The solution of boundary value problems with asymmetric boundary conditions by means of finite fourier transforms

New finite Fourier transforms and the corresponding infinite series are introduced which bring the solution of boundary value problems with asymmetric endpoint conditions within the domain of integral transform theory. Multiple finite transforms based upon these new kernels are given along with the appropriate infinite series. Using these multiple transforms the solution of the two-dimensional Laplacian for all possible combinations (15) of Dirichlet and Neumann boundary conditions is shown. Faltung theorems are established for some of the kernels. Finally, as an illustration of the theory the solution of the electrostatic problem under various boundary conditions is given.