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...

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...

Solving nonlinear equations is a problem that needs to be dealt with in the practical engineering application. This paper uses Block–Broyden method for solving large-scale nonlinear systems, and two preconditioners are applied for solving the underlying linear systems, including SSOR preconditioner as well as ASSOR method, which is based on SSOR. It discusses their implementation processes and ...

In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not ...