Solving the difference initial-boundary value problems by the operator exponential method

In this paper a technique is suggested to integrate linear initial boundary value problems with exponential quadrature rules in such a way that the order in time is as high as possible. A thorough error analysis is given for both the classical approach of integrating the problem firstly in space and then in time and of doing it in the reverse order in a suitable manner. Time-dependent boundary conditions are considered with both approaches and full discretization formulas are...

In this study, we give the variation of parameters method from a different viewpoint for the Nth order inhomogeneous linear ordinary difference equations with constant coefficient by means of delta exponential function . Advantage of this new approachment is to enable us to investigate the solution of difference equations in the closed form. Also, the method is supported with three difference eigenvalue problems, the second-order Sturm-Liouville problem, which is called also ...

Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as systematic ways of matching up to the operator solution of the partial differential equation. By completely abandon the idea of approximating derivatives directly, the theory provides a unified description of explicit finite-difference schemes for ...

The method, proposed in the given work, allows the application of well developed standard methods used in quantum mechanics for approximate solution of the systems of ordinary linear differential equations with periodical coefficients.

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider matrix elliptic second order differential operators $\mathcal{A}_{D,\varepsilon}$ and $\mathcal{A}_{N,\varepsilon}$ with the Dirichlet or Neumann boundary condition on $\partial \mathcal{O}$, respectively. Here $\varepsilon>0$ is the small parameter. The coefficients of the operators are periodic and depend on $\mathbf{x}/\varepsilon...

Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new auxiliary functions, which have definite properties. Several examples show that proposed approach can be useful in solving different physical problems.

This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.

In this paper a special type of difference equations is investigated. The impulses start abruptly at some points and their action continue on given finite intervals. This type of equations is used to model a real process. An algorithm, namely, the monotone iterative technique is suggested to solve the initial value problem for nonlinear difference equations with non-instantaneous impulses approximately. An important feature of our algorithm is that each successive approximati...

In this work, we introduce a new difference equation which is discrete analogue of Diffusion differential equation and analyze some essential spectral properties, Diffusion difference operator is self-adjoint, eigenvalues of this problem are simple and real, eigenfunctions corresponding to distinct eigenvalues, of this problem are orthogonal. Also, some useful sum representation for the linearly independent solutions of Diffusion difference equation with Dirichlet boundary co...

A technique is described in this paper to avoid order reduction when integrating reaction-diffusion initial boundary value problems with explicit exponential Rosenbrock methods. The technique is valid for any Rosenbrock method, without having to impose any stiff order conditions, and for general time-dependent boundary values. An analysis on the global error is thoroughly performed and some numerical experiments are shown which corroborate the theoretical results, and in whic...