Topics in Laplace and Fourier Transform

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The Laplace transform* of a suitably defined function f of a real variable t is a related function F of a real variable s. The use of Laplace transforms provide a powerful method of solving differential and integral equations.

We have already noted that the main purpose of studying Laplace transforms is for solving various types of differential equations. During the process of solving a differential equation, we shall also require to find a function when its Laplace transform is known.

Integral equations occur very frequently in the fields of mechanics and mathematical physics. The development of the theory of integral equations began with the works of Italian mathematician V. Volterra (1896) and the Swedish mathematician I. Fredholm

In the present chapter, we shall study the method of solving a system of differential equations using Laplace transformations of functions. Our knowledge of finding Laplace transforms and inverse Laplace transforms would help us to solve the systems of differential equations.

Fourier* transforms play an important part in the theory of many branches of science. The use of Fourier transforms is indispensible in solving the problems in the field of mathematics, physics and engineering. Many types of integrals and differential equations can be solved by using Fourier transforms.

When the function involved in a physical (or geometrical) problem involves more than one independent variable, a partial differential equation is obtained. The solution of this partial differential equation gives the required function. Fourier transforms are a very important tool in solving such partial differential equations.