Fast High Order Difference Schemes For The Time Fractional Telegraph Equation

Fractional derivative has been very important in different fields such as Departments of mechanics(viscoelastic theory and viscoplasticity),biochemistry(models of polymers and proteins),electrical engineering(ultrasonic transmission),pharmaceuticals(human body models),mechanical loads and so on.Many researchers utilize fractional differential equations(FDEs)to study some natural models,but the exact solutions of FDEs are too difficult to be detected in reality.Even if people were able to attain the exact solutions of FDEs,it is difficult to apply them to the reality.Therefore,it is necessary to conduct research on numerical methods of FDEs and other related efficient algorithms.In this paper,we consider finite difference method to construct high-order finite difference schemes solving the fractional Telegraph equation.Because of the two Caputo fractional derivatives with two different orders,we need to utilize mathematic methods to approximate the fractional derivatives.In addition,we also consider the high-order and efficient schemes to improve the efficiency of the computation.At the same time,the complexity of the schemes may cause difficulties in terms of analysis.According to the above problems,we propose fast and high-order numerical methods and a compact difference scheme to solve the fractional Telegraph equation.The content is arranged as follows:In the first charter,we discuss the history of differential and fractional differential equation,the preliminary theory of fractional differential equation,the development of differential equations and the importance of studying fractional difference equations.Then we talk about the researches and their achievements over the past decade,in terms of their approximations,efficient algorithms,numerical solution of serval FDEs,so on and so forth.In the end,we introduce the fractional Telegraph equation that we study in this paper and discuss the difficulties that the equation has.In the second charter,we first introduce several important fractional derivatives,and then we basically construct the high-order difference scheme,using the FL2-1_? formula with the Fast evaluation to approach two different types of Caputo fractional derivatives and we use the second order method to approximate the equation in the spatial direction,with the orderO(?~2+h~2)in general,where ? and h are the temporal step and the spatial step respectively.Then we give the unconditional stability and convergence analysis for the proposed scheme with some creative methods,utilizing the energy method.In the end,we use the MATLAB to conduct numerical experiments,presenting some data to verify the validity of our analysis and the efficiency of our scheme.In the third charter,we introduce a fast and high-order compact scheme with the order O(?~2+h~4).Then based on the similar method,we make use of some important lemmas to rigorously prove that the proposed compact scheme is unconditional stable and convergent.Moreover,we present our numerical example to verify the theorical analysis.