New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations

Analytical approach to Boussinesq equation with space‐ and time‐fractional derivatives

“…One of the most remarkable features of the HPM is that usually only few perturbation terms are sufficient for obtaining a reasonably accurate solution. Considerable research works have been conducted recently in applying this method to a class of linear and non-linear equations [20][21][22][23][24][25][26][27][28][29]. The interested reader can see References [30][31][32][33] for last development of HPM.…”

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

Mesoscopic Simulation of the Two-Component System of Coupled Sine-Gordon Equations with Lattice Boltzmann Method

“…There are many analytical methods solving the two-component system of coupled sine-Gordon equations, such as the modified decomposition method [ 52 ], the homotopy analysis method [ 53 ], the hyperbolic auxiliary function method [ 54 ], the homotopy perturbation method [ 55 ], the rational exponential ansatz method [ 56 ], the variational iteration method [ 57 ] and the modified Kudryashov method [ 58 ]. However, to our best knowledge, there are few numerical method to solve this coupled system.…”