Lie analysis and laws of conservation for the two-dimensional model of Newell–Whitehead–Segel regarding the Riemann operator fractional scheme in a time-independent variable

Abstract

The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in $2D$-space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized.

HIGHLIGHT

• The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived.

• The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time.

• The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem.

• Several graphical countenances were utilized to award a visual performance of the obtained solutions.

Keywords:

• Extended series solution
• fractional differential operator
• Newell–Whitehead–Segel model
• symmetry analysis

Disclosure statement

The authors declare that they have no conflicts of interest.

Data availability statement

No datasets are associated with this manuscript. The datasets used for generating the plots and results during the current study can be directly obtained from the numerical simulation of the related mathematical equations in the manuscript.