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

##### 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
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