Solution of the heat equation with a discontinuous coefficient with nonlocal boundary conditions by the Fourier method
DOI:
https://doi.org/10.70474/4sw3g811Keywords:
Heat equation with discontinuous coefficients, spectral problem, non-self-adjoint problem, Riesz basis, classical solution, Fourier methodAbstract
This paper substantiates the solution by the method of separation of variables of the initial-boundary value problem for the heat equation with a discontinuous coefficient, under periodic or anti-periodic boundary conditions. Using the Fourier method, this problem is reduced to the corresponding spectral problem. The eigenvalues and eigenfunctions of this spectral problem are found. It is shown that the spectral problem is non-self-adjoint and a conjugate spectral problem of this original spectral problem is constructed. Further, it is proved that the system of eigenfunctions forms a Riesz basis. For this purpose, a self-adjoint spectral problem is constructed and its eigenvalues and eigenfunctions are found. In conclusion, using biorthogonality, the main theorem on the existence and uniqueness of a classical solution to the problem is proven.
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