A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations

Abstract

In the present note, a new modification of the Adomian decomposition method is developed for the solution of fractional-order diffusion-wave equations with initial and boundary value Problems. The derivatives are described in the Caputo sense. The generalized formulation of the present technique is discussed to provide an easy way of understanding. In this context, some numerical examples of fractional-order diffusion-wave equations are solved by the suggested technique. It is investigated that the solution of fractional-order diffusion-wave equations can easily be handled by using the present technique. Moreover, a graphical representation was made for the solution of three illustrative examples. The solution-graphs are presented for integer and fractional order problems. It was found that the derived and exact results are in good agreement of integer-order problems. The convergence of fractional-order solution is the focus point of the present research work. The discussed technique is considered to be the best tool for the solution of fractional-order initial-boundary value problems in science and engineering.

Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy

“…The method's use for fractional differentia equations has recently been broadened [20][21][22][23]. Researchers are expected to work on the resolution of fractional-order diffusion-wave equations (FDWEs) using the ADM approach, which is a novel technology [24]. FDWEs are the most important type of anomalous diffusion equation derived from classical diffusion-wave equations [25].…”

“…Finally, Section 5 describes the conclusions we have drawn from the study. Definition 1: provides the Reimann-Liouville (RL) integral operator of arbitrary order 𝜏(𝜏 ≥ 0) for a function 𝜒(𝛽) [28,29]…”

Immediate solution for fractional nonlinear oscillators using the equivalent linearized method

“…[25][26][27][28] Therefore, the researchers have given full attention to developing strong and novel techniques to handle this complicated class of fractional mathematical tools. 29 In this regard, a new technique is urgent to simplify and reduce the hard work required for obtaining an asymptotic solution that is closer to the exact numerical solution.…”

Solving Nonlinear Fractional Partial Differential Equations Using the Elzaki Transform Method and the Homotopy Perturbation Method

“…The homotopy perturbation method (HPM) was introduced by Ji-Huan He in 1999 [9] for solving differential and integral equations. The HPM is applied to algebraic equations [10], nonlinear reactiondiffusion-convection problem [11], singular boundary and initial value problems [12,13], nonlinear wave equations [14], and other modifications which can be seen in [15][16][17][18][19][20][21][22][23][24][25].…”