SOLVING SPDDE USING FOURTH ORDER NUMERICAL METHOD

Copyright © 2021 the author(s). This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper we present a fourth order numerical method to solve singularly perturbed differentialdifference equations. The solution of this problem exhibits layer behaviour at one end. A fourth order finite difference scheme on a uniform mesh is developed. The effect of delay and advance parameters on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.


INTRODUCTION
Mathematically, any ordinary differential equation in which the highest derivative is multiplied by a small positive parameter and containing at least one shift term(delay or advance) is known as singularly perturbed differential-difference equation (SPDDE). Such problems arise frequently in the study of human pupil light reflex [1], control theory [2], mathematical biology [3], etc. The mathematical modelling of the determination of the expected time for the generation of action potentials innerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential-difference equation with small shifts. Different numerical methods were proposed to solve singularly perturbed problems by Roberts [4], Bender and Orszag [5], O'Malley [6], and Miller et al. [7]. In [8], Lange and Miura considered boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts. The analyses of the layer equations using Laplace transform lead to novel results. Numerical study for approximating the solution of SPDDE given by Kadalbajoo and Sharma [9] with mixed shifts. In [10] Kadalbajoo and Kumar presented a technique based on piecewise uniform mesh and quasilinearization process for SPDDE with small shifts.
Chakravarthy and Rao [11] proposed a modified fourth order Numerov method is presented for solving singularly perturbed differential-difference equations of mixed type. Authors constructed a special type of mesh, so that the terms containing shift lie on nodal points after discretization.
This finite difference method works nicely when the delay parameter is smaller or bigger to perturbation parameter. In [12], Ravi Kanth and Murali has given a numerical method based on parametric cubic spline for a class of nonlinear singularly perturbed delay differential equations.
Quasilinearization process is applied to reduce the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations.
To handle the delay term, they have constructed a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. RaviKanth and Murali [13] discussed an exponentially fitted spline method for singularly perturbed convection delay problems

DESCRIPTION OF THE METHOD
Think about SPDDE along with little delay and additionally advance parameters of the kind: By utilizing Taylor series almost the aspect x, the deviating argument conditions may be taken as Using Eq.(4) and Eq. (5) in Eq. (1) we receive an asymptotically equal singularly perturbed boundary value problem of the type: Since0 < << 1 and 0 < << 1, The transition from Eq.(1) to Eq.(6) is admissible. Further details on the validity of this transition is found in El'sgol'ts and Norkin [14].

NUMERICAL EXAMPLES
The exact solution of Eq. (1)  The numerical results are given in Table 1 & Fig.1.  The numerical results are given in Table 2 & Fig. 2.     The numerical results are given in Table 5 & Fig. 5. The numerical results are given in Table 6 & Fig. 6.     1103 SOLVING SPDDE USING FOURTH ORDER NUMERICAL METHOD

DISCUSSIONS AND CONCLUSION
We have discussed fourth order numerical method to solve singularly perturbed differential-difference equations exhibiting one end layer behaviour. To analyze the effect of the parameters on the solution, the numerical results have been plotted using graphs. From the graphs (Fig.1 -Fig. 4), we observed that when the solution of the boundary value problem exhibits layer behaviour on the left side, the affect of delay or advance on the solution in the boundary layer region is negligible, while in the outer region it is considerable and the change in advance affects the solution similarly as the change in delay effects, but reversely. From the graphs (Fig. 5 -Fig. 8), 1106 V. VIDYASAGAR, K. MADHULATHA, B. RAVINDRA REDDY we observed that when the solution of the boundary value problem exhibits layer behaviour on the right side, the changes in delay or advance affect the solution in the boundary layer region as well as the outer region. The thickness of the layer increases as the size of the delay increases while it decreases as the size of the advance increases.