Roumen Anguelov

Nonstandard finite difference method by nonlocal approximation

Two types of monotonic properties of solutions of differential equations are discussed and general finite difference schemes, which are stable with respect to these properties are investigated. Apart from being elementary stable, these schemes are also shown to preserve qualitative properties of nonhyperbolic fixed points of the differential equations. From the practical point of view, a systematic procedure based on nonlocal approximation, is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.

Dedekind Order Completion of C (X) by Hausdorff Continuous Functions

The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set Hft (X) of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set C b(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and C b(X) are characterised as subsets of Hft(X). This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs.

Solving large classes of nonlinear systems of PDEs

The essentials of a new method in solving very large classes of nonlinear systems of PDEs, possibly associated with initial and/or boundary value problems, are presented. The PDEs can be defined by continuous, not necessarily smooth expressions, and the solutions obtained cab be assimilated with usual measurable functions, or even with Hausdorff continuous ones. The respective result sets aside completely, and with a large nonlinear margin, the celebrated 1957 impossibility of Hans Lewy regarding the nonexistence of solution in distributions of large classes of linear smooth coefficient PDEs.

The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions

Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis.

Order convergence structure on C(X)

This paper brings together three concepts which have not been related so far, namely, the concept of order convergence, the concept of convergence space and the concept of Hausdorff continuous functions. The order convergence on a poset P, which is generally not a topological convergence, can be studied through the concept of convergence space. Indeed, under certain mild assumptions there exists a convergence structure on P which induces the order convergence. In particular, the result is true for any vector lattice. The primary focus is on the set C(X) of all continuous real functions on a topological space X. The vector lattice C(X) gives a typical example when the order convergence cannot be induced by a topology, thus justifying our interest in the convergence vector structure inducing the order convergence. The completion of the respective convergence vector space is obtained through Hausdorff continuous functions.

Qualitatively stable finite difference schemes for advection-reaction equations

A systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection-reaction equations. In the case of hyperbolic fixed-points, a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions. Non-hyperbolic fixed-points are described with the help of two new monotonic properties the construction of schemes, which preserve these properties, being done by nonlocal approximation of nonlinear terms in the reaction terms.

On nonstandard finite difference schemes in biosciences

We design, analyze and implement nonstandard finite difference (NSFD) schemes for some differential models in biosciences. The NSFD schemes are reliable in three directions. They are topologically dynamically consistent for onedimensional models. They can replicate the global asymptotic stability of the disease-free equilibrium of the MSEIR model in epidemiology whenever the basic reproduction number is less than 1. They preserve the positivity and boundedness property of solutions of advection-reaction and reaction-diffusion equations.

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R 0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough, R 0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a specific algebraic decomposition into infective and non-infected humans and mosquitoes. Stability results are extended and the possibility of backward bifurcation is clarified. A dynamically consistent nonstandard finite difference scheme is designed.

Comparison of Some Standard and Nonstandard Numerical Methods for the MSEIR Epidemiological Model

The paper presents some non‐standard finite difference methods for the MSEIR model. Following a short description of the methods, they are compared with the standard methods usually implemented in scientific software. It is shown that the non‐standard schemes are superior to the standard ones with respect to their qualitative properties and the CPU running time. We show that non‐local approximations of non linear terms in the equations can be of utmost importance to capture the dynamic of the system.