Abstract
A new technique of the Adomian decomposition method is developed and applied in this research article to solve two-term diffusion wave and fractional telegraph equations with initial-boundary conditions. The proposed technique is used to solve problems of both fractional and integer orders of the telegraph equations. The fractional-order solutions provide useful information about the data transmission from one point to another. The solutions are obtained in the form of an infinite series, demonstrating a high rate of accuracy from fractional to integer orders of the problems. The technique's accuracy is verified by drawing various fractional and integer order plots and tables. The fractional-order plots demonstrate that the solutions have a higher rate of accuracy, and the different dynamical behaviors of the problems are revealed as a result. It is discovered that the new Adomian decomposition method is the best option for solving initial-boundary value problems. The new approximations of each solution improve the method's accuracy. As a result, it is suggested that the method can be applied to other problems with initial-boundary conditions.