The a posteriori error estimate in fractional differential equations using generalized Jacobi functions

Abstract

In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators.

Keywords

• fractional initial value problems
• fractional boundary value problems
• generalized jacobi functions
• petrov-galerkin spectral methods
• a posteriori error estimators