Difference approximations of boundary value problems for functional differential equations

The midpoint difference method applied to boundary value problems for functional differential equations is studied. Convergence is shown to be to the order of the truncation error, the same result that holds in the ordinary differential equation case.     

Nonlinear boundary value problems for second order differential inclusions

In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations, which satisfy the generalized Hartman condition weaker than that considered in some papers. Using techniques from multivalued analysis, theory of monotone operators and fixed points, we prove the existence of solutions in both “convex” and “nonconvex” cases. Our framework can be incorporated with Dirichlet, Neumann, and mixed boundary problems.     

Nonlocal boundary value problems in differential and difference settings

We study nonlocal boundary value problems of the first and second kind for the heat equation with variable coefficients in the differential and difference settings. By the method of energy inequalities, we find a priori estimates for the differential and difference problems.     

Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.     

Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

A fractional-derivative two-point boundary value problem of the form \({\tilde{D}}^\delta u=f\) on (0, 1) with Dirichlet boundary conditions is studied. Here \({\tilde{D}}^\delta \) is a Caputo or Riemann–Liouville fractional derivative operator of order \(\delta \in (1,2)\). The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition \(u(0)=0\) and the special right-hand-side condition \(f(0)=0\) are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.     

Stability and convergence of difference schemes for boundary value problems for the fractional-order diffusion equation

A family of difference schemes for the fractional-order diffusion equation with variable coefficients is considered. By the method of energetic inequalities, a priori estimates are obtained for solutions of finite-difference problems, which imply the stability and convergence of the difference schemes considered. The validity of the results is confirmed by numerical calculations for test examples.     

On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri and B. Ricceri we prove existence results improving earlier theorems by Gupta and Marano.     

First-order approximations for differential inclusions

It is shown, under a mere continuity assumption, that the union of affine functions generated by the right-hand side of a differential inclusion, is a little oh approximation of the attainable set. Explicit estimates are given. An application to polygonal approximations is displayed.Research supported by a grant from the Basic Research Fund, The Israel Academy of Science and Humanities.Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Bifurcations of a parametrized family of boundary value problems for second order differential inclusions

We study the existence of non-trivial solutions of the following family of differential inclusions of second order (S) $$\left\{ \begin{gathered} y''(t) \in F(p, t, y(t), y'(t)) t \in [0,a] , \hfill \\ (y(0), y'(0), y(a), y'(a)) \in b(p) , \hfill \\ \end{gathered} \right.$$ where \(F:P \times [0,a] \times \mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n } \) is a Carathéodory multifunction with non-empty compact convex values and b: P→G_2n(?⁴ⁿ) is a continuous map from a CW-complex P to the Grassmann manifold G_2n(?⁴ⁿ). We show that if (X,A) is a finite CW-pair in P, A contractible in X, b: (X, A)→(G_2n(?⁴ⁿ), pt) is such that and F satisfies the Nagumo growth conditions at some point p₀ ε X, then the system (S) has a bifurcation from infinity in X; i.e. there exists a sequence of non-trivial solutions of S whose norms in the space C¹ tend to infinity.     

Rate of convergence of finite difference approximations for degenerate ordinary differential equations

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations:

where is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.

     

Ritz approximations to two-dimensional boundary value problems

Summary Various Ritz solutions to the plane strain elasticity and the steady state heat flow boundary value problems for a polygonal domain are considered. Historically, two basic approaches have been used in partitioning into finite elements, (i) complete triangulation and (ii) rectangles with boundary triangles. In each case, the Ritz solution is the unique function (or vector of functions) which minimizes an energy functional over a finite dimensional vector spaceS. We consider as choices forS, piecewise linear and cubic functions for complete triangulations; and piecewise bilinear and bicubic functions for the case in which is a union of rectangles and boundary triangles. For the elasticity problem,L ₂ convergence of the components of stress and strain is established for each choice of the spaceS. L ₂ convergence of the displacement vector is also shown for a wide class of boundary conditions. Convergence of the temperature is proven for the heat flow problem also. Numerical comparisons are made of the Ritz solutions based upon each spaceS of trial functions.     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

Finite difference methods and their convergence for a class of singular two point boundary value problems

Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x y)=f(x,y), y(0)=A, y(1)=B, 0<<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all (0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM ₁ based on just one evaluation off. For uniform mesh we obtain two methodsM ₂ andM ₃ each based on three evaluations off. For =0,M ₁ andM ₂ both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h ²)-convergence established and illustrated by numerical examples.     

Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

Some results for fractional boundary value problem of differential inclusions

In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay.