We consider multivariate sequences, deened by a partial linear recurrence equation together with appropriate boundary conditions. The domain of deenition of these sequences is the rst orthant of the integer lattice, restricted in some dimensions to an initial segment of the nonnegative integers. By means of the kernel method we obtain an explicit expression for the generating function of the so...

The theory of the nonlocal linear boundary value problems is still on the level of examples. Any attempt to encompass them by a unified scheme sticks upon the lack of general methods. Here we are to outline an algebraic approach to linear nonlocal boundary value problems. It is based on the notion of convolution of linear operator and on operational calculus on it. Our operators are right inver...

Solutions are obtained for the boundary value problem, y(n) + f(x, y) = 0, y(i)(0) = y(1) = 0, 0 ≤ i ≤ n − 2, where f(x, y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. §

It is shown that the non-homogeneous Dirichlet and Neuman problems for the 2-order Seiberg-Witten equation admit a regular solution once the H-condition 3.1.1 is satisfied. The approach consist in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation.

In this chapter we discuss discrete variable methods for solving BVPs for ordinary differential equations. These methods produce solutions that are defined on a set of discrete points. Methods of this type are initial-value techniques, i.e., shooting and superposition, and finite difference schemes. We will discuss initialvalue and finite difference methods for linear and nonlinear BVPs, and th...

In the first part of this chapter we focus on the question of well-posedness of boundary-value problems for linear partial differential equations of elliptic type. The second part is devoted to the construction and the error analysis of finite difference schemes for these problems. It will be assumed throughout that the coefficients in the equation, the boundary data and the resulting solution ...