New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order. (English) Zbl 1393.65015
Summary: In this paper, a class of new compact difference schemes is presented for solving the fourth-order time fractional sub-diffusion equation of the distributed order. By using an effective numerical quadrature rule based on boundary value method to discretize the integral term in the distributed-order derivative, the original distributed order differential equation is approximated by a multi-term time fractional sub-diffusion equation, which is then solved by a compact difference scheme. It is shown that the suggested compact difference scheme is stable and convergent in \(L^\infty\) norm with the convergence order \(\mathcal{O}(\tau^2 + h^4 +(\Delta \gamma)^p)\) when a boundary value method of order \(p\) is used, where \(\tau\), \(h\) and \(\Delta\gamma\) are the step sizes in time, space and distributed-order variables, respectively. Numerical results are reported to verify the high order accuracy and efficiency of the suggested scheme. Moreover, in the example, comparisons between some existing methods and the suggested scheme is also provided, showing that our method doesn’t compromise in computational time.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M38 | Boundary element methods for initial value and initial-boundary value problems involving PDEs |
| 65D30 | Numerical integration |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
distributed-order derivative; boundary value method; finite difference method; quadrature rule; stability and convergence; high accuracyReferences:
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