Jacobi-Davidson Method For Fractional Eigenvalue Problems

Fractional eigenvalue problems originate from fields such as fractal geometry,fractal dimension,and Brown motion.The discretization of fractional eigenvalue problems are investigated and a numerical algorithm for fractional eigenvalue problems is proposed.The main contents can be stated as follows:For the first-order fractional eigenvalue problems,the difference scheme is given by the numerical approximation method based on piecewise linear interpolation of Caputo fractional derivative.The Jacobi-Davidson method for solving the first-order fractional eigenvalue problems is proposed.In order to solve the correction equation effectively,we can turn the problem into solving Toeplitz linear systems.We construct the Strang circulant matrix as a preconditioner and analyze the properties of the preconditioned coefficient matrix.Then we propose the preconditioned generalized minimal residuals method(PGMRES)to solve this linear systems.For the second-order fractional eigenvalue problems,the difference scheme is given based on the difference scheme of the first-order fractional eigenvalue problems.The Jacobi-Davidson method is proposed to solve the second-order fractional eigenvalue problems.In order to speed up the convergence of the algorithm,we propose the preconditioned generalized minimal residual method(PGMRES)to solve the correction equation.Numerical examples show the efficiency of the proposed algorithms.