Numerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivative

Abstract

In this manuscript, fractional nonlinear coupled Whitham‐Broer‐Kaup equation associated with Atangana‐Baleanu fractional derivative is considered. General conditions under which a system solution exists and is unique are presented using the fixed‐point theorem method. For numerical simulations, coupled fractional modified Boussinesq equations and coupled fractional approximate long wave equations are investigated using the homotopy perturbation transform technique (HPTT). The suggested technique is an elegant compilation of the Laplace transform technique with the homotopy perturbation approach. The physical behavior of the obtained solutions has been presented graphically as well as in tables for diverse fractional order. Comparative simulations have been performed to validate the efficiency and accuracy of the suggested technique.

Stability and numerical analysis of the generalised time-fractional Cattaneo model for heat conduction in porous media

“…In addition, it is pretty helpful in simulating and comprehending natural phenomena due to its inheritance and recall qualities. Many researchers studied and analysed different fractional mathematical models such as the fractional competition model of bank data analysed with CF derivative [ 1 ], the fractional Fitzhugh–Nagumo equation is studied in [ 2 ] by homotopy perturbation technique, the fractional model of CFWBK equation is studied in [ 3 ] by ABC fractional derivative, the fractional multi-dimensional telegraph equation is examined in [ 4 ] by Laplace transform, the fractional coupled burger equation is investigated in [ 5 ], the nonlinear fractional model of Zakharov–Kuznetsov equation is studied in [ 6 ] via Sumudu transform, the fractional model of hearing loss due to Mumps virus is studied in [ 7 ] via Caputo–Fabrizio operator, the fractional-order SEIR epidemic of measles is studied in [ 8 ] by using Genocchi polynomials, the 2019-nCOV outbreaks are studied in [ 9 ] via non-singular derivative, the fractional predator–prey dynamical system is studied in [ 10 ], the heat equations arises in diffusion process are studied in [ 11 ] using new Yang-Abdel-Aty-Cattani fractional operator.…”

“…Therefore, there is a need for the development of a numerical technique that can get approximate analytic solutions to these models. Many techniques have been applied to study these fractional models such as the Adomian decomposition technique [ 38 ], finite elements technique [ 3 ], Adam–Bashforth method [ 39 ], homotopy analysis transform technique [ 40 ], Shehu transform technique [ 41 ], fractional homotopy perturbation transform method [ 42 ], collocation method [ 43 ], fractional reduced transform method [ 44 ], fractional variation iteration method [ 45 ], Sumudu transform technique [ 46 ], q -HAM [ 47 ], Sumudu perturbation transform technique [ 48 ] and hybrid numerical technique [ 49 ].…”

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

“…First,ourself receive the solutions for existence and uniqueness. And other research solved timefractional generalised analytical-approximate solutions PC equations for waves publicity of an elastic rod using the q-homotopy analysis of the transform method [13][14][15], Modulation instability analysis [16], Hydro-magnetic [17], solitary wave [18], Carreau fluid [19], kink wave [20], Existence and Uniqueness [21,24], Hilbert space [22], natural reduced differential transform method [23], iterative Laplace transform method [25], tanh-coth and the sine-cosine methods [26], explicit fourthorder time stepping methods [27], decomposition method [28], weierstrass elliptic function method [30], modified F-expansion methods [31], global existence [32], generalized exponential rational function (GERF) technique [33][34], bernoulli sub-equation function method [35], lie group method [36], fractional natural decomposition method [37], modified exponential method [39], residual power series method [40], adams-bashforth scheme [41], laplace transform [42][43], conformable derivative [44,47], Mittag-Leffler function [45], caputo derivatives [46], Sumudu transform [48],and so on [49][50][51][52][53]…”

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised Pochhammer-Chree equation in mathematical physics

“…First,ourself receive the solutions for existence and uniqueness. And other research solved timefractional generalised analytical-approximate solutions PC equations for waves publicity of an elastic rod using the q-homotopy analysis of the transform method [13][14][15], Modulation instability analysis [16], Hydro-magnetic [17], solitary wave [18], Carreau fluid [19], kink wave [20], Existence and Uniqueness [21,24], Hilbert space [22], natural reduced differential transform method [23], iterative Laplace transform method [25], tanh-coth and the sine-cosine methods [26], explicit fourthorder time stepping methods [27], decomposition method [28], weierstrass elliptic function method [30], modified F-expansion methods [31], global existence [32], generalized exponential rational function (GERF) technique [33][34], bernoulli sub-equation function method [35], lie group method [36], fractional natural decomposition method [37], modified exponential method [39], residual power series method [40], adams-bashforth scheme [41], laplace transform [42][43], conformable derivative [44,47], Mittag-Leffler function [45], caputo derivatives [46], Sumudu transform [48],and so on [49][50][51][52][53]…”