Research Of Exponential Time Adams Methods For Semi-linear Fractional Differential Equations

Today, with the progress of science and technology, fractional differentialequations have been widely used in various fields, such as physical, chemistry,biomedicine, mathematical finance, image processing, materials analysis, signalprocessing and so on, and more and more people pay their attention to the developmentof them. Unfortunately, majority of fractional differential equations are difficult to givetheir analytic solution, due to the computational complexity of fractional calculus.Therefore, we have to study the numerical solution of fractional differential equations.Currently, the theoretical results of fractional differential equations are far less thaninteger-order differential equations.As an important branch of fractional differential equations, the theory ofsemi-linear fractional differential equations start soon, and there isn’t a monographsystematically introduced them. Since they contain linear and nonlinear terms, theresearch for them is difficult. Compared with this, there exist many numerical methodsfor integer-order differential equations, and exponential integrators are popular.However, some numerical methods applicable to integer-order differential equationsmay not be effectively applied to the case of fractional order. Therefore, how to makeadaptation of these methods is worth further studying.The main aim of this paper is to discuss the exponential time differencing methodsfor a class of semi-linear fractional differential equations. As a branch of exponentialintegrators, such methods for first-order ordinary differential equations on time variablehave a rich theoretical results. By the variation of constants formula, we devise twomethods of this kind which are named as generalized exponential timeAdams-Bashforth method and generalized exponential time Adams-Bashforth-Moultonmethod, and generalized Mittag-Leffler function is included. Under the various conditions,the convergence properties are investigated, and the convergence orders are given.Some numerical experiments are presented to validate the theoretical findings.