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A new Legendre collocation method for solving a two-dimensional fractional diffusion equation. (English) Zbl 1468.65161

Summary: A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials \(P_{L, n}(x) P_{L, m}(y)\), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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