Study On The Numerical Algorithm For Fractional-order Differential Equations

Fractional-order differential equations is an class of equation that contains fractional derivatives. In recent years, with the increasing number of fractional differential equations be used to describe of thermodynamics, viaco-elasticity, acoustics, electro-chemical, rheological, fractal and other areas of application.The devolopment history of from fractional-order process,it not only shows deep research background and good application biasis,but also have a very broad space for development. Therefore, the fractional-order differential theory, the properties and the solving method research appears more important, it for its further study to lay a solid foundation.The main contents of this paper is as follows:①research on the theory of fractional-order. The correclated throey of fractional-order calculus is introduced, fractional calculus of various definitions, the relationship between define and basic operation nature. Moreover, the definition of the fractional-order calculus is compared with that of interger-order calculus detailedly.②"Higher-order"fractional-order differential equation convert multivariable fractional-order differential equations. The general form of fractional order differential equations of existence and uniquence problem into multivariable fractional order differential equations of existence and uniquencess problems,and illustrate higher-order fractional-order differential euqation of the existence and uniquence of solutions.③Fractional order differential control system research and simulation. In excluding delays binary fractional control system equations, order number of cases for different scores, using G-L definition, combined with numerical iterative thought, find out numerical solution of this systems, then prove the convergence of this system. In the dual with delays fractional control systems equations, order number of cases for different scores, equations contains delayτ, using the step length h to segmentationτ, using G-L definition, combined with numerical iterative thought, find out numerical solution of this system, then prove the convergence of this system. In multivariate without delay the fractional-order control system, the multiple fractional-order equations into matrix equations, constructed iteration format, find out the numerical solution of fractional systems, then prove the convergence of this system. With delays in the plural of fractional-order control system, on the system of variableτsegmentation, make the systenm into matrix equations,then structure formed iterative format, find out the fractional-order system solutions, and proved the convergence of solution. The solution to all system simulation with MATLAB software.