Abstract
The stable difference schemes for the fractional parabolic equation with Dirichlet and Neumann boundary conditions are presented. Stability estimates and almost coercive stability estimates with ln (1/(τ + |h|)) for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one‐dimensional fractional parabolic partial differential equations.
Highlights
Theory and applications, methods of solutions of problems for fractional differential equations have been studied extensively by many researchers 1–18
We introduce the Hilbert space L2h L2 Ωh of the grid function φh x {φ h1p1, . . . , hmpm } defined on Ω, equipped with the norm
The method of proofs of Theorems 2.1–2.3 enables us to obtain the estimate of convergence of difference schemes of the first and second orders of accuracy for approximate solutions of the initial-boundary-value problem
Summary
Theory and applications, methods of solutions of problems for fractional differential equations have been studied extensively by many researchers 1–18. The first and second orders of accuracy stable difference schemes for the numerical solution of problem 1.1 are presented. The solutions of difference scheme 2.7 and 2.11 satisfy the following stability estimate: max 1≤k≤N uhk. The solutions of difference scheme 2.7 satisfy the following almost coercive stability estimate: max uhk − uhk−1. The method of proofs of Theorems 2.1–2.3 enables us to obtain the estimate of convergence of difference schemes of the first and second orders of accuracy for approximate solutions of the initial-boundary-value problem. Our interest in the present paper is studying the difference schemes 2.7 and 2.11 by numerical experiments Applying these difference schemes, the numerical methods are proposed for solving the one-dimensional fractional parabolic partial differential equation.
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