Numerical Analysis of anH1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

Jinfeng Wang,Min Zhang,Hong Li,Yang Liu,Meng Zhao

doi:10.1155/2014/371413

Abstract

We discuss and analyze an H 1-Galerkin mixed finite element (H 1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H 1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H 1-GMFE method. Based on the discussion on the theoretical error analysis in L 2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H 1-norm. Moreover, we derive and analyze the stability of H 1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.

Highlights

In this paper, our purpose is to present and discuss a mixed finite element method for the time fractional telegraph equation∂02,αtu (x, t) + 2κ∂0α,tu (x, t) − Δu (x, t) + βu (x, t) (1)= f (x, t), (x, t) ∈ Ω × J, with boundary condition u (x, t) = 0, (x, t) ∈ ∂Ω × J, (2)and initial conditions u (x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω, (3)where Ω(⊂ Rd, d = 1, 2, 3) is a bounded domain with boundary ∂Ω and J =
We discuss and analyze an H1-Galerkin mixed finite element (H1-GMFE) method to look for the numerical solution of time fractional telegraph equation
We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and formulate an H1-GMFE scheme with two important variables

Summary

Introduction

∂02,αtu (x, t) + 2κ∂0α,tu (x, t) − Δu (x, t) + βu (x, t) (1). = f (x, t) , (x, t) ∈ Ω × J, with boundary condition u (x, t) = 0, (x, t) ∈ ∂Ω × J, (2). Initial conditions u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω, (3). The coefficients κ > 0 and β ≥ 0 are two constants, f(x, t) is a given source function, u0(x) and u1(x) are two given initial functions, and the time Caputo fractional-order derivatives ∂0α,tu(x, t) and ∂02,αtu(x, t) are defined, respectively, by ∂0α,tu (x, t) = t ∫ ∂u (x, ∂τ τ) (t dτ − τ)α (4)

Objectives

Methods

Results