Solving matrix differential and integro-differential equations using differential transformation method and convolutions

The differential transform method (DTM) was introduced to solve linear and nonlinear initial value problems which appear in electrical circuit analysis. In this method we construct approximate solutions which is close to the exact solutions that differentiable and having high accuracy with minor error. However, DTM is differ with the traditional high order Taylor series where we need long computation time and derivatives. Thus the DTM is applied to the high order differential equations as alternative way to get Taylor series solution. In final stage this method yields truncated series solution in the practical applications and most of time coincides with the Taylor expansion. In this work we study the differential equations systems by using the differential transformation method (DTM). Further, we apply the convolutions to matrices and study their fundamental properties using differential transformation method. We also provide many different applications of matrix convolutional equations such as coupled matrix convolution equations by DTM. In the applications, we proved that the solutions converge to the exact solutions. Finally we propose to generate matrix integro-differential equations by using convolutions and differential equations