Concept of Generalized Solution of Nonlinear Operator Equation

The Numerical Solution of Integral Equations on the Half-Line

Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations

Recurrence relations for a Newton-like method in Banach spaces

Two theorems on an iteration method of order three -- usually referred to as Chebyshev's method -- for the approximate solution of nonlinear operator equations in Banach spaces are given. Very simple and weak assumptions are obtained as well as error estimates which are much better compared with known results. Some worked examples of operator equations of different types are given.

We consider the problem of approximate determination of isolated bounded-norm solutions of nonlinear operator equations in a Hilbert space. Closed balls are constructed, such that the existence and uniqueness conditions are satisfied in each ball.

We can compute with bounding sets of numbers, vectors, or functions using the techniques of interval mathematics. The techniques can be used to computationally verify sufficient conditions for existence, uniqueness, and convergence and to construct upper and lower bounds on solutions of nonlinear operator equations. Rounding errors are taken into account. We can compute with set-valued functions and operators on them. The techniques are also useful in search procedures for finding safe starting regions for iterative methods and for constructing natural stopping criteria.

Fixed points and approximate solutions for nonlinear operator equations

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.

Some theorems about the existence of solutions of nonlinear operator equation and the global homeomorphism in abstract space by the global existence theorem of an initial value problem are given.They are generalizations of some well-known results in these areas.

A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss. The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions. Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.

Seven common errors in finding exact solutions of nonlinear differential equations

Solution of nonlinear partial differential equations from base equations

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

The auxiliary equation method is used to construct the finite new exact solutions of nonlinear evolution equations. To search for infinite sequence soliton-like exact solutions of nonlinear evolution equations, characteristics of constructivity and mechanization of auxiliary equation method are analyzed and summarized. Therefore, the quasi-Bcklund transformation between new solutions of a kind of auxiliary equation with Riccati equation is presented, then (2+1)-dimensional modified dispersive water-wave system is taken as an applicable example to find infinite sequence soliton-like new exact solutions by choosing two kinds of formal solutions of nonlinear evolution equations with the help of symbolic computation system Mathematica, where included are the infinite sequence smooth soliton-like solutions, compact soliton solutions and peak soliton-like solutions.