A Mixed Spectral Method For Heat Conduction Equation In Unbounded

Many problems in engineering can be came down to solving the problem of heat transfer equation in unbounded domains. There will bring about much more difficulties for numerical simulation of equations in unbounded domains. The simplest method for solving them numerically is to set certain artificial boundaries, impose some artificial boundary conditions and then resolve the corresponding approximate problems by using the finite difference method, the finite element method and the spectral method in bounded domains. But additional errors may be caused by this treatment. To avoid the unnecessary error, pioneers propose some direct approach, such as Hermite spectral method. These will bring about some difficulties in the theoretical analysis and numerical calculation by using the weight function χ(x) = eαx2, α ?= 0 of the existing literature.Thus it seems better to solve such problems directly using the Hermite function with the scaling factor and the weight function χ(x) ≡ 1 as the base function, which can be better to match the asymptotic behaviours of solution, improve the accuracy of numerical solution.The purpose of this work is to develop the spectral method for heat transfer equation in unbounded domains.In chapter 2, some one-dimensional Hermite orthogonal approximation results and fundamental results of one-dimensional Legendre orthogonal approximation are introduced. These results are the mathematical foundation of our paper. Based on which we build up the orthogonal approximation theory with the orthogonal polynomial or orthogonal polynomial function as the basis in unbounded domains.In chapter 3, firstly, generalized Hermite functions with the scaling factor are used as basic functions to expand numerical solution to approximate the solution of linear heat transfer on the whole line. Algorithms scheme is constructed on the basis of the Hermite spectral method. The convergence of the scheme is proved. Numerical results demonstrate its efficiency and high accuracy of this approach. Secondly, the mixed spectral method for heat transfer in an infinite plate is studied combined with Legendre orthogonal approximation. Some mixed Hermite-Legendre orthogonal approximation results are established. The mixed Hermite-Legendre spectral scheme is constructed for non-isotropic heat transfer in unbounded domains. Its convergence is proved. Numerical results demonstrate the spectral accuracy of this approach. Compared with the Hermite- Legendre spectral method in [7], our proposed algorithms have the more accurate numerical solution.In chapter 4, a generalized Hermite spectral method for Fisher’s equation with different asymptotic behaviours is proposed. A fully discrete scheme is given using second order finite difference scheme in time. The convergence and stability of the proposed scheme are analyzed.Numerical experiments substantiate the theoretical analysis and show the efficiency of our approach.The chapter 5 is for some concluding remarks and the problems which can be studied in the future.