An efficient finite difference/Hermite–Galerkin spectral method for time-fractional coupled sine–Gordon equations on multidimensional unbounded domains and its application in numerical simulations of vector solitons

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

“…This section describes the suggested method to obtain solutions of the equations considered in (3) with the conditions (6) and (8). Then, we put relations (26)- (30) into Equation 3, and the following results are obtained: where…”

“…Some applications of the Prabhakar function can be seen in mathematics and physics as a fractional Poisson process [16], Havriliak-Negami relaxation functions [18,19], irregular case of the dielectric relaxation responses [20], a model of anomalous relaxation in dielectrics of fractional order [21], fractional thermoelasticity [10], telegraph equations [22], thermodynamics [23], and fractal time random [24]. By placing α 1 ðp, qÞ = 2, α 2 ðp, qÞ = 2 in Equation 3, the coupled nonlinear sine-Gordon equations of fractional variable orders given in (3) change into the classical coupled nonlinear sine-Gordon equations which are defined by Equation (2), and the classical coupled nonlinear sine-Gordon equations have many applications in physics as nonlinear models [25,26], plasma [27], quantum [28], optics [29], and mathematics [13,30,31]. Getting analytic solutions to fractional differential equations in general are not easy; therefore, numerical methods are used to obtain the solutions of this type of equations.…”

Reproducing kernel Hilbert pointwise numerical solvability of fractional Sine-Gordon model in time-dependent variable with Dirichlet condition

“…The physical constructions of TFSGM concern an interdisciplinary field connecting mathematical analysis, numerical mathematics, semiclassical physics, and quantum physics, which are designed and analyzed to simulate physical behaviors in modern wave theory, models of particle physics, stability of fluid motions, nonlinear optics, differential geometry, and propagation of fluxon [1][2][3][4][5][6][7][8][9]. But in return, the sources why we examine the fractional kind of SGM are that: in the real phenomena, the following state of a physical mode relies on not exclusive its existing state but also on its historical states.…”

Approximation of three‐dimensional nonlinear wave equations by fundamental solutions and weighted residuals process

“…As might be expected, the high applications give rise to the question of how best to solve nonlinear wave equations numerically. Several methods rely on mesh-free methodology, in which a spacious review can be found in [10,[16][17][18][19][20][21]. The MFS, boundary element (BEM), and finite element (FEM) are mainly helpful for collocation-based approximation, although the methods BEM or FEM have some deficiencies in transferring the distorted elements or re-meshing [22].…”

“…Other approach employs the Galerkin method in two or three dimensions the unbounded or bounded space domain. Among them, we can find the Hermite‐Galerkin spectral method on three dimensions of sine‐Gordon equation [10, 11] and local Petrov‐Galerkin method on two dimensions [12]. Since the function $$$ g(C) $$$ is $$$ \mathit{\sin}(C) $$$, the system called sine‐Gordon (SG) equation as the one of the nonlinear wave theory, and a number of studies of these waves has been published.…”