Numerical Solution For Five Types Of Variable Order Fractional Differential Equations With Bernstein Polynomials

In recent years, it opens a new chapter for variable order fractional derivative inthe successful applied to model the direction of viscoelastic materials and viscousfluids. So solving the variable order fractional equations will becomes a new hotspot.It is the first attempt to seek the solutions of variable order fractional differentialequations numerically. The specific method in this paper is as follows: combining thedefinitions and properties of variable order fractional derivative and Bernsteinpolynomials, we derive some types of operational matrixes. With the operationalmatrixes, the original equations are transformed into the products of several dependentmatrixes which can also be regarded as the system of equations after dispersing thevariable. By solving the equations, the numerical solutions are acquired. It is not onlythat the computer procedure is relatively simple, but also only a small number ofBernstein polynomials are needed to obtain a satisfactory result. The paper isorganized as follows:Firstly, the paper describes the historical background and the research status offractional calculus and variable order fractional calculus. Then the definitions andproperties of fractional calculus, variable order fractional calculus Bernsteinpolynomials are given.Secondly, we seek the numerical solutions of the one-dimensional linear,nonlinear variable order fractional differential equations, the derivative operationalmatrix and integral operational matrix of first-order of Bernstein polynomials arederived. Combining with multiple operator matrixes, the calculating format of theoriginal equation is obtained.Thirdly, we solve the variable order time fractional diffusion equation in terms ofBernstein polynomials and make comparison with the numerical approximationscheme by difference methods, numerical examples explain the proposed method isefficient.Finally, we solve the variable order fractional differential equations defined on the extended interval, namely the interval limits on0,1or0,10,1will be extendedto0, R or0, R10,R2. Then it needs changing the coefficient matrix of Bernsteinpolynomials.