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\end$$ Boundary value problems of partial differential equations concerned with temperature as the unknown may be solved by a finite Fourier transform method.
The temperature at points other than the boundary, if they should be needed, can be obtained by summing the Fourier coefficients.
Similarly, partial differential equations are changed into ordinary differential equations by applying these transformations.
Two transformations which are particularly useful in solving boundary value problems are the finite Fourier sine and cosine transformations.
If you are unfamiliar with the Fourier Transform, check out my introduction here.
The method presented here works for initial value problems where the PDEs is solved with respect to some initial values known a priori.\end$$ $$\begin &\frac\int_^\sin nx\int_^F_(x-y) G_(y) \,dy\,dx -\frac \int_^\sin nx \int_ ^F_(x-y) G_(y) \,dy\,dx \ &\quad = \frac \int_^\sin nx \int_^F_(x-y) G_(y) \,dy\,dx \ &\quad = \frac \int_^\sin nx \int_^F_(x-y) G _(y) \,dy\,dx \ &\quad = \frac S\=S\biggl\ , \end$$ Since we will be concerned with functions of two independent variables, it is necessary to comment briefly on the characteristic of the finite Fourier transform of such a function.The finite Fourier sine transformation of Similar changes must be made in the other formulas in the applications which follow.Many other transforms exist which may be used to solve PDEs [4, 5].A feature which makes the finite transform a very economical method, is that the inverse transform may be solved only for regions of interest [6–8].In my previous post, PDEs using Fourier Analysis I, I investigated solving PDEs, in particular the wave equation, on a membrane using Fourier Series.In this article, I shall formulate a generalised method of solving PDEs, that is founded on the Fourier Transform.The particular transformation discussed in this paper is the finite Fourier transform, which is applicable to equations in which only the even order derivatives (of the function) with respect to transformed variable will be treated.The finite Fourier transform method is one of various analytical techniques in which exact solutions of boundary value problems can be constructed.The most widely used methods for the solution of boundary value problems are based on finite differences.These methods require certain assumptions about where the finite difference equals the derivative which by necessity have to be most loosely made on the boundaries.