A Study Of Method For Approximate Solution And Numerical Solution Of Fractional Differential Equations

Fractional calculus is a promotion of the integer calculus. Because of the good memoriesand hereditary of fractional calculus, fractional calculus theory has been widely applied tovarious fields of nature and science, especially in the wide application of control theory,viscoelastic theory, electronic chemicals, fractal theory and so on.At the same time a large number of available research results greatly promote the researchprogress of fractional calculus, and some scholars get involved in this emerging area of research.In the use of fractional model, there have been a series of fractional differential equations, so thestudy of fractional differential equations has great significance.Based on the previous study，Imainly done the following aspects:（1） For several linear and nonlinear differential equations problems in Caputo sense, usingof Mittag-Leffler function, we give a new method of approximate solutions. While specificexamples are given to further verify the feasibility of the method.（2） Considering the following boundary value problemD_α+^αy=f（x,y），y（a）=y₀wher0﹤α﹤1and fractional derivative is in Riemann-Liouville sense, we give a numerical methodfor solving fractional differential equations which is based on the Euler’s method. The maincharacteristic behind the approach is that Euler method has intuitive geometric meaning. Thealgorithm is presented and the convergence of the algorithm is proved.（3） For further improvements on the basis of the Euler algorithm, we presents the improvedEuler algorithm, the accuracy of the algorithm improved1order. Using mathematical softwareMatlab for three instances, we give the figure of improved Euler method and Euler method andanalytical solutions to compare, then to illustrate the applicability of the algorithm as well as theadvantages and disadvantages of the algorithm.