Types Of Differential-difference Equations Research Articles

Novel approach to solve singularly perturbed boundary value problems with negative shift parameter

Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for h≥ε in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence.

Delay differential equations as a tool for mathematical modelling of population dynamic

The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University. 
 Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture).
 In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson.
 Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system.
 This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function.
 Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.

Meromorphic solutions of certain type of differential-difference equations

We consider meromorphic solutions to general differential-difference equation of type L ( z , f ) + ∑ j = 1 q c j ( z ) f n j ( z + ζ j ) = ∑ j = 1 r γ j e λ j z , where L ( z , f ) ≢ 0 is a linear differential-difference polynomial of f with small function coefficients, and 1 < n 1 < ⋯ < n q , γ j ( 1 ≤ j ≤ r ) are nonvanishing constants, c j ( z ) ( 1 ≤ j ≤ q ) are nonvanishing small functions. If the equation admits a transcendental meromorphic solution f of finite order satisfying that λ ( f ) < σ ( f ) , λ ( 1 / f ) < σ ( f ) , then representation of f and relations between coefficients in ψ ( z , f ) and γ i , λ j can be obtained.

ON ENTIRE SOLUTIONS OF CERTAIN TYPE OF DIFFERENTIAL-DIFFERENCE EQUATIONS

In this paper, we deal with differential-difference equations of the form $$ f(z)^2+p(z)f(z+c)+h(z)f'(z)+g(z)=d_1e^{\lambda z}+d_2e^{-\lambda z} $$ where $p(z),~ h(z),~ g(z)$ are polynomials, and $c,~ d_1,~d_2, ~\lambda\in \mathbb{C}$ are constants with $d_1 d_2\lambda\not= 0$. By utilizing Nevanlinna's value distribution theory, some sufficient conditions on the nonexistence of entire solutions regarding the equations are provided.

EXACT SOLUTIONS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS BY BÄCKLUND TRANSFORMATION OF RICCATI EQUATION

Based on a Bäcklund transformation of the Riccati equation and its known soliton solutions, we obtain in this paper some exact traveling-wave solutions, including triangle function solutions and hyperbolic function solutions, of a hybrid lattice equation. The proposed method can be easily extended to locate exact solitary wave solutions for other types of differential-difference equations.

On Robust Stability of Positive Infinite-Dimensional Linear Systems

It is shown that for linear infinite-dimensional continuous-time positive systems the complex and real stability radius with respect to the single perturbation structure coincide. Furthermore, a simple formula for the complex stability radius of positive systems is derived by using the associated transfer function. These results generalize the finite-dimensional counterparts due to Hinrichsen and Son. The results are illustrated with an example dealing with a certain type of differential-difference equations.