Nilpotent and recursive flows

In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coeficients. Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a generalization of the ‘method of variation of constants’ which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many applications are presented     

Two new approaches to computing Hopf bifurcation problems

We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter ε; practical experience shows that one gets convergence for ε sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures.     

非线性多孔介质流动模型问题的分歧

本文讨论一类多孔介质流动模型非线性边值问题．对这类问题以及广泛的一类算子方程，我们得到了渐近歧点的存在性     

Green functions for boundary value problems about divergence of elongated plate

Bucking of a thin flexible elongated plate in supersonic gas flow along Ox-axis, compressed or extended by boundary stresses at the edges x = 0 and x = 1 is governed by a boundary value problem for nonlinear ordinary integro-differential equation with two bifurcation parameters: external stress and Mach number. On two examples of boundary conditions B w″(0) = 0, w‴(0) = 0, w(1) = 0, w′(1) = 0 and B ′ w(0) = 0, w′(0) = 0, w″(1) = 0, w‴(1) = 0. Green functions for the linearized problems are constructed. Arising technical difficulties are overcoming with the aid of the bifurcation curves representation through the roots of relevant characteristic equations. In the known literature [1] marked, that Green functions for aeroelasticity problems were not constructed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation of nonlinear problems modeling flows through porous media

This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems. This work is supported in part by NSF of Shandong Province and NNSF of China     

Numerical solution of Hopf bifurcation problems

We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ?. It has isolated solutions in the ?-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.     

Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen

Summary We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).

     

The liouville property and boundary value problems for semilinear elliptic equations on noncompact Riemannian manifolds

We address the asymptotic behavior of solutions to semilinear equations on noncompact Riemannian manifolds. Under study is the relation between the solvability of some boundary and exterior boundary value problems as well as conditions for the fulfillment and stability of Liouville-type theorems for the solutions to semilinear equations on these manifolds.     

Solutions of n-point boundary value problems associated with nonlinear summary difference equations

Variation of parameter methods play a fundamental rôle in understanding solutions of perturbed nonlinear differential as well as difference equations. This paper is devoted to the study of n-point boundary value problems associated with systems of nonlinear first-order summary difference equations by using the nonlinear variation of parameter methods. New variational formulae, which provide connections between the solutions of initial value problems and n-point boundary value problems, are obtained. An iterative scheme for computing approximated solutions of the boundary value problems is also provided.     

零点不可导时锥映射的歧点与正特征元的全局结构

研究了半序Banach空间中一类非线性锥映射歧点的存在性与正特征元的全局结构.与已知文献不同的是,不要求算子在零点沿着锥Frechet可微. 作为应用,讨论了一类椭圆型偏微分方程边值问题正解的歧点与全局结构.     

Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.     

Positive solutions of semi-positone nonlinear operator equations in Banach spaces with lattice structure

By using global bifurcation theories we obtain an existence result for positive solutions of some nonlinear operator equations. The main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions.     

Multiplicity Results for Some Nonlinear Elliptic Equations

This paper deals with the existence of multiple solutions for some classes of nonlinear elliptic Dirichlet boundary value problems. The interplay of convex and concave nonlinearities is studied both for second order equations and for problems involving thep-Laplacian. The bifurcation of positive solutions for some quasilinear eigenvalue problems is also discussed.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Nonlinear three-point boundary value problems

In the present work, we investigate nonlinear three-point boundary value problems (second-order differential equations with boundary conditions imposed at three consecutive points) in an effort to understand the conditions that must be imposed upon nonlinear perturbations of a linear problem in order to obtain a solution of a three-point problem. Both the noncritical case and the critical case are considered. In the first case, existence and uniqueness results are obtained, and in the second case we deal with local bifurcation problems when the nonlinearities are quadratic and cubic.     

Unbounded connected component of the positive solution set of nonlinear operator equation and its applications

In this paper, by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters. As a result, we obtain some existence results for positive solutions of the nonlinear operator equation. The main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Inertial Manifolds for Weakly and Strongly Dissipative Hyperbolic Equations

One considers mixed boundary value problems for a quasilinear hyperbolic equation with a weak, as well as strong, dissipation. The nonlinear function in the equation is assumed Lipschitz continuous. For each of these problems one obtains the conditions on the Lipschitz constant that ensure the existence of inertial manifolds.     

Mono-implicit Runge–Kutta formulae for the numerical solution of second order nonlinear two-point boundary value problems

Mono-implicit Runge–Kutta (MIRK) formulae are widely used for the numerical solution of first order systems of nonlinear two-point boundary value problems. In order to avoid costly matrix multiplications, MIRK formulae are usually implemented in a deferred correction framework and this is the basis of the well known boundary value code TWPBVP. However, many two-point boundary value problems occur naturally as second (or higher) order equations or systems and for such problems there are significant savings in computational effort to be made if the MIRK methods are tailored for these higher order forms. In this paper, we describe MIRK algorithms for second order equations and report numerical results that illustrate the substantial savings that are possible particularly for second order systems of equations where the first derivative is absent.