Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel–Zakharov–Kuznetsov–Burgers equation

Abstract

In this paper, we consider the (3 + 1)-dimensional time-fractional Schamel-Zakharov-Kuznetsov-Burgers (SZKB) equation. With the help of the Riemann-Liouville derivatives, the Lie point symmetries of the (3 + 1)-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi-Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

“…where a i (i = 0, 1, … , n) are constants to be detected and the positive integer n can be obtained by balancing the nonlinear terms and the highest order derivatives and in Equation (36). Also, the function 𝜓(𝜉) satisfies the following transformation as below:…”

“…step 3: Now, substituting Equation (37) along with (38) into (36) and equating all the coefficients of all powers of (𝜑(𝜉)) i by zero, we get a system of algebraic equations. Solve this system via the Mathematica program to determine the value of a i (i = 0, 1, … , n).…”

“…When 𝛼 = 1, Equation ( 1) is reduced to the (3 + 1)-dimensional generalized Z-K equation. 35,36 The main purpose of this paper is to construct a new set of explicit solutions for Equation (1). Also, we aimed to find the Lie symmetries for Equation (1) using the Lie group analysis.…”

“…Also, we aimed to find the Lie symmetries for Equation (1) using the Lie group analysis. [37][38][39] Then, these symmetries are used to derive conservation laws 35,36,[40][41][42][43] for Equation (1) via formal Lagrangian. In addition, a reduced form of this equation is given by the similarity variables, which are obtained from the symmetries.…”

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

“…Therefore, this new conservation law plays an increasingly important role in solving the conservation laws of FPDEs. More details about conservation laws can be found in [29][30][31][32].…”