The present paper sheds some light on constructing a new fourth order, fourstage parallel Rosenbrock method (NPROS4) and investigating its convergence and stability properties in view of solving stiff ordinary differential equations. The expression for local truncation error (LTE) has been derived and the parameters are chosen to minimize the coefficients in the LTE term. It is pertinent to point out here that this method possesses more free parameters to improve the stability region. The present method can also be efficiently implemented in a parallel computer having four processors with less error than the existing methods of the same order. Numerical examples are given to illustrate the efficiency of the present developed method.

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In this paper, we have solved Generalised pantograph equation which is special delay differential equation (DDE) using Natural Continuous Extension Runge-Kutta two stage fourth order Method (NCERKM). A modest effort is taken to derive NCERKM quadrature formula. Cubic Hermite Interpolation is incorporated to estimate the delay term. Numerical Results are given for various coefficients arrived.

Numerous amazing physical phenomena are much of time demonstrated by non-linear Delay Differential Equations (DDE) in the field of Science and Technology. The Mackey-Glass equation illustrates non-linear phenomena in physiological control systems in which the dynamics of the density of blood cells is considered. The Mackey-Glass equation is the representative example of delay induced chaotic behavior. The dynamics of Mackey-Glass equation examined utilizing Natural Continuous Extension (NCE) of ImplicitRunge-Kutta method of fourth order with Cubic Spline interpolation used to approximate the delay term. The commitment in this paper focuses with the derivation of coefficients for continuous Implicit Runge-Kutta method for fourth order. This method considers the minimum number of stages than Runge-Kutta Fourth order method. Numerical solution of Mackey -Glass equation simulated for a different delay yielding the periodic and chaotic dynamics.

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