A PDE approach to space-time fractional parabolic problems. (English) Zbl 1337.26014
Solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative have been studied, where the Caputo fractional time derivative is discretized and analyzed in a general Hilbert setting. Also, an implicit fully discrete scheme, first-degree tensor product finite elements in space, and an implicit finite difference discretization in time are analyzed. The authors are interested in the numerical approximation of an initial boundary value problem for a space-time fractional parabolic equation. Relation (1.1) gives this basic concept. Whereas, Equation (1.2) suggests the definition of the fractional derivative in time in an open interval \((0, 1)\), which is considered as the left-sided Caputo fractional derivative of certain order with respect to the variable \(t\). The main difficulty or one of the difficulties that arises in the study of the problem given in (1.1) is the non-locality of the fractional time derivative and the fractional space operator. An approach to overcome the non-locality in space, a seminal result is suggested in Equation (1.5). The non-locality of the operator considered in the paper raises the difficulty to design an efficient technique to treat numerically the left-sided Caputo fractional derivative of certain order. The approaches, among several similar, to overcome this difficulty are finite differences, finite elements, and spectral methods. Fractional powers of elliptic operators through spectral theory are discussed in Sections 2.1 and 2.2, whereas the solutions of Equations (1.1) and (1.5) are to find in Section 2.3. Space and time regularity, with adequate conditions, are established. Section 3.4 discusses semi-discrete error estimation scheme. Section 4.1 contains the truncation method proposed to study the space discretization of the relation (2.7), the finite element approximation of the solution of (1.5) and has the properties of the weighted elliptic projector. Employing the fully discrete scheme for the solution of (4.4), error estimation is given in Section 5.
Reviewer: Deshna Loonker (Jodhpur)
MSC:
| 26A33 | Fractional derivatives and integrals |
| 65J08 | Numerical solutions to abstract evolution equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65R10 | Numerical methods for integral transforms |
Keywords:
fractional integrals; fractional derivatives; fractional diffusion; weighted Sobolev spaces; error estimates; finite elements; elliptic operators; non-localityReferences:
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