On the modified Taylor's approximation for the solution of linear and nonlinear equations

Abstract

Fourier series and Taylor's expansions are commonly used in the fields of science and engineering. A whole range of interesting physical problems can be solved using one or the other of these standard techniques. Yet, there is a class of problems whose solutions exhibit near sinusoidal or repetitive behavior that cannot be solved using the above expansions. It is for these types of problems that the modified Taylor expansion has been developed. It is a method that combines the advantages of the repetitive behavior of sinusoidal functions and polynomial series. This technique has not been employed for tackling equations whose solutions are near sinusoidal. We discuss the method and apply it to a number of interesting problems that show its utility. Examples include the solution of differential, integral, integro-differential and functional equations. In addition, we propose an algorithm for the numerical integration of a function which exhibits near periodic behavior, namely the Bessel function. Most of the symbolic and numerical computations have been performed using the Computer Algebra System––Maple.