Mixed Jacobi-Fourier Spectral Method For Fisher Equation

The spectral method as one of the numerical methods for solving differential equations de-veloped rapidly in the past three decades. The main merit is the high accuracy. So it has been usedto many fields in science and engineering. The early works of the spectral method are availablefor periodic problems and problems defined on rectangular domains. However, many problemsarising in science and engineering are singular problems, and some problems in unbounded do-main can be reformulated as singular ones in bounded domain. This often undermine the merit ofspectral methods. Some authors developed the Jacobi orthogonal approximation in certain Hilbertspaces and used for singular problems. Recently, some authors proposed the generalized Jacobiapproximations. These provide efficient means to overcome the above disadvantage and enlargethe applications of the spectral method.In this thesis, we propose the mixed Jacobi-Fourier spectral method for solving the Fisherequation in a disc. To deal with the singularity at the center of the region, we consider thepolar condition. We first recall some results on the Jacobi approximation and the Fourierapproximation. Then, some mixed Jacobi-Fourier approximation results are established, whichplay an important role in numerical simulation of various problems defined in a disc. Next, wepropose the spectral scheme for the Fisher equation in a disc and prove it’s generalized stabilityand convergence. Numerical results demonstrate the efficiency of this new algorithm. Thisapproach has several merits: We use the polar condition and the generalized Jacobi approximationin the radial direction to avoid the singularity. It also reduces the difficulty of the theoreticalanalysis and provide a sparse system of unknown coefficients of expansion of numerical solution.Moreover, the numerical solution possesses the spectral accuracy in space.