Efficient computation of delay differential equations with highly oscillatory terms.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation

Unconditional Stability for Multistep ImEx Schemes: Theory

This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.Comment: 33 pages, 7 figure

Parameter Class for Solving Delay Differential Equations

In this paper a parameter class of Linear multistep method are applied to solve delay differential equations of the form y’(t) = f(t; y(t); y(t- t(t))), (t >0) subject to the initial condition y(t) = j (t) for tmin =<t <= 0 , t > 0 . The stability properties when the methods were applied to the test equation with a fixed delay t ; y’(t) = λy(t) + µy(t - t ); t ³ 0; are studied λ; µ are complex constants and Ф(t) is a continuous complex-valued function. The stability regions are plotted and numerical results are introduced

A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods

Fifty Years of Stiffness

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.Comment: 24 pages, 11 figure

F-stable and F[α,β]-stable integration/interpolation methods in the solution of retarded differential-difference equations

AbstractThe equation u̇(t)=F0u(t)+∑i=1mFiu(t−τi) is presented as an archtype (vector-valued) equation for assessing the quality of integrator/interpolator pairs used to solve retarded differential-difference equations. F-stability and F[α,β]-stability, defined with respect to this archetype equation, are proposed as desireably properties of integrator/interpolator pairs. Relationships of these properties to stability and order properties of multistep integrators and to boundedness and order properties of Lagrange interpolators are developed