Multipoint boundary value problems with discontinuities I. Algorithms and applications

An algorithm, referred to as the initial value adjusting method with discontinuities, is presented for the numerical solution of multipoint boundary value problems arising from systems of ordinary differential equations in which jump discontinuities are permitted and for which both the dynamics and boundary conditions may be nonlinear. Numerical results are given for several examples and the algorithm is also applied to a noisy dynamical system in which the states are estimated by using a variational technique.     

On quadratic convergence of the initial value adjusting method for nonlinear multipoint boundary value problems

A natural interpretation of “maj” and “inv” q-counting of multiset permutations in terms of walks on a lattice with multilane highways is presented. This is applied to give a short combinatorial proof of two theorems of MacMahon and to rederive a recent result of Gessel.     

On the applicability of the Adomian method to initial value problems with discontinuities

In this paper we extend our results of L. Casasús, W. Al-Hayani [The decomposition method for ordinary differential equations with discontinuities, Appl. Math. Comput. 131 (2002) 245–251] to initial value problems with several types of discontinuities, giving relevant examples of linear and nonlinear cases.     

Multipoint boundary value problems for ODEs. part I

The method of quasilinearization is applied to multipoint boundary value problems of ordinary differential equations. It is shown that monotone iterations converge quadratically to the unique solution of our problem.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

Well-posedness of hyperbolic initial boundary value problems

Assuming that a hyperbolic initial boundary value problem satisfies an a priori energy estimate with a loss of one tangential derivative, we show a well-posedness result in the sense of Hadamard. The coefficients are assumed to have only finite smoothness in view of applications to nonlinear problems. This shows that the weak Lopatinskii condition is roughly sufficient to ensure well-posedness in appropriate functional spaces.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

Initial value adjusting method and graph theoretical analysis for the solution of nonlinear multipoint boundary value problems with varying system dimensions

The initial value adjusting method for the solution of nonlinear multipoint boundary value problems in which the system dimensions vary over subintervals is proposed. To reduce the computer storage requirements and the excessive amount of computer time, an algorithm based on a digraph and its associated Boolean matrices is also proposed. The effectiveness of these algorithms is illustrated by an application to a five compartment model from pharmacokinetics.     

Convergence analysis of a monotone method for fourth-order semilinear elliptic boundary value problems

This work is concerned with the convergence of a monotone method for fourth-order semilinear elliptic boundary value problems. A comparison result for the rate of convergence is given. The global error is analyzed, and some sufficient conditions are formulated for guaranteeing a geometric rate of convergence.     

Convergence of a Bakhvalov grid adaptation method for singularly perturbed boundary value problems

In this paper, a Galerkin finite element method for non-self-adjoint boundary value problems on Bakhvalov grids is considered. Using the Galerkin projectionmethod, the convergence of a sequence of computational grids with an unknown boundary of the boundary layer is proved. Some numerical examples are presented.     

On one method of approximation of initial boundary value problems for the Navier-Stokes equations

Solutions of the initial boundary value problems for Navier-Stokes equations are approximated by solutions of the initial boundary value problem $$\partial _t u(t) + u_k (t)\partial _k u(t) - v\Delta u(t) - \frac{1}{\varepsilon }\triangledown div u(t) + \frac{1}{2}u(t)div u(t) = f(t),u(0) = u_0 in \Omega , u(t) = 0 on \partial \Omega $$ We study the nearness of solutions of these problems in suitable norms and also the nearness of their minimal global B-attractors. Bibliography:11 titles.     

Spectral methods for initial boundary value problems with nonsmooth data

Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution. Received August 1, 1995 / Revised version received August 19, 1997     

Singular initial and boundary value problems with sign changing nonlinearities

A variety of new existence results are presented for both singularinitial and boundary value problems where the nonlinearity is allowed to change sign. Our theory is then applied to anexample which arises naturally in nonlinear mechanics (thatis, in the membrane response of a spherical cap). Received 17 May, 1999.     

A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems

In this paper, the variational iteration method (VIM) is used to study the singular Emden-Fowler initial value problems and boundary value problems arising in physics and astrophysics. The VIM overcomes the singularity at the origin. The Lagrange multipliers for all cases of the equations are determined. The work is supported by analyzing few initial value problems and boundary value problems where the convergence of the results is emphasized.     

Correct initial boundary value problems for dispersive equations

This paper deals with correctness of initial boundary value problems for general dispersive equations of finite odd orders. For the Kawahara and KdV equations we prove existence, uniqueness and stability of strong global solutions in a bounded domain for different signs of a coefficient of the highest derivative as well as their asymptotics when the coefficient of the higher-order derivative in the Kawahara equation approaches zero.     

An initial value theory for linear causal boundary value problems

A new formalism in the theory of linear boundary value problems involving causal functional differential equations is presented. The approach depends on the construction of a differentiable family of boundary problems into which the original boundary value problem is imbedded. The formalism then generates an initial value problem which is equivalent to the family of imbedded problems. An important aspect of the method is that the equations in the initial value algorithm are ordinary differential equations rather than functional differential equations, although nonlinear and of higher dimension. Applications of the theory to differential-delay and difference equations are given.     

Second-order boundary value problems with nonhomogeneous boundary conditions (II)

Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

We prove that the whole plane is divided by a “continuous decreasing curve” Γ into two disjoint connected regions Λ^E and Λ^N such that the above problem has at least one solution for (λ₁,λ₂)Γ, has at least two solutions for (λ₁,λ₂)Λ^EΓ, and has no solution for (λ₁,λ₂)Λ^N. We also find explicit subregions of Λ^E where the above problem has at least two solutions and two positive solutions, respectively.