20.1. INTEGRAL TRANSFORMS

The integral transform f( s) of a function f( x) is defined by the equation

where K( s, x) is a known function of s and x, called the kernel of the transform ; s is called the parameter of the transform and f( x) is called the inverse transform of f( s).

Some of the well-known transforms are given below :

1. Laplace Transform. K( s, x) = e ^sx

   
   

2. Complex Fourier Transform

   
   

3. Fourier Sine Transform

   
   

4. Fourier Cosine Transform

   
   

5. Hankel Transform. K( s, x) = x J _n( sx)

   
   

   where J _n( sx) is the Bessel function of the first kind and order n.

6. Mellin Transform. K( s, x) = x ^{s 1}

   
   

We have already discussed the Laplace transform and its application to the solution of ordinary differential equations. In the present chapter, we shall discuss other transforms and their application to the solution of partial differential equations.

The effect of applying an integral transform to a partial differential equation is to reduce the number of independent variables by one. The choice of a particular transform is decided by the nature of the boundary conditions and the facility with which the transform f( s) can be inverted to give f( x).

20.2. THE FOURIER INTEGRAL THEOREM

Consider a function f( x)...