Numerical Methods For Several Kinds Of Fractional Differential Equations Based On Their Integral Transformations

In recent decades,fractional calculus has been widely applied in many fields,such as mathematics,physics and engineering.As is known,fractional differential equations can describe viscoelastic materials that exhibit a behavior between the pure viscous liquid and the pure elastic solid.However,similar to the nonlinear integer-order ordinary differential equations,it is often difficult(or even impossible)to analytically obtain the exact solutions of nonlinear fractional differential equations.Thus,numerical methods are needed to solve nonlinear fractional differential equations.The main difficulty to construct numerical methods for fractional differential equations is the discretization of the fractional derivatives or fractional integrals.It is challenging to construct numerical solutions and carry out theoretical analysis due to the nonlocal property of fractional operators.Therefore,constructing the numerical solutions to the fractional differential equations has become a hot research area.In Chapter 2,we first construct an efficient scheme for nonlinear Caputo fractional differential equations with the initial value and the fractional degree 0<α<1.Then,the unconditional stability and the superlinear convergence with the order 1+αof the proposed scheme are strictly proved and discussed.Due to the nonlocal property of fractional operators,the proposed scheme is time-consuming for long-time simulations.Thus,a fast implement of the proposed scheme is presented based on the sum-of-exponentials(SOE)approximation for the kernel tα-1 on the interval[h,T]in the Riemann-Liouville integral,where h is the stepsize.Some numerical experiments are provided to support the theoretical results of the proposed scheme and demonstrate the computational performance of its fast implement.In Chapter 3,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations.Firstly,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Secondly,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN-1-αN-2 of the proposed scheme are strictly established,where aN-1 and aN-2 are the maximum and the second maximum fractional index in the considered multi-term fractional nonlinear ordinary differential equations,respectively.Finally,two numerical examples are provided to support the theoretical resultsIn Chapter 4,we extend the numerical scheme of Chapter 3 to the multi-term fractional partial differential equations.Then we establish a classical L1 scheme and a scheme based on their equivalent fractional integral equations.Finally,the two schemes are compared by numerical experiments.Obviously,the latter scheme performs better when 2αN-1-αN-2>1.