Infinite Fourier Sine and Cosine Transforms for Solving Linear Partial Differential Equations

Abstract

Fourier transforms are integral transforms obtained from Fourier series coefficients. Integral transforms are considered as one of a very important applications due to their super ability to separate complex mathematical connections that the function lies under their effects; or to obtain another form from the original function that can easily and mathematically be handle with. Fourier transforms depend on Fourier series that have great importance in approximating functions in forms of infinite series from the functions of sine and cosine simultaneously or representing each one separately. Fourier transforms are of two types: the first type is termed Fourier finite Transforms and the second type is called infinite Fourier Transforms. In both types, there are what are called as Fourier Sine and Cosine Transforms and their inverse transforms. The importance of Fourier transforms lies in its ability to transform a function that lies under its effect into another function with another independent variable and other domain. Since most of the physical (natural) phenomena can be described by a partial differential equation, however, the solution of these partial differential equations is considered difficult under the different restriction that it imposed. This study aimed at obtained easy and fast solutions for the partial differential equations by applying Fourier sine and cosine infinite transforms, where the partial differential equation is transformed into ordinary differential equation to be solved as an ordinary differential equation, according to its type. This study tackled how a function is represented under some conditions to take the form of infinite series of sine and cosine, called Fourier series and then how to transform these series to Fourier integrals that should be transformed by certain techniques to what are called Fourier transforms. It was clear through this study that Fourier infinite sine and cosine transforms and their inverse transforms are able, with their capabilities, to provide a solution for the partial differential equation by a simple and easy way. Therefore, the study recommends the expansion in studying and applying them in solving nonlinear partial differential equations ,the non computed integrals in this study are difficult to be computed by normal way of integrations , therefore we recommend to evaluate them by residue theorem .