Numerical solution of initial value problems by rational interpolation method using Chebyshev polynomials

Abstract

<jats:p>In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval.Keywords and Phrases: Chebyshev polynomial, Rational Interpolation, Minimaxpolynomial, Initial Value Problems and Ordinary Differential Equations (ODEs)</jats:p>

A New Modified Algorithm Based Upon Exponential and Polynomial Functions for Solving Initial Value Problems

“…Some methods are explicit, and some are implicit methods. In continuation of this effort, many researchers have derived new numerical integration methods of explicit and implicit nature in an attempt to obtain better approximate results than various of the available ones in the present literature such as [1,2,5,6,10,12], just to mention a few. In [10] author developed a new numerical technique for the solution of initial value problems of first-order ODEs whereas in the present study, we have proposed a new modified algorithm based upon the exponential and polynomial function for the solution of initial value problems of first-order ODEs, the paper is organized as follows: the Section 2 starts with the derivation of the polynomial and exponential type function for a proposed new modified numerical algorithm, the section 3 to proves that proposed new modified numerical algorithm is fourth-order convergent and discuss the local truncation error, consistency and larger stability region of the proposed algorithm structured within its subsections.…”