Solution of Second-Order IVP and BVP of Matrix Differential Models Using Matrix DTM

Abstract

We introduce a matrix form of differential transformation method (DTM) and apply for nonlinear second-order initial value problems (IVPs) and boundary value problems (BVPs) of matrix models which are given by ${\mathbf{u}}^{″}\left(t\right)=f\left(t,\mathbf{u}\left(t\right),{\mathbf{u}}^{\prime }\left(t\right)\right)$ and subject to initial conditions $\mathbf{u}\left(a\right)={\mathbf{u}}_0,{\mathbf{u}}^{\prime }\left(a\right)={\mathbf{u}}_1$ and boundary conditions $\mathbf{u}\left(a\right)={\mathbf{u}}_0,\mathbf{u}\left(b\right)={\mathbf{u}}_1$, where ${\mathbf{u}}_0,\hspace{0.17em}{\mathbf{u}}_1\in R^{r\times q}$. Also the convergence of present method is established. Several illustrative examples are given to demonstrate the effectiveness of the present method.