Analytical and numerical solution of multi-term nonlinear differential equations of arbitrary orders
The authors study a nonlinear multi-term fractional differential equation (FDE) of the form \[ D^\alpha x = f(t, D^{\alpha_{0}} x, D^{\alpha_{1}} x, \cdots D^{\alpha_{n}}x), \quad x(0) = 0, \] where \(t \in J = [0,T]\), \(T \in \mathbb{R}^+\), \(\alpha_0 = 0,\) and the fractional derivatives are considered in the Riemann-Liouville sense. Here, \(\alpha_i \in [0,1)\), \(\alpha > \{\alpha_0, \alpha_1, \dots, \alpha_n\}\). It is proved that, if the function \(f\) satisfies a Lipschitz condition uniformly in \(t\), then the above IVP has a unique continuous solution on \(J\). Using the Adomian polynomials formula, the convergence of the Adomian decomposition method is established. Based on this convergence analysis, the maximum absolute truncated error of the Adomian series is estimated. Numerical solutions are obtained via a numerical method introduced by [\textit{I. Podlubny}, Fractional differential equations. Mathematics in Science and Engineering. 198. San Diego, CA: Academic Press (1999; Zbl 0924.34008)] and the results are compared with those of the ADM solutions.
nonlinear fractional differential equation
scientific article; zbMATH DE number 5728304
scientific article