Polynomial approximation of the numerical solutions of second order linear differential equations

Abstract

In this work the method of finding the solution of second order linear differential equations in the Chebyshev basis polynomial representation is studied and presented. In order to determine the coefficients of the solution of the differential equation which is assumed to have an orthogonal polynomial representation, the derivatives up to second order of each of the basis polynomials in its orthogonal representation has to be computed. The method reduces the problem into solving algebraic equations that approximate the coefficients of the particular integral. Comparison between the numerical solutions and the exact solutions of certain equations shows that in the case when the source function is not a polynomial equation, the results do not converge to the expected exact solution. Better approximation of the source function in terms of Chebyshev basis is required for the exponential or other trigonometric and transcendental functions.