The residue method of the non self-adjoint initial boundary value problem of the linear heat equation

The Fourier series method is one of the basic tools for a solution of the linear heat equation. As known, the spectral problem related with a mixed self-adjoint problem has real eigenvalues, and corresponding eigenfunctions of these eigenvalues make a complete system. In this case, the expansion formula of any continuous function of these eigenfunctions holds, that is fundamental in the application of the Fourier series method.There are many problems which described by differential equations with non self-adjoint boundary conditions. In this case, the eigenfunctions are incomplete and the application of the Fourier method becomes difficult.In the thesis, the linear heat equation with a singular source function subject to boundary condition involving the higher derivatives with respect to time coordinate is studied.In the second section, the necessary mathematical backgrounds from a spectral theory of differential operators are introduced.In the final section, the solution of the mentioned problem is found using the Fourier and Residue methods, respectively.