Error analysis of the reduced RBF model based on POD method for time-fractional partial differential equations. (English) Zbl 1452.65273
Summary: In this paper, we present a new reduced order model based on radial basis functions (RBFs) and proper orthogonal decomposition (POD) methods for fractional advection-diffusion equations with a Caputo fractional derivative in time. In the proposed scheme, the number of basis functions in the usual RBFs method reduces by the POD technique. Therefore, the computational cost of the RBF-POD method decreases in comparison with usual RBFs method, while the accuracy completely maintains. In the sequel, we provide a complete error analysis in the \(L_2\) norm between the exact solution and the RBFs solution, as well as between the exact solution and the proposed RBF-POD model by using the properties of the native space and projection operators. Also, the obtained error estimation is used to choose the number of POD bases for constructing the RBF-POD model with the required accuracy. Numerical examples are given to confirm the accuracy and efficiency of the proposed scheme.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 65D12 | Numerical radial basis function approximation |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
radial basis function (RBF); proper orthogonal decomposition (POD); fractional differential equations; reduced order modelReferences:
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