Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE

Abstract

We apply homotopy perturbation transformation method (combination of homotopy perturbation method and Laplace transformation) and homotopy perturbation Elzaki transformation method on nonlinear fractional partial differential equation (fpde) to obtain a series solution of the equation. In this case, the fractional derivative is described in Caputo sense. To avow the adequacy and authenticity of the technique, we have applied both the techniques to Fractional Fisher’s equation, time-fractional Fornberg-Whitham equation and time fractional Inviscid Burgers’ equation. Finally, we compare the results obtained from homotopy perturbation transformation technique with homotopy perturbation Elzaki transformation.

A Jacobi wavelet collocation method for fractional fisher's equation in time

“…2 and 3. When the tables and figure are analyzed, it is obvious that we obtain pretty better results than the results in both [28,29].…”

“…Substituting the eqs. (28), (29), and (32) to he eq. (1), and substituting the collocation points x i and t i to the new equation, we get an algebraic equation system.…”

Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

“…Very recently, Cherif and Djelloul coupled Elzaki integral transform and variational iteration method on PDEs of fractional order; Dhunde and Waghmare coupled the double Laplace transform with the new iterative method on NPDEs; Jena and Chakraverty coupled the Elzaki transform with the homotopy perturbation on time‐fractional Navier‐Stokes equation, Again in Reference , they implemented the residual power series approach on the fractional Black‐Scholes option pricing equations, Alderremy et al proposed a novel decomposition method involving the Elzaki transform and Adomian decomposition method on nonlinear fractional differential equations, Singh and Sharma implemented EHTPM and HPTM method on nonlinear fractional PDEs as a comparative study of these two methods. Series solutions were obtained using all of these methods of which converges rapidly to the exact solutions of the problems/models being solved.…”

“…[36][37][38][39] In the quest of seeking exact solutions to model/equations, mathematicians have also come up with hybrid methods (coupling two distinct methods) to obtain exact solutions of some models, namely, Sumudu decomposition method, [40][41][42] homotopy perturbation transformation (HPTM) method, 43 homotopy-variational iteration method, 44 Elzaki differential transform, 45 Elzaki projected differential transform, 46,47 Elzaki homotopy transformation perturbation method (EHTPM), 48,49 Laplace-Adomian decomposition method, 50 Elzaki iterative method, 51 Laplace-variational iteration method, [52][53][54] new iterative transform method, 32,55 and homotopy analysis transform methods. 32 Very recently, Cherif and Djelloul 56 coupled Elzaki integral transform and variational iteration method on PDEs of fractional order; Dhunde and Waghmare 57 coupled the double Laplace transform with the new iterative method on NPDEs; Jena and Chakraverty 58 coupled the Elzaki transform with the homotopy perturbation on time-fractional Navier-Stokes equation, Again in Reference 7, they implemented the residual power series approach on the fractional Black-Scholes option pricing equations, Alderremy et al 2 proposed a novel decomposition method involving the Elzaki transform and Adomian decomposition method on nonlinear fractional differential equations, Singh and Sharma 59 implemented EHTPM and HPTM method on nonlinear fractional PDEs as a comparative study of these two methods. Series solutions were obtained using all of these methods of which converges rapidly to the exact solutions of the problems/models being solved.…”

Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

“…In the various eld of engineering and science, it is very important to nd the approximate or the exact solution of some nonlinear partial di erential equations [1]. There are several potent methods such as Homotopy perturbation [2,3]; homotopy perturbation transformation method (HPTM) [4] and homotopy perturbation sumudu transformation method (HP-STM) have been proposed to obtain the approximate or the exact solutions of nonlinear equations [5][6][7][8][9].…”