In this study, the nonlinear term in two-dimensional Bratu equation has been replaced by its Taylor’s expansion. Hence, resulting partial differential studied using Lie group method. The symmetry reductions that reduce equations to ordinary are determined theory. resultant were analytically solved, and solutions obtained closed form for some specified parameter values, while others solved numer...

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu′′ + 2u′ + f (u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based o...

Solution by M.C. Nucci (Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy) Lie’s monumental work on transformation groups, [12], [13] and [14], and in particular contact transformations [15], has provided systematic techniques for obtaining exact solutions of differential equations [16]. Many books have been dedicated to this subject and its generalizations (...

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu′′+2u′+f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slo...

The present work deals with the construction of a high‐order efficient technique for solving one‐dimensional nonlinear Bratu‐type and Lane‐Emden‐type problems describing various physical, chemical biological phenomena in applied science engineering. Convergence analysis method is studied. Numerical experiments are carried out to demonstrate efficiency, accuracy performance proposed method. show...

Solving nonlinear equations is a problem often needed to be dealt with in the practical engineering application. This paper discusses preconditioning methods based on Block Broyden method. It first introduces the Block Broyden method and preconditioning technique. Then it presents four different preconditioners for the Block Broyden method and discusses the implementation process. It also analy...

We present a derivation and implementation of asymptotic boundary conditions to be imposed on "artificial" boundaries for nonlinear elliptic boundary value problems on semi-infinite "cylindrical" domains. A general theory developed by the authors in [11] is applied to establish the existence of exact boundary conditions and then to obtain useful approximations to them. The derivation is based o...