The approximation of the distributional solution of heat and wave problems by spectral decomposition connected with elliptic differential operators / Abdulkasim Akhmedov

Centre for Mathematical Sciences

Thesis (Doctor of Philosophy) -- Universiti Malaysia Pahang Al Sultan Abdullah - 2025

Includes bibliographical references

Partial differential equations of elliptic type have important applications in physical sciences and engineering, especially in the field of non-relativistic quantum theory, which is built up using the spectral analysis of Schrodinger operator. It is quite difficult to imagine the modern achievements in the field of geometry and stochastic analysis without application of spectral theory of second order elliptic differential operators. Many qualitative and quantitative techniques like Sturm-Liouville theory, separation of variables, Fourier and Laplace transforms, perturbation theory, eigenfunction expansions, variational methods are used to study and develop the spectral theory of the elliptic differential operators. In this research the problems concerning the spectral expansions connected with elliptic differential operators in the space of singular distributions is investigated, which describes the vibration process made of thin elastic membrane stretched tightly over a rectangular frame. Additionally, spectral expansions of the distributions connected with elliptic differential operators are applied to solve heat equations governing circular plates with singular initial heat sources by the numerical approximation techniques. The aim of this research is to study the regularization of spectral expansions and optimization of the regularization of the solution of heat and wave problems. This research starts with establishment of the sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material. Then, the numerical approximation by Riesz means to the solution of wave problems with initial condition involving Delta function is constructed. Finally, validation of the convergence of spectral decomposition by Riesz Means of order 𝑠𝑠>𝑁𝑁−12+1 for the case 𝑁𝑁=3 is attained. The challenge lies in obtaining appropriate estimates for the spectral decomposition of the elliptic operator within classes of periodic distributions, which ensures the convergence of Fourier series distributions. The process of estimating the spectral decompositions of the distributions connected with the elliptic differential operators requires very complicated calculations which depends on the functional structure of the distributions. It is well known that the theory of the eigenfunction expansions of the differential operators closely connected with the convergence problems of Fourier series and integrals. The one of the most important summation methods which is called spherical summation method connected with the eigenfunction expansions of the Laplace operator, while the questions on convergence of the multiple Fourier series summed over the elliptic levels can be investigated by using the spectral theory of the elliptic differential operators. Sufficient and necessary conditions for multiple Fourier series and integrals, corresponding to distributions from classes of Sobolev, to be equiconvergent has been established in the case of plate vibration problem for steady state heat transfer and heat transfer through insulated plate subjected to different boundary and initial conditions. Furthermore, the Riesz summation method is used to approximate the solution of the wave and heat equations with distributional boundary conditions and sufficient conditions for convergence of the multiple Fourier series of functions from Sobolev classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigation of multiple Fourier series in contemporary harmonic analysis extensively incorporates techniques from functional analysis, mathematical physics, modern operator theory, and spectral decomposition.