Lie Symmetry Analysis to General Fifth-Order Time-Fractional Korteweg-de-Vries Equation and Its Explicit Solution

Abstract

In this research paper, we have discussed a systematic approach to solve the general time fractional fifth-order Korteweg-de-Vries equation (KdV) by Lie Symmetry Analysis. Similarity reduction transformed the fractional-order partial differential equation (FPDE) into a nonlinear fractional ordinary differential equation with new independent variable. Erdelyi-Kober fractional differential and integral operator depend on parameter 'α' implemented to reduce into fractional ordinary differential equation (FODE). At last, explicit solution is obtained by power series solution, which arises in modeling many physical phenomena.

Application of Symmetry Analysis and Conservation Laws to Fractional-Order Nonlinear Conduction-Diffusion Model

“…Now for further solution of FODEs, we want to explore the explicit power series solution [30,31], which can be applied to solve FODE (37).…”

“…Jafari et al [16] explored the numerical scheme to study the system of fractional PDEs. Gandhi et al [30][31][32][33][34] provided the explicit solution of fifth-order and fourth-order fractional systems by Lie symmetry analysis. He has been discussed about brain cancer tumor growth model and its analytic solution by the application of fractional reduced differential transform method on Burgess equation.…”

“…components λ t and λ x of conserved vector fields in(31) can be expressed by . adjoint equation (59) satisfied for all solutions of (2), is said to be nonlinear self adjoint.…”

Conservation Laws and Explicit Solution of system of Fractional-Order Coupled Nonlinear Hirota Equations by Lie Symmetry Analysis

“…Chauhan and Arora [29] has obtained the complete analysis of time fractional Kupershmidt equation. Recently, Gandhi et al [40,41,50] have applied symmetry reduction on multi-ordered time-fractional KdV equations and Hirota-Satsoma-coupled Korteveg-de-Vries equations to obtain the explicit solutions with convergence and conservation laws; he concluded that the fractional-order parameter  can control the output of solution of fractional mathematical models and in physical and mathematical aspects, the conservation laws play very crucial role to discuss the consistency of system. Zhang et al [45] promoted the 3 conservation laws of Fokkar-Plank equation with power diffusion.…”

Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws

“…Chauhan and Arora [41] obtained the complete analysis of the time fractional Kupershmidt equation. Recently, Gandhi et al [42][43][44] applied symmetry reduction on multiordered time-fractional KdV equations and Hirota-Satsoma-coupled Korteveg-de Vries equations to obtain the explicit solutions with convergence and conservation laws. Zhang et al [45] exploited the conservation laws of the Fokkar-Plank equation with power diffusion.…”