New analytical solutions by the application of the modified double sub-equation method to the (1 + 1)-Schamel-KdV equation, the Gardner equation and the Burgers equation

Abstract

In this research we present the application of the modied double sub-equation guess solution together with the analytical solutions of the Riccati equation to obtain new analytical exact solutions to the (1+1)- Schamel-KdV equation, the (1 + 1)-dimensional Gardner equation (or combined KdV-mKdV) and the nonlinear evolution (1 + 1)-dimensional Burgers equation. Results show some conditions between the allowed values of the interaction coecients and the parameters of the allowed analytical solutions of the double sub-equation guess solution. Additionally, it has been revealed that handle the complexiton process with two dierent travelling variables have as an important consequence that the mixing between trigonometric and hyperbolic solutions is only observed for the analytical solutions of the nonlinear evolution (1 + 1)-dimensional Burgers equation, while no mixing between trigonometric and hyperbolic solutions is observed for the solutions of the (1 + 1)-Schamel-KdV ((1 + 1)-S-KdV) equation or the (1 + 1)-dimensional Gardner equation. Numerical simulations are provided in 3-D graphs and 2-D plots representing the general characteristics of the resulting analytical solutions.

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

“…There are several methods to solve and find the exact solutions to the evolution equations involving non-linearity. For example, the -expansion method [6] , [7] , [8] , [9] , the exp-function method [10] , the modified exp-function method [11] , [12] , the tanh–coth expansion method [13] , [14] , [15] , the improved method [16] , [17] , the -expansion method [18] , [19] , [20] , [21] , the simple equation method (SEM) [22] , [23] , the Lie symmetry approach [24] , [25] , the sine-Gordon method [26] , the modified Sardar sub-equation method [27] , [28] , the generalized Kudryashov method [29] , [30] , the Riccati-Bernoulli sub-ODE method [31] , [32] , the improved generalized Riccati mapping method [33] , the modified double sub-equation method [34] , the generalized exponential rational function (GERF) method [35] , [36] and there are many more. Beside these integer order PDEs there are lots of techniques for investigating fractional order PDEs such as [37] , [38] etc.…”

Some Latest Families of Exact Solutions to Date–Jimbo–Kashiwara–Miwa Equation and Its Stability Analysis

“…Many intricate natural phenomena have been described using nonlinear partial differential equations (NPDEs). Many researchers concentrate on this topic, since finding the exact solutions to these equations have enhanced our comprehension of how they function, how they are applied, and how they are created [1,2]. For the generalized Schrödinger equation, Hosseini et al used a modified Jacobi elliptic expansion approach to discover exact solutions [3].…”