The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability

Exact Solution of Two-Dimensional Fractional Partial Differential Equations

“…The mathematical physics governing by nonlinear partial deferential dynamical equations have applications in physical science. The analytical solutions for these dynamical equations play an important role in many phenomena in optics; fluid mechanics; plasma physics and hydrodynamics [7][8][9][10]. In recent years, many authors have investigated partial differential equations of fractional order by various techniques such as homotopy analysis technique [11,12], variational iteration method [13][14][15], homotopy perturbation method [16], homotopy perturbation transform method [17], Laplace variational iteration method [18][19][20], reduce differential transform method [21], Laplace decomposition method [22] and other methods [23][24][25][26][27].…”

N‐lump and interaction solutions of localized waves to the (2 + 1)‐dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid

“…[6][7][8][9][10][11][12][13][14][15] A class of semi-implicit finite difference weighted essentially nonoscillatory schemes for solving the nonlinear heat equation along with utilizing the implicit Runge-Kutta methods and the local Taylor expansion has been investigated by Hajipour et al 16 Also, Hajipour et al 17 developed an accurate discretization technique to solve a class of variable-order fractional reaction-diffusion problems, and solvability, stability, and convergence of their proposed method were derived. Khater et al have worked on diverse topics of nonlinear PDEs with determining the exact solutions including the bright and dark soliton, solitary wave, periodic solitary wave elliptic function, solitary and rogue waves, periodic, breather, the multiple-soliton solution, Lax pair, infinitely many conservation laws, and soliton solutions containing the equations along with well-known methods such as two-dimensional Ginzburg-Landau equation via the function transformation method, 18 three-dimensional weakly nonlinear shallow water waves regime utilizing the extended modified mapping method, 19 the longitudinal wave equation with the extended trial equation method, 20 the Davey-Stewartson system by using the simplest equation method, 21 the Kadomtsev-Petviashvili (KP) and modified KP dynamical equations via the generalized extended tanh method and the F-expansion method, 22 the nonlinear two-dimensional water waves of Olver dynamical equation by applying the extended mapping method, 23 the complex Ginzburg-Landau equation by modified simple equation method, 24 the generalized nonlinear fifth-order KdV water wave equations using the extended direct algebraic method, 25 three-dimensional Zakharov-Kuznetsov-Burgers equation with modified extended mapping method, 26 the nonlinear higher-order extended KdV equation by utilizing of integrals of the modal function, 27 the nonlinear Ginzburg-Landau equation with the F-expansion method, 28 and nonlinear Kadomtsev-Petviashvili dynamic equation via the sech-tanh, sinh-cosh, extended direct algebraic, and fraction direct algebraic methods. 29 The nonlinear (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov (ANNV) equation has been investigated for finding the new lump-type solutions by the Hirota bilinear form, ...…”

Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

“…For deliberative speedy development of symbolic computation systems [1][2][3][4][5], the search for the exact solutions of nonlinear equations has attracted a lot of attention [6][7][8][9] as the exact solutions make it possible to research nonlinear physical phenomena comprehensively and facilitate testing the numerical schemes [10][11][12][13][14]. In the last two decades, various approaches have been proposed and applied to the nonlinear equations of PDEs, such as homogeneous balance method [15,16], extended tanh-function method [17][18][19][20], Jacobi elliptic function expansion method [21], simple equation method [22][23][24], (G/G′)-expansion method [25][26][27], Hirotas bilinear method [28], Exp function method [29], general projective Riccati equation method [30], modified simple equation method [31][32][33], improved direct algebraic technique, [34,35], auxiliary scheme [36] and so on [37][38][39][40][41][42][43]…”