A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions. (English) Zbl 1540.65402
Summary: We present a linear and nonlinear comprehensive class of variable-order (VO) time-fractional partial differential equations in multi-dimensions with Caputo VO fractional operator. This class comprises, as special cases, several linear and nonlinear equations of important applications in diverse fields, such as, the VO time-fractional telegraph equation, the VO time-fractional diffusion-wave equation and the VO time-fractional Klein-Gordon equation. A novel and accurate spectral tau method based on the shifted Gegenbauer polynomials (SGPs) is presented for the numerical treatment of the aforementioned class of equations. Actually, applying the tau method for solving this class of equations is very difficult, especially in the presence of the nonlinear term of these equations and the nature of the VO derivatives. To overcome this difficulty, new operational matrices for the VO fractional derivative and the approximation of the multiplication of the space-time Kronecker product vectors in multi-dimension are derived. These new matrices have the ability to remove the aforementioned difficulties that facing for applying the tau method. Several linear and nonlinear numerical examples are examined and compared in multi-dimensions with other methods to verify the validity, applicability and accuracy of the proposed methodology.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
variable-order; linear and nonlinear fractional partial differential equations; shifted Gegenbauer polynomials; spectral tau methodReferences:
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