Abstract
In this paper, we establish a finite difference scheme for a class of time fractional parabolic equations with variable coefficient, where the time fractional derivative is defined in the sense of the Caputo derivative. The local truncating error, unique solvability, stability, and convergence for the present scheme are discussed by use of the Fourier analysis method, which shows that the present finite difference scheme is unconditionally stable and possesses spatial fourth-order accuracy. Theoretical analysis is supported by two numerical examples, and the maximum errors and the convergence order are checked.
Highlights
Nonlinear partial differential equations are widely used to describe many complex phenomena in various fields including the scientific work and engineering fields
Fractional differential equations (FDEs) containing the fractional derivative are widely used as models to express many important physical phenomena such as fluid mechanics, plasma physics, classical mechanics, quantum mechanics, nuclear physics, solid state physics, chemical kinematics, chemical physics, and so on
In order to better illustrate the described physical phenomena, we need to obtain the solutions of fractional differential equations (FDEs)
Summary
Nonlinear partial differential equations are widely used to describe many complex phenomena in various fields including the scientific work and engineering fields. In [ , ], the authors presented compact finite difference schemes with convergence order O(τ –α + h ) ( < α < ) for fractional subdiffusion equation with spatially variable coefficient, whereas Wang et al [ ] proposed a Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations and proved the well-posedness and optimal-order convergence of this method. Chen et al [ ] presented a fast semiimplicit difference method with convergence order O(τ + h) for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients and developed a fast accurate iterative method by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices, whereas Wang [ ] established a compact finite difference method with convergence order O(τ –α + τ + h ) ( < α < ) for a class of time fractional convection-diffusion-wave equations with variable coefficients. Some conclusions are presented at the end of this paper
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