Abstract
A numerical approach is suggested for the layer behaviour differential-difference equations with small and large delays in the differentiated term. Using the non-polynomial spline, the numerical scheme is derived. The discretization equation is constructed using the first-order derivative continuity at non-polynomial spline internal mesh points. A fitting parameter is introduced into the scheme with the help of the singular perturbation theory to minimize the error in the solution. The maximum errors in the solution are tabulated to verify the competence of the numerical method relative to the other methods in literature. We also focussed on the impact of large delays on the layer behaviour or oscillatory behaviour of solutions using a special mesh with and without fitting parameter in the proposed scheme. Graphs show the effect of the fitting parameter on the solution layer.
Highlights
Differential- difference equations are problems in which the time evolution of the state variable can depend on specific past in some arbitrary way, i.e., the rate of physical system change depends on the state of the physical system and on its history [1]
Such problems occur when many practical phenomena are modelled such as thermo-elasticity [2], hybrid optical system [3], in population dynamics [14], red blood cell system [17], in models for physiological processes [18], predator-prey models [19], optimal control theory [7] and in the potential in nerve cells by synaptic inputs in dendrites [26]
Ravikanth and Murali [23] have proposed a fitted method to solve the problems through tension splines, which only contains a delay in the differentiated term
Summary
Differential- difference equations are problems in which the time evolution of the state variable can depend on specific past in some arbitrary way, i.e., the rate of physical system change depends on the state of the physical system and on its history [1] Such problems occur when many practical phenomena are modelled such as thermo-elasticity [2], hybrid optical system [3], in population dynamics [14], red blood cell system [17], in models for physiological processes [18], predator-prey models [19], optimal control theory [7] and in the potential in nerve cells by synaptic inputs in dendrites [26]. Ravikanth and Murali [23] have proposed a fitted method to solve the problems through tension splines, which only contains a delay in the differentiated term
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