Higher-order Lidstone boundary value problems for elliptic partial differential equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

On Numerov's method for a class of strongly nonlinear two-point boundary value problems

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.     

A reliable algorithm for solving tenth-order boundary value problems

In this paper we present an efficient numerical algorithm for solving linear and nonlinear boundary value problems with two-point boundary conditions of tenth-order. The differential transform method is applied to construct the numerical solutions. The proposed algorithm avoids the complexity provided by other numerical approaches. Several illustrative examples are given to demonstrate the effectiveness of the present algorithm.      

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.     

Multipoint approach to the two-point boundary value problem

This paper treats nonlinear, two-point boundary value problems of the form $\text{x}\text{?}\text{? ?(x, t) = 0}$, in which the Jacobian matrix ?_x(x, t) is characterized by large positive eigenvalues. The resulting numerical difficulties are reduced by treating the two-point boundary value problem as a multipoint boundary value problem. A totally finite-difference approach is employed, thus bypassing the integration of the nonlinear equations, which characterizes shooting methods.The approach employed consists of extending to multipoint boundary value problems the modified-quasilinearization method developed by Miele and lyer for two-point boundary value problems. Basic to the method is the consideration of the performance index P, which measures the cumulative error in the differential equations, the boundary conditions, and the interface conditions.A modified-quasilinearization algorithm is generated by requiring the first variation of the performance index δP to be negative. This algorithm differs from the ordinary-quasilinearization algorithm because of the inclusion of the scaling factor or stepsize α in the system of variations. The main property of the modified-quasilinearization algorithm is the descent property: if the stepsize α is sufficiently small, the reduction in P is guaranteed. Convergence to the desired solution is achieved when the inequality P ? ? is met, where ? is a small, preselected number.The variations per unit stepsize $\text{Δx(t)}\text{α}\text{= A(t)}$ are governed by a system of mn nonhomogeneous, linear differential equations subjected to p initial conditions, q final conditions, and (m ? 1)n interface conditions, with p + q = n, where n is the dimension of the vector x and m is the number of subintervals. Therefore, the total number of boundary conditions and interface conditions is mn. The above system is solved employing the method of particular solutions: m(n + 1) particular solutions are combined linearly, and the coefficients of the combination are determined so that the linear system is satisfied.Two numerical examples are presented, one dealing with a linear system and one dealing with a nonlinear system. The examples illustrate the effectiveness as well as the rapidity of convergence of the present method.     

Iterative methods for a fourth order boundary value problem

We consider a fourth order nonlinear ordinary differential equation together with two-point boundary conditions and provide a-priori error estimates on the length of the interval (b?a) so that the Picard's iterative method, the approximate Picard's iterative method and the quasilinear iterative method convergence to the solution of the problem.     

An <Emphasis Type="BoldItalic">O</Emphasis>(<Emphasis Type="BoldItalic">h</Emphasis><Superscript><Emphasis Type="Bold">6</Emphasis></Superscript>) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h ²) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h ⁶) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.     

高阶微分积分方程的单调迭代法及其应用

首先利用上下解方法以及微分不等式理论给出了n阶微分积分方程的初值问题解的存在性及其单调迭代法,然后将所得结果应用到n阶微分方程的两点边值问题,得到了n阶非线性两点边值问题解的存在性及其单调迭代法,所得结果推广了已有的结果．     

Note on a proposed weighted residual method for solving nonlinear differential boundary problems

A proposed iterative method for solving nonlinear differential two-point boundary value problems is generalized and shown to be equivalent to using the standard method of adjoints on a linearized form of the nonlinear boundary-value problem.     

Resolvent growth and Birkhoff-regularity

We prove a long standing conjecture in the theory of two-point boundary value problems that unconditional basisness implies Birkhoff-regularity. It is a corollary of our two main results: minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant, and sparseness of nth-order roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis.Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff's solutions of the equation l(y)=λy. This property was discovered earlier by the author.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

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.     

L ∞-convergence of collocation and galerkin approximations to linear two-point parabolic problems

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L _∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.     

Computational experience with the Chow—Yorke algorithm

The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.This work was partially supported by NSF Grant MCS 7821337.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Boundary value problems with causal operators

In this paper, we apply the monotone iterative method for nonlinear two-point boundary value problems with causal operators. We formulate sufficient conditions under which such problems have extremal or quasisolutions in a corresponding sector. We also investigate differential inequalities.     

The iterative solutions of 2nth-order nonlinear multi-point boundary value problems

The aim of this paper is to investigate the existence of iterative solutions for a class of 2nth-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, (n + 2)-point boundary condition and 2(n − m)-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution. Two examples are presented to illustrate the results.     

Approximation methods and alternative problems

Projection methods applied to abstract problems of the form Ax = Nx, where A is a linear operator with a nontrivial null space and N is a nonlinear operator, both on a normed space, are studied. Convergence results are obtained and then are applied to periodic two-point boundary value problems using splines as approximations.     

A nonlinear singular diffusion equation with source

In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which depends on initial data in L¹ continuously.