The use of Fourier Transform Techniques in Solving Heat and Wave Equations

Abstract

In mathematics, a transform is often some expression that is used to convert one type of problem into another one that is easier to solve. Integral transforms have been used for almost two centuries in solving many problems in applied mathematics, engineering science and physics science. The use of an integral transform is somewhat similar to that of logarithms. That is, a problem containing multiplication or division can be reduced to one containing the simpler operations of addition or subtraction by taking logarithms. After the solution has been obtained in the logarithm domain, the original solution can be obtained by finding an antilogarithm. In the same method, a problem involving derivatives can be reduced to a simpler problem consisting only multiplication by polynomials in the transform variable by taking an integral transform, solving the problem in the transformed domain, and then finding an inverse transform. Historically, the origin of the integral transforms includes the Laplace and Fourier transforms. Fourier transforms is a natural extension of Fourier series to a function represented over the whole x-axis, since many problems described for infinite or semi infinite range, the Basic idea of the Fourier transforms technique is to reduce the number of the independent variables. For the one dimensional (time – dependent) of heat and Wave Equations, it transform the partial differential equation into an ordinary differential equation, the solution of the original problem is obtained by inverting back from the Fourier transform. The main goal of this study is to build analytical skills of Fourier transforms technique and to give a comprehensive coverage of the use of Fourier transforms in solving the heat and wave equations under different initial and boundary conditions. Careful attention is given to the appropriate transformation to be used. The study started by describing Fourier series for a functions defined on a finite interval (-L,L) and then its shown that the Fourier series can be represented by the sums called Riemann sums and so to an integration called Fourier Integral, from which Fourier cosine and sine Integrals are produced. The Fourier transform and inverse Fourier transform are defined, a very interesting properties of Fourier transform and its inverse introduced, similar to Fourier cosine and sine integrals Fourier cosine and sine transforms are defined, through important properties and explained examples, a relationship between the Fourier and Laplace transformation have been introduced in this study . As for future works it is recommended to use the Matlab package to study Fourier transforms, and to solve the Integral equations by using Fourier transforms technique. Finally study the effect of Fourier transforms technique in image processing field.