Optimal System and Exact Solutions of Monge-Ampere Equation

Tooba Feroze,Mohsin Umair

doi:10.26713/cma.v12i4.1516

Abstract

A solution that remains unchanged when transformed under Lie group of point symmetries of the differential equation is an invariant solution of the differential equation. Optimal system of Lie group of point symmetry generators provide all possible invariant solutions of differential equation. Here, using optimal system of Lie point symmetry generators group invariant solutions are obtained. Using these solutions, exact solutions of non-homogeneous Monge-Ampere equation have been presented here.

Highlights

The Monge-Ampere equation uxxu yy − u2xy + f (x, y) = 0, (1.1)is a semi-linear non-homogeneous partial differential equation with f (x, y) as non-homogeneous part of the equation
In 1781, Gaspard Monge originally formulated and analyzed the problem of optimal transportation, initiating a profound mathematical theory, which connects the different areas of differential geometry, nonlinear partial differential equations, linear programming and probability theory
Continuing in the same way, we find the optimal system of one-dimensional sub algebras of non-homogeneous Monge-Ampere equation (1.1) with f (x, y) = ex as

Summary

Introduction

Is a semi-linear non-homogeneous partial differential equation with f (x, y) as non-homogeneous part of the equation. In 1781, Gaspard Monge originally formulated and analyzed the problem of optimal transportation, initiating a profound mathematical theory, which connects the different areas of differential geometry, nonlinear partial differential equations, linear programming and probability theory. It was later studied by Minkowski (1864-1909) [14, 15], Lewy (1904-1988) [12], Bernstein (1918-1990) [1] and many others. This system helps us to reduce the semi-linear non-homogeneous Monge-Ampere equation (1.1) into ordinary differential equations Solutions of these reduced equations give new set of group invariant solutions. We find (a) the optimal system by (i) calculating the commutator table for symmetry generators of given differential equation; (ii) constructing adjoint representation table, by conjunction of adjoint map with already calculated commutator relation table; and (iii) construct the conjugacy classes. (b) Using these optimal algebras, equation (1.1) is reduced to ordinary differential equation whose solutions lead to the solutions of equation (1.1)

Lie symmetries and Commutator Relation Table

Reduction and Solution

Conclusion