Study of nonlinear time-fractional hyperbolic-like equations with variable coefficients via semi-analytical technique: Differential J-transform method

This work proposes a semi-analytical new hybrid approach, so-called differential J-transform method (DTM), to evaluate the behavior of n-space dimensional fractional-nonlinear hyperbolic-like wave equations, where time-fractional derivative is considered in Caputo format. The DTM is the hybrid method in which projected differential transform is implemented after imposing the recently introduced integral transform, i.e., so-called transform [W. Zhao and S. Maitama, J. Appl. Anal. Comput. 10, 1223 (2020)]. The efficiency and applicability of the proposed DTM had been tested by considering three different test examples of the Caputo time-fractional nonlinear hyperbolic-like wave equations in terms of absolute error norms, and the different order DTM solutions are compared with exact solution behaviors and the existing results, for the large time level tau is an element of[0,10]. In addition, the convergence analysis of DJTM is studied theoretically and verified it numerically as well as graphically, which confirms that the numerical experiments via DJTM for distinct fractional orders support the theoretical findings excellently, and the presented DJTM results converge to their exact solution behavior, very fast. The evaluated series approximations are expressed in the compact form of Mittag-Leffler functions.