Research On High-Precision Numerical Algorithms For Caputo Fractional-Order Differential Equations

Recent years,fractional calculus theory has developed rapidly,more and more fractional-order models appear in various engineering fields.The initial value of the Caputo fractional-order derivative has clear physical meaning,therefore the fractional-order model in practical engineering is described as the Caputo fractional-order differential equation.The analytical solutions of some Caputo fractional-order differential equations could be solved,however,they are too complicated to be used in practical engineering.Therefore,the algo-rithm for solving the numerical solution is an urgent problem,this paper mainly studies the high-precision numerical algorithm for solving Caputo fractional-order differential equations.First,the high precision numerical algorithm is designed based on Lubich's fractional linear multistep method,for calculating various kinds of fractional-order differential and in-tegral.A simple method is proposed to construct the generating function,and the recursive formula is induced with the generating function for computing the coefficients in fractional linear multi step method.The nonzero initial condition of the original function could affect the computation accuracy,therefore,the original function is decomposed into two parts to eliminate the influence.The one has zero initial condition,the other one has nonzero initial condition.To the function with nonzero initial condition,its fractional-order differential and integral could be computed by the analytical formula.To the function with zero initial condi-tion,its fractional-order differential and integral could be computed by the recursive formula.The fractional-order differential and integral of the original function could be calculated by combining the two parts.In addition,the matrix algorithm for computing fractional-order differential and integral is improved,the improved matrix algorithm is the basis for solving the implicit nonlinear Caputo fractional-order differential equations.Second,the numerical algorithm is designed for solving linear Caputo fractional-order differential equations.In the algorithm,the auxiliary function is constructed with the initial condition of the original equation,the original equation is transformed into a new equation with zero initial condition.And then,the new equation could be solved with the closed-form solution formula or the matrix algorithm.The linear Caputo fractional-order differential equation could be solved by the algorithm,however,the computation accuracy is constrained by the number of the initial conditions.To improve the computation accuracy,the Taylor series algorithm is designed to evaluate the needed initial conditions,the auxiliary function is constructed with the new initial conditions,and then,the more accurate numerical solution could be solved by the closed-form solution formula or the matrix algorithm.The compu-tation accuracy of the improved algorithm is no longer constrained by the number of initial conditions,the computation results in the examples show that the computation accuracy is improved significantly by the proposed algorithm.Third,the numerical algorithm for solving nonlinear Caputo fractional-order differential equations is proposed.The predict-correct algorithm is designed for solving the explicit non-linear Caputo fractional-order differential equations.In the algorithm,the nonlinear equation is transformed into a linear equation approximately,the predictor could be obtained by solv-ing the linear equation.Substituting the predictor into the original equation,the corrector could be obtained by solving the equation again.A practical example of the implicit nonlin-ear Caputo fractional-order differential equation is presented for the first time.The equation has not any explicit form,therefore it can not be solved by predict-correct algorithm.A high-precision matrix algorithm is designed to solve this kind of the equations.At the end of this section,the mistake is found in the other numerical algorithm for the Caputo fractional-order differential equation set,the correcting method is also proposed.The computation results in the examples show that the numerical algorithm proposed in this paper is correct.At last,the block diagram scheme is proposed for modeling and solving the Caputo fractional-order differential equation.Because of the influence of nonzero initial conditions,the block diagram of the Caputo fractional-order derivative can not be modeled as Caputo definition.To solve this problem,time domain compensation scheme and integrator chain scheme are designed,the two schemes are tested by various kinds of Caputo fractional-order differential equations.The testing results show that the integrator chain scheme is simpler,and its computation result is more accurate.Therefore integrator chain scheme is a general method,various kinds of the equations could be solved by this method,including implicit nonlinear Caputo fractional-order differential equation.The block diagram scheme is im-portant supplement to the numerical algorithm,and it is of great significance in practical engineering.Caputo fractional-order differential equations could be solved by the numerical algo-rithms proposed in this paper,and the high-precision numerical solutions could be obtained.In addition,some Caputo fractional-order differential equations with the known analytical so-lutions are presented,they could be used as the benchmark problems for testing the numerical algorithms.