Solutions of differential equations in Fourier transform space

Almost everything (a time-dependent function or signal, electromagnetic waves, sound waves, stock price changes, etc.) can be described as a waveform. The Fourier Transform is a powerful tool for manipulating and evaluating these forms. It can be analyzed in two different ways: continuous and discrete. Both transforms map an object between two orthogonal spaces. The Fourier transform for continuous variables is given by: F(k)=1/√2π ∫_(-∞)^∞▒〖f(x) e^(-ikx) dx〗 and the inverse Fourier transform f(x)=1/√2π ∫_(-∞)^∞▒〖F(k) e^ikx dk〗. The Fourier transform is denoted by the mapping f(x)→F(k), and the inverse Fourier transform by the mapping F(k)→f(x). In this thesis, we present a new solution method for differential equations by considering that the given equation is in Fourier space. We utilize the properties of Fourier transforms and the Dirac Delta function. By the new method we present, we obtain the solutions of some differential equations.