A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Application of invariant imbedding to linear multipoint boundary value problems with general boundary conditions

Two extensions of the usual application of invariant imbedding to the solution of linear boundary value problems are presented. The invariant imbedding formulation of a linear two point boundary value problem in which functional relationships are given between the variables at either one or both of the boundary points is presented. Also, extension of invariant imbedding to linear multipoint boundary value problems is given. Using these extensions singly or in combination, a general multipoint boundary value of linear ordinary differential equations can be solved. In addition, the problems of infinite initial conditions and / or indeterminate initial derivatives are resolved. Numerical examples demonstrate the feasibility and accuracy of the method.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

Invariant imbedding and Riccati transformations

From its inception, the theory of invariant imbedding has been concerned with the study of various relations between the inputs and outputs of various physical processes. Where the processes could be modelled by differential or integro-differential equations, these ideas have led to the heuristic development of various functional relationships for the solutions of these equations. In this work, we show that for a general class of two point boundary value problems these relations can be obtained from mathematical arguments rather than physical ones. The principal result is the establishment of the equivalence of solving a family of two point boundary value problems and that of determining the existence of two transformations on the set of solutions of the given differential equations. We refer to these transformations as Riccati transformations. They are shown to be determined by a set of initial value problems which generalize the invariant imbedding equations obtained by previous authors. We work in the coordinate free setting of a Banach space. The usefulness of this approach is shown as we are able to readily extend our results to nonlocal and multipoint boundary conditions. An indication is made of how a similar theory applies to a class of problems for difference equations.     

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations with a boundary perturbation

The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations

The existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations. The arguments are based only upon the positivity of the Green's functions and the Krasnosel'skii fixed point theorem. As an example, a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity.     

Constructing orthogonal curvilinear meshes by solving initial value problems

Summary The purpose of this paper is to develop methods for constructing orthogonal curvilinear meshes suitable for solving partial differential equations over plane regions with smooth, curved boundaries. These curved meshes cover an annular strip along the boundary of the region which is included in the mesh. In this strip difference approximations of partial differential equations and boundary conditions can be set up as easily as they can for halfspace problems. The rest of the region and a suitable part of the annular strip can be covered by a square or rectangular mesh. In the present paper we consider the problem of determining curved meshes by solving nonlinear hyperbolic initial value problems which are formally related to the Cauchy-Riemann equations.This work was sponsored by the Swedish Institute for Applied Mathematics (ITM)     

Complete second order differential equations in Banach spaces with dynamic boundary conditions

In this paper, we deal with the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions. General wellposedness theorems are obtained (for the first time), which are used to solve the corresponding inhomogeneous problems. Examples of applications to initial boundary value problems for partial differential equations are also presented.     

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.     

反应扩散方程解的渐近性态

研究了一类反应扩散方程初始边值问题. 利用微分不等式理论得到了问题解的渐近性态.     

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.     

The Singularly Perturbed Nonlinear Nonlocal Initial Boundary Value Problems for Reaction Diffusion Equations

The singularly perturbed nonlinear nonlocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.     

具有边界摄动的非线性非局部反应扩散方程的奇摄动问题

A class of nonlinear nonlocal singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained; secondly, by using the stretched variable, the composing expansion method and the expanding theory of power series, the initial layer is constructed; and finally, by using the theory of differential inequalities the asymptotic behavior of solutions for initial boundary value problems is studied, and including some relational inequalities the existence and uniqueness of solutions for the original problem and the uniformly valid asymptotic estimation are discussed.     

Convergence of monotone iterations to initial value problems of functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. We use this technique to initial value problems of functional differential equations showing that corresponding linear iterations converge to the unique solution of our problem and this convergence is superlinear     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

Neumann initial–boundary value problem for Benjamin–Ono equation

We consider the Neumann initial–boundary value problem for Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

Mixed initial–boundary value problem for the Benjamin–Ono equation

We consider the mixed initial–boundary value problem for the Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.