Fractional Differential Equations

This work deals with a higher order numerical approximation for analyzing a class of multi-term time fractional partial integro-differential equations involving Volterra integral operators. The solutions to these problems have a mild singularity at the initial time, due to which an initial layer appears, which becomes more sharper as the highest order time fractional derivative decreases. This behaviour reduces the rate of convergence by standard approaches. We start the present work by considering the existence and uniqueness of a class of generalized partial integro-differential equations and then, present the L1 discretization on a graded mesh in time which is adapted towards the initial time level. This discretization leads to a higher order accuracy than the solutions obtained on a time uniform mesh. The convergence analysis corresponding to the Volterra integral operator is nontrivial as it uses a repeated quadrature rule. This analysis can also be extended for weakly singular kernels. The stability analysis of the present scheme with a sharp error estimation is also provided. The analysis with extensive experiments shows that a higher rate of accuracy can be attained for several suitable choices of the grading parameters for solving several classes of time fractional integro-differential equations. 

Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis

In this paper, we consider an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. The L1 method on graded meshes is employed in temporal direction. In the spatial direction, we develop the orthogonal spline collocation method. The stability and convergence of the above mentioned method are proved theoretically, and it's still valid if α 1 → 1 − . A new robust error bound is obtained, which is a more natural and desirable result and does not blow up as α 1 → 1 − . Some numerical results are also provided to verify our theoretical analysis and demonstrate the effectiveness of the OSC method. 

A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation

A time-fractional initial-boundary value problem is considered, where the differential equation has a sum of fractional derivatives of different orders, and the spatial domain lies in $${\mathbb {R}}^d$$ with $$d\in \{1,2,3\}$$. A priori bounds on the solution and its derivatives are stated; these show that typical solutions have a weak singularity at the initial time $$t=0$$. A standard finite element method with mapped piecewise bilinears is used to discretise the spatial derivatives, while for each time derivative we use the L1 scheme on a temporal graded mesh. Our analysis reveals the optimal grading that one should use for this mesh. A novel discrete fractional Gronwall inequality is proved: the statement of this inequality and its proof are different from any previously published Gronwall inequality. This inequality is used to derive an optimal error estimate in $$L^\infty (H^1)$$. It is also used to show that, if each mesh element is rectangular in the case $$d=2$$ or cubical in the case $$d=3$$, with the sides of the element parallel to the coordinate axes, then a simple postprocessing of the computed solution will yield a higher order of convergence in the spatial direction. Numerical results are presented to show the sharpness of our theoretical results.

https://link.springer.com/content/pdf/10.1007/s10915-019-01115-w.pdf

Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem

An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties

Abstract An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is ( 0 , 1 ) d {(0,1)^{d}} for some d ∈ { 1 , 2 , 3 } {d\\in\\{1,2,3\\}} . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where d = 1 {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

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Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem