Summation of the spectral expansions associated with the solvability of the heat and wave problems

Abstract

A relation between the theory of multiple Fourier series and partial differential equations was divulged in the beginning of the nineteenth century and it is known as spectral theory of the differential operators. The spectral theory of the differential operators is a significant part of mathematical sciences and it has applications in many branches of engineering. The development of the spectral theory of the differential operator is started since the time when Fourier studied heat conduction problem in a rod and the solution is found as a form of sin series. To adjust the obtained solution leads to study of the problems of the convergence of that series solutions and it depends on the initial and/or boundary data. Obtained series solutions of the problems may not be convergent. Then the problem of summability will occur. Regularization of the divergent series solution is accurate numerical interpretations of the solutions of the problems. In this research, to find the equiconvergence of the spectral expansions and to find the solution of the heat and wave problem regularization was required. In the first phase, we studied a special elliptic partial sum of order 2(m+1) of multiple Fourier series and integral in the spaces of singular distributions. We discussed the equiconvergence in summation of the Fourier series and integral of the linear continuous functional for specific conditions in the Lioville space. Therefore, we proved a precise equiconvergence relation between index of the Bochner-Riesz means of the expansions and power of the singularity of the distributions with compact support in summation associated with the elliptic operator. After that, we studied the vibration problem made of thin elastic membrane stretched tightly over a square frame. The deflection of the membrane during the motion is small compared to the size of the membrane. And for heat transfer problems the plate is made of some thermally conductive material. We discussed different types of heat transfer problems such as, steady state heat transfer problem, heat transfer insulated plate problems. Solution of wave and the heat transfer problems are subjected to the boundary conditions and initial conditions and had a form of double Fourier series. The coefficient of the Fourier series found from the initial conditions. Convergence of the corresponding Fourier series depends on smoothness or singularity of initial conditions. In our case, initial conditions were the Dirac delta function and it diverges. Thus for the solutions of the corresponding heat and wave problems some regularizations of the Fourier series solutions are required. Here, based on the singularity we considered the Reisz method of summation as regularization of the Fourier series solutions of the heat and wave problems. When we increased the order of the Reisz means, the solutions were convergence but the numerical calculations were increased. So, to minimize the calculations of the regularized Fourier series solutions, we optimized the regularization of the solutions of the plate vibration and heat transfer problems. For optimization of the regularized Fourier series solutions, we took minimum order of the Reisz means. The minimum order was s > (N - 1)/2 – l. After optimization, we used a numerical computing programming (MAT LAB) for the numerical solutions. Here, we found the optimization of the regularization of the series solutions at a fixed point of the plates at initial time and critical index. After critical point we achieved the good convergence.