Browsing by Author "Adilkhanova Zh."

Now showing 1 - 1 of 1
  • Thumbnail Image
    ItemOpen Access
    A symmetric analysis in LP space in two dimensions
    (Faculty of Engineering and Natural Sciences, 2023) Adilkhanova Zh.
    The p-Laplacian equation is a nonlinear partial differential equation of third order that arises as Euler-Lagrange equation of the gradient of function in L p norm which was first studied by Gunnar Aronsson in the late 80s [1]. Since then many explicit classical solutions and their generalisations are found. In this paper we find only the classical solutions of p-Laplacian equation in spatial dimensions, i.e. the p-harmonic functions in two dimensions. The p-harmonic functions are found by the use of Lie symmetry analysis method which deals with invariant solutions under some transformations of the solution of the partial differential equations. We obtain Lie algebra generators of the p-Laplacian equation, and the corresponding symmetry reductions of the p-Laplacian equation to ordinary differential equations. Finally, we use the Lie symmetries to construct invariant solutions of p-Laplacian that already known and some new ones in explicit form. Moreover, by using the Lie symmetries we can construct new solutions from known solutions of the p-Laplacian equation. In this article we use Lie symmetry analysis to find two-dimensional a new solution to Laplace’s p-equation. The p-Laplace equation is nonlinear partial differential equation (PDE), arising as an equation Euler-Lagrange expression of the gradient of a function in the L p norm. Using symmetry Lie, we reduce PDEs to ordinary differential equations. We find already known and new solutions of p-Laplace. It turns out using symmetric Lie analysis, we find the symmetry of the given u. We have 8 cases where we ended up getting rid of x,y,u. And in the end what remains is g and s. We were able to find the symmetry for u. In this research article, we employ the Lie symmetry analysis technique to discover two-dimensional solutions of the p-Laplacian equation. The p-Laplacian equation is nonlinear partial differential equation (PDE) that arises as EulerLagrange equation of the gradient of function in L p norm. By using Lie symmetries we reduce PDEs to ordinary differential equations. We find solutions of p-Laplacian that already known and some new. Lie symmetry analysis is a powerful mathematical tool used to investigate symmetries and simplify the solutions of differential equations. In this paper, the authors apply Lie symmetry analysis to the p-Laplacian equation, which is a nonlinear PDE that arises in the context of gradient optimization problems. iv The p-Laplacian equation can be written as: ∇ · (|∇u| p−2∇u) = 0 where ∇ represents the gradient operator and u is the unknown function. The parameter p determines the nonlinearity of the equation. By applying Lie symmetry analysis, the authors are able to identify symmetries of the p-Laplacian equation, which correspond to transformations that leave the equation invariant. These symmetries allow the authors to reduce the PDE to a system of ordinary differential equations (ODEs), which are often easier to solve. Using this approach, the authors are able to find two-dimensional solutions of the p-Laplacian equation. They also compare their results with known solutions in the literature and find some new solutions. Overall, this paper demonstrates the effectiveness of Lie symmetry analysis in simplifying and solving nonlinear PDEs, specifically the p-Laplacian equation. The identified solutions can have various applications in fields such as physics, engineering, and mathematical modeling.