Combination of discrete technique on graded meshes with barycentric rational interpolation for solving a class of time-dependent partial integro-differential equations with weakly singular kernels. (English) Zbl 1538.65619
Summary: In this paper, a combination of a time discrete technique on graded meshes with barycentric rational interpolation is employed to solve a class of one- and two-dimensional time-dependent partial integro-differential equations with weakly singular kernels. Equations of this type are also commonly called fractional evolution equations, whose solutions have some singular behaviours near the initial time. This makes that it’s difficult to reach an expected convergence result for the numerical treatment concerning time variable on uniform meshes. In the temporal direction, a generalized Crank-Nicolson difference scheme and the product integration formula on graded meshes are used to discretize the differential term and the integral term, respectively. Moreover, it’s proved that the time discrete scheme is unconditionally stable and quadratically convergent. Further, a fully discrete numerical approach is structured with spacial directional discretization in terms of barycentric rational interpolation. The results of numerical experiments display the efficiency of the method and confirm the theoretical analyses. In addition, the present scheme can reach second-order convergence in time both for smooth initial value and non-smooth initial value.
MSC:
| 65R20 | Numerical methods for integral equations |
| 65D05 | Numerical interpolation |
| 45K05 | Integro-partial differential equations |
| 35R11 | Fractional partial differential equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
time-dependent partial integro-differential equations; weakly singular kernels; graded meshes; barycentric rational interpolationSoftware:
Differentiation Matrix SuiteReferences:
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