A Numerical Approach for Solving of Fractional Emden-Fowler Type Equations
Josef Rebenda, Zdeněk Šmarda
TL;DR
Problem: solving singular fractional Emden-Fowler type equations with Caputo derivatives presents numerical difficulties near $t=0$. Approach: introduce the Fractional Differential Transformation (FDT) to yield convergent $\alpha$-power series solutions, selecting $\alpha=\frac{1}{q}$ for $\beta=\frac{p}{q}$ and initializing coefficients from the initial data. Contribution: derive the forward and inverse FDT, establish transformation rules, and formulate a recurrence involving $\Gamma$-functions; in the model problem $_0^{C}D_t^{2\beta} u + \frac{2}{t^{\beta}} $_0^{C}D_t^{\beta} u + u=0$ with $u(0)=1$, $u'(0)=0$, the scheme recovers the exact solution $u(t)=\frac{\sin t}{t}$ in the integer-order limit. Significance: the FDT framework provides a simple, fast-converging numerical tool for fractional ODEs with singular behavior and can be extended to broader fractional-differential equations.
Abstract
In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated in the form of convergent series with fast computable components. The numerical results show that the approach is correct, accurate and easy to implement when applied to fractional differential equations.