Qualitative Theory Of Short Memory Fractional Order Equations And Applications

The fractional calculus operators hold non–locality or memory effects and they frequently appear in nonlinear phenomena with memory.Recently,much effort has been dedicated to definitions of fractional calculus,theory of fractional differential equations and numerical calculation.However,the real-world applications of fractional calculus still have some challenges.There are two main concerns:(1)There are many definitions of fractional calculus with different physical meanings.(2)Computational cost is high in numerical calculus of fractional differential equations.Particularly for long–term numerical calculation,the efficiency is poor.It becomes challenging to balance high costs and good results during the real–world applications.So this thesis chooses short memory differential equations as the research topic and makes efforts to the following aspects:(1)This thesis first revisits the method to define the classical R–L integral.The method using 9)–fold integral and delta differential equations is suggested to define a general fractional integral.Through the boundedness theorem of the general fractional integral,the mathematical constraint conditions of general kernel functions are presented.The general definition not only can be reduced to several existing fractional integrals but also can find some new ones.Furthermore,The function space and often used propositions like the fractional mean theorem and Taylor series are provided.Besides,the physical meaning of the general kernel functions is given by the classical continuous time random walk theory.(2)The numerical method is considered for the general fractional differential equations.A non-equivalent partition is introduced in the numerical scheme.The general fractional differential equations are equivalently given in form of weakly singulary integral equations.The rectangle and Adams formulae are proposed to develop a general predictor–correct scheme.The convergence order is proved in a general function space and some numerical examples are given for 2 < ? 3.This shows that the general fractional derivative is well defined.(3)A concept of short memory fractional differential equations is proposed.It gives up those initial time information and increases the effects of the neighborhood of the present state.More importantly,it also reduces computational costs.Sections 5 and 6 discuss existence conditions and numerical methods.Fully using of new features of short memory fractional differential equations,variable–order fractional derivative is constructed where the fractional order is a piecewise constant order on each sub–interval.Then the existence conditions of one class of variable–order fractional differential equations are discussed.Some successful applications to the fractional relaxin equation and neural networks are considered in Section 6.The efficiency of the new concept is shown.(4)The above results are extended to fractional difference equations.The short memory method is adopted to improve the recurrent neural network model.The stability conditions are investigated.