Numerical Solutions of Second Order Two Point Boundary Value Problems. Legendre Polynomials, Homotopy Continuation Method and Their Applications

      Аннотация: Now days many physical problems which arise in different fields of Science and Engineering can be solved by the aid of differential equation which is either ordinary differential equation or partial differential equation. Although, both linear and nonlinear second order ordinary differential equations together with two point boundary value problems play a vital role in solving such kind of problems mentioned above, their solution may not be obtained analytically as easy as one can expect. In this book shifted Legendre polynomial approximation on a given arbitrary interval has been designed to find an approximate solution of a given second order linear or nonlinear two point boundary value problems of ordinary differential equations. The unknown Legendre coefficients of the nonlinear systems which are the solutions of the system have been solved by continuation method, whereas the unknown Legendre coefficients of the linear systems which are the solutions of the system have been solved by one of the direct methods available for solving them. The method is computationally economical and is able to find the solution on any arbitrary interval. Lastly numerical examples are done.

Ключевые слова: Ordinary Differential Equations (ODEs), Boundary Value Problems (BVPs), Legendre Operational Matrix of Differentiation, Linear and Nonlinear ODEs, Shifted Legendre Polynomials, Homotopy Continuation Method