A stable operational matrix based computational approach for multi-term fractional wave model arise in a dielectric medium. (English) Zbl 07853106
Summary: The semi-discretization technique combined with an operational matrix approach is proposed to solve the fractional order wave equation that arises in a dielectric medium. In this approach, Caputo’s derivative terms of order \(\alpha\) and \(\beta\) are approximated by the difference scheme of order \(\mathcal{O}(\tau^{3 - \alpha})\) and \(\mathcal{O}(\tau^{3 - \beta})\), \(1 < \beta < \alpha < 2\), respectively, to transform the proposed fractional order wave equation into a system of second order ordinary differential equations (ODEs). To solve the ODEs, the operational matrix method is used which has several advantages over the several ODE solvers. The convergence of the approximation taken in the spatial direction at the \(k\)th level of time is established. Moreover, the scheme is unconditionally stable with the rate of convergence \(\mathcal{O}(\tau^{3 - \alpha})\) and \(\mathcal{O}\left(\frac{1}{2^{N + 1}(N + 1)!}\right)\) in the time and spatial direction, respectively. Finally, test examples are included to show the efficiency and accuracy of the proposed method and to support the theoretical results.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35Rxx | Miscellaneous topics in partial differential equations |
| 65Lxx | Numerical methods for ordinary differential equations |
Keywords:
fractional order wave equation; semi-discretization; operational matrix; convergence analysisSoftware:
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