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Application of generalized differential transform method to multi-order fractional differential equations. (English) Zbl 1221.34022

Summary: In a recent paper [Appl. Math. Comput. 197, No. 2, 467–477 (2008; Zbl 1141.65092)] the authors presented a new generalization of the differential transform method that extend the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form \(y^{(\mu )}(t)=f(t,y(t),y^{(\beta_{1})}(t),y^{(\beta_{2})}(t),\ldots ,y^{(\beta_{n})}(t))\) with \(\mu >\beta _{n}>\beta _{n-1}>\ldots >\beta _{1}>0\), combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Citations:

Zbl 1141.65092
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