Legendre Wavelets Method For Solving Three Kinds Of System Of Fractional Calculus Equations

Mathematical modeling of many practical phenomena in different disciplines such asengineering and physics leads to a linear or nonlinear fractional system. And the behaviorof the process in these systems need to described by fractional differential or fractionalintegral, so how to solve the equations is a vital way to deal with these systems. As thecontinue development of Fourier analysis, wavelet analysis has made great contributionsin many fields of science in recent decades. It has the advantages of smoothness andlocally compact support, and has more detailed analysis ability of time-frequency thanFourier analysis, and can better deal with the local singularity problem. So, in this paper,three kinds of systems of fractional calculus equations are solved by the method ofLegendre wavelets. We apply the characteristics of Legendre wavelets, and combine withthe idea of operator matrix, to transform the original equations into linear or nonlinearalgebraic equations which can be solved easily.Firstly, we give a brief introduce of wavelet analysis and the development offractional calculus equations, also the research progress of this kind of problem at present.Then present the relevant definition and the properties of fractional calculus and Legendrewavelets.Secondly, in the third and fourth chapter, we using Legendre wavelets method and itsfractional integral operator matrix to simplify the original problem into a system of linearor nonlinear algebraic equations, so we can solve it by the least square method or Newtoniterative method easily. And the fractional integral operator matrix of Legendre waveletscan be got by combining with the properties of Block Pulse functions (BPFs). Theconvergence analysis of algorithm is given in chapter three. In chapter four we give theerror analysis of technique in this paper by discussing two cases of exact solution isknown and unknown. The numerical examples show that the method is effective andaccurate.Finally, based on the definitions and properties of Legendre wavelets, we adopt thedefinition of Caputo fractional differential operator to gain a new fractional differential operator matrix of Legendre wavelets. Then apply the new operator matrix to solve asystem of nonlinear fractional singular Volterra integro-differential equations. We alsogive the error analysis of the system and three examples.