Stable and functional solutions of the Klein-Fock-Gordon equation with nonlinear physical phenomena

Abstract

The present article uses a modified G ′ G -expansion method and the generalized Kudryashov method on Klein-Fock-Gordon (KFG) equation and receives some stable and functional solutions. The obtained results are checked by putting backwards into the physical model and are very beneficial over various existing processes. The diverse variety of stable and functional outcomes such as kink-type shape, bright and dark lump shape, bright and dark singular kinky shape, periodic bright and dark lump shape, multiple bright and dark lump shape, the lump with rough wave shape, the rough wave shape and the kinky shape are taken. The above procedure could also be employed to get stable and functional solutions for other integral and fractional nonlinear models in physics, mathematics, and engineering.

On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

“…. , N), v, λ, µ, C 1 and C 2 obtained into (16), the exact solutions of Equation (15), which is expressed by the hyperbolic functions, can be verified.…”

“…, N), v, λ and µ are constants to be determined later. The positive integer N can be found by considering the homogeneous balance principle between the highest-order derivatives and the nonlinear terms in Equation (15).…”

“…Recently, a large number of methods that have been used to obtain exact solutions of NPDEs have become essential due to the existence of symbolic computation packages which aid researchers in the investigation of the dynamical behavior and the physical meaning of these solutions. Hirota's direct method [5][6][7][8], the tanh method [9], the exp-function method [10,11], (G /G)-expansion method [12,13], the extended (G /G)-expansion method [14,15], the extended trial equation method [16], the first integral method [17], the sine-cosine method [18] and the auxiliary equation method [19] are among these methods.…”

General Traveling Wave Solutions of Nonlinear Conformable Fractional Sharma-Tasso-Olever Equations and Discussing the Effects of the Fractional Derivatives

“…Arrange equation (15) to get 16) is a familiar three-differential equation, and its general solution can be easily obtained, denoted as p = c/ρk 3 ≠ 0, q = C 0 /ρk 4 ; then, equation ( 16) has a characteristic equation…”

“…It can be seen from literature [1][2][3][4][5][6][7] that the first step of these authors was to reduce the STO equation of the fractional order to a nonlinear ordinary differential equation by using fractional complex transformation and then to solve the reduced equation by various methods to obtain the solution of the original fractional-order equation. In addition to the methods provided in literature [1][2][3][4][5][6][7], there are other techniques that can be used to obtain wave solutions of different structures [14][15][16][17][18][19]. It can be seen from equations ( 1)-( 3) that the most general equation is equation ( 1), so we only discuss the accurate general traveling wave solution of equation (1).…”