Abstract
In this paper, the boundary value problem for second order singularly perturbed delay differential equation is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by cubic spline v h defined by its values v i = v h (t i ) at grid points t i , i = 0, 1, . . . , N. The necessary for construction the cubic spline and missing the first derivatives at the boundary are replaced by the derivatives of the corresponding interpolating polynomials matching the grid points values nearest to the boundary points. An approximation of the solution is obtained by minimization techniques applied to a function whose arguments are the grid point values of the sought spline. The results of numerical experiments with two boundary value problems for the second order singularly perturbed delay differential equations as well as their comparison with the results of other methods employed by other authors are also provided.
Highlights
The problems we consider in this paper are boundary value problems for second order singularly perturbed delay differential equations of the form y (x) = f x, y(x), y (x), y α(x), a ≤ x ≤ b, y(x) = φ(x) for x ≤ a, y(b) = ψ, (1.1) (1.2)Z
We presented a fixed-point approach to solve boundary value problem for second order singularly perturbed delay differential equation (1.1)–(1.2)
The proposed method can be extended in a way similar to that used in [3] so that it can be applied to systems of singularly perturbed delay differential equations with general linear boundary conditions
Summary
F : D → R, D = (t, z1, z2, z3) : a ≤ t ≤ b, −∞ ≤ zi ≤ +∞ , α : [a, b] → [β, b], β ≤ a, φ : [β, a] → R, are continuous and φ(a) = 0 It was discussed in [3] as an approximate method of solution of more general boundary value problems for the system of DDEs of the form dy(t) dt = f t, y(t), y(t − τ1), . An approximate solution yh was recovered through the integral form of the solution to the above-mentioned linear problem and assumed the form of a cubic (or quadratic) spline It was interesting for us whether we can apply this approach to solving boundary value problems for singularly perturbed DDEs with such a success and without additional difficulties which introduce the usage of nonuniform grids. The operator A : L2[a, b] → L2[a, b] defined by (2.12) is continuous
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
