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Mohebbi, Akbar; Abbaszadeh, Mostafa; Dehghan, Mehdi

A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. (English) Zbl 1287.65064

J. Comput. Phys. 240, 36-48 (2013).
Summary: The aim of this paper is to study the high-order difference scheme for the solution of modified anomalous fractional sub-diffusion equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grünwald-Letnikov discretization of the Riemann-Liouville derivative to obtain a fully discrete implicit scheme. We analyze the solvability, stability and convergence of the proposed scheme using the Fourier method. The convergence order of the method is \(\mathcal{O}(\tau +h^4)\). Numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme.

Cited in 77 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations

Keywords:

modified anomalous fractional sub-diffusion equation; compact finite difference; Fourier analysis; solvability; unconditional stability; convergence; Grünwald-Letnikov discretization; Riemann-Liouville derivative; numerical examples
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\textit{A. Mohebbi} et al., J. Comput. Phys. 240, 36--48 (2013; Zbl 1287.65064)
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