The use of Fourier integrals and Fourier transforms in applications is quite extensive. Therefore, we will make no attempt to consider all the various applications involving these integrals, but rather briefly discuss how they are used in several representative areas of application.
The basic aim of the transform method is to transform the given problem into one that is easier to solve. In the case of an ordinary differential equation (ODE) with constant coefficients, the transformed problem is algebraic. The effect of applying an integral transform to a partial differential equation (PDE) is to exclude temporarily a chosen independent variable and to leave for solution a PDE in one less variable. The solution of the transformed problem in either case will be a function of the transform variable s and any remaining independent variables. Inverting this solution produces the solution of the original problem.
The exponential Fourier transform may be applied to derivatives of all orders that may occur in the formulation of a given problem. However, since it incorporates no boundary conditions in transforming these derivatives, it is best suited for solving DEs on infinite domains where the boundary conditions usually only require bounded solutions. On the other hand, the Fourier cosine and Fourier sine transforms are well suited for solving certain problems on semiinfinite domains where the governing DE involves only even-order derivatives.
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