Truncation method with point transformation for exact solution of Liouville Bratu Gelfand equation

• Published in: Computers & mathematics with applications (1987) Vol. 76; no. 5; pp. 1219 - 1227
• Main Authors: Saleh, R., Mabrouk, S.M., Kassem, M.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.09.2018; Elsevier BV
• Subjects: Boundary conditions; Bäcklund transformation; Differential equations; Exact solutions; Liouville Bratu Gelfand equation; Mathematical analysis; Nonlinear differential equations; Nonlinear equations; Numerical analysis; Painlevé analysis; Singular manifold method; Topological manifolds; Transformations (mathematics); Painlevé analysis; Singular manifold method; Liouville Bratu Gelfand equation; Bäcklund transformation
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2018.06.016

In this paper, we construct the exact solution of Liouville Bratu Gelfand (LBG) equation. This equation has Painlevé property under a point transformation. Painlevé analysis based on the truncated singular manifold method is satisfied. Bäcklund transformation is obtained and is applied to reduce the problem to a system of nonlinear differential equations. This system is reduced to a factorized form due to the symmetry of the equation. By solving this factorized form, we obtain exact solution for LBG equation satisfying the boundary conditions. The solution is plotted for different values of λ and is compared with previous numerical results. According to our intensive search there is no exact solution of the problem. The obtained solution shows that the transformation does not change the properties of the equation.