For potential problems, the temperature at the boundary should be as accurate as possible.
Many boundary value problems can be solved by means of integral transformations, such as the Laplace transform function, which transform a differential equation into an algebraic equation in which the boundary conditions are automatically considered.
After solving the algebraic equation, one finds the solution of the original equation by means of the inverse transformation.
From the definition of the transformations it is apparent that they are linear.
The usefulness of the finite Fourier sine and cosine transformations in solving differential equations is primarily due to the fact that differentiation of $$\begin f_(n) g_(n) =& \int_^F_(\alpha) \cos n \alpha \,d\alpha \int_^ G_(\beta) \sin n \beta \,d\beta \\ =& \frac \int_^ F_(\alpha)\cos n \alpha \,d\alpha \int_^ G_(\beta) \sin n \beta \,d\beta \\ =& \frac \int_^ \int_^F_(\alpha)G _(\beta)\cos n\alpha \sin n\beta \,d\alpha \,d\beta.
The transform exists for all bounded, piecewise continuous functions over a finite interval.
Solved Problems On Fourier Transform
In recent years, the finite Fourier transform method has been applied to a wide class of boundary value problems in many interesting mathematics, physics, chemistry and engineering areas [1–3].
\end$$ $$\begin &\frac\int_^\sin nx\int_^F_(x-y) G_(y) \,dy\,dx -\frac \int_^\sin nx \int_ ^F_(x-y) G_(y) \,dy\,dx \ &\quad = \frac \int_^\sin nx \int_^F_(x-y) G_(y) \,dy\,dx \ &\quad = \frac \int_^\sin nx \int_^F_(x-y) G _(y) \,dy\,dx \ &\quad = \frac S\=S\biggl\ , \end$$ 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.
Also the finite Fourier transform method differs from the usual Fourier transformation method in that the solutions are obtained without performing the inverse Fourier transforms.
In principle, the finite Fourier transform method may be extended to analog simulations of heat equations in three space variables, and it may also be a very efficient technique for the solution of multidimensional heat equations.