excellent book for courses on numerical ordinary differential equations at the upper-undergraduate and graduate levels. It is also a useful reference for academic and research professionals in the fields of computational and applied mathematics, computational physics, civil and ...
Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena.
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts.
Apr 28, 2017 · Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.