CHAPTER IX The Integral Transform Method 9.1 The Laplace Transform 9.2 The Fourier Transform 9.3 Finite Fourier Transform 9.4 The Hankel Transform 9.5 Generalization of the Integral Transform Method 9.6 Conjugate Integral Transform 698 The Integral Transform The integral transform T of a function fx( ) with variable xD∈ turns it into another function Fs( ) of variable s T :f( x) → F( s) The original function fx( ) can be reconstructed from the transformed function Fs( ) with the help of the inverse transform T -1 :F( s) → f( x) The general form of integral transform is given by the formula F( s) = ∫ K( x,s) f( x) dx D where the function of two variables K( x,s) is called the kernel of the integral transform. The particular form of the kernel K( x,s) defines the type of integral transform. There are many integral transforms which are applied in engineering mathematics: the Laplace, the Fourier, the Fourier sine, the Fourier cosine, the Hankel, the Mellin, the Legendre, the Laguerre, the Hermite and many other integral transforms. The main property of the suitable integral transform for a particular differential equation or initial and boundary value problems for a differential equation is the ability of an integral transform to eliminate the corresponding derivative from the equation reducing it to a simpler form. The procedure of application of the integral transform for solution of differential equations consists in the following steps: - apply an integral transform to the whole equation; - solve it for the transformed function; - apply the inverse transform to retrieve the solution of the differential equation. Schematically this procedure can be described in the following way: let the differential equation (DE) for the unknown function u include its nth derivative with respect to the variable ξ , and let T :u→ U, T -1 :U→ u be the appropriate integral transforms. Then DE includes direct solution n ∂ u u n ∂ξ transform inverse transform eliminates T T -1 yields the derivative solution of DE transformed equation for U solve for U U The choice of the appropriate integral transform depends on the type of the domain of differential equation and on the form of boundary or initial conditions . .