A numerical method to boundary value problems for second order delay differential equations

Abstract

Summary. In the first part of this paper we are dealing with theoretical statements and conditions which lead to existence and uniqueness of the solution of a nonlinear boundary value problem with delay. Next we apply this method successfully to a numerical example. The computations have been carried out at the computer Siemens 4004.

An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

“…BVPs for non-neutral delay differential equations [8,12,28,29,37,38,49,52] BVPs for non-neutral differential equations with deviating arguments [1,3,[9][10][11]19,20,50,51,53] BVPs for non-neutral differential equations with a state-dependent deviating argument θ(t, y(t)) [4] BVPs for non-neutral singularly perturbed differential equations with deviating arguments [31][32][33][34][35] BVPs for the determination of periodic solutions of non-neutral delay differential equations [15,16,18,39,54] BVPs for the determination of periodic solutions of non-neutral delay differential equations with a state-dependent delay [40] BVPs for the determination of periodic solutions of non-neutral differential equations with deviating arguments [6,7] BVPs for non-neutral Fredholm integro-differential equations [21,22,45] BVPs for non-neutral Fredholm integro-differential equations with weakly singular kernels [46][47][48] BVPs for non-neutral Volterra integro-differential equations [23] BVPs for the determination of periodic solutions of neutral delay differential equations [5,17] BVPs for neutral differential equations with st...…”

“…Under the assumption that F(·, v, β) ∈ U for any (v, β) ∈ V × R d 0 , the problem (12) can be restated in the form PAF by introducing the operator F :…”

An iterative numerical method for singularly perturbed reaction–diffusion equations with negative shift

“…Using the iterative procedure suggested in [17] for a second order ordinary delay differential equation and finite difference schemes and a finite element method available in the literature for non-delay singularly perturbed differential equations, we, in this paper, propose an iterative method to find a numerical solution for the following singularly perturbed delay differential equations of reaction-diffusion type:…”

Boundary Value Problems for Systems of Functional Differential Equations

“…If r is fixed and r ∈ (0, 1), then we need two extra values y h (a + rh) and y h (t 1 − rh) to solve (11) (see (16)); for r = 1 2 only the value y h a + 1 2 h is needed.…”

“…We can indicate that a shooting method (see for example [12], [19], [22]) and a finite difference method (see [5], [9], [12], [16], [17]) are frequently used for finding a numerical solution of problems of type (1). A collocation method [7] or iterative sequences [11] can also be used (see also [6]). …”