Fourier Transforms
Fourier Transforms
Fourier Transforms
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3.3 Applications of <strong>Fourier</strong> <strong>Transforms</strong>The principal uses of <strong>Fourier</strong> <strong>Transforms</strong> are the solution of partial differential equations,possessing a particular structure, and the processing and properties of time series. It is alsopossible to invoke transform methods to solve certain types of integral equations and toenable the summation of certain types of slowly-converging series. Some of theseproperties are shown in the following examples.3.3.1 Solution of Ordinary Differential EquationsAlthough not especially useful for the solution of ordinary differential equations (ODEs), itis instructive to show how transform methods work on ODEs before considering thebroader structure of partial differential equations (PDEs). This is best illustrated by anexample.Example 1Obtain a solution of the Ordinary Differential Equation (ODE)d 2 ydx 2 − 2 y gx ,which → 0asx →, assuming that gx is absolutely integrable.The general solution to a linear ODE is often written in the formyx y 1 x y 2 x ,where y 1 x and y 2 x are the Complimentary Function (CF) and Particular Integral (PI)respectively. The complementary function isy 1 x Ae x Be −x ,for arbitrary constants A and B; these solutions do not → 0asx →. Thus only the PI ispotentially capable of meeting this requirement. For convenience we write yx instead ofthe correct y 2 x.Note that it is assumed that ≠ 0. If 0 can occur, then it is necessary to consider thiscase separately and the CF solution is y 1 x Ax B and which is also incapable ofmeeting the large x requirement unless y 1 x ≡ 0.Taking the FT of the equation gives12