Numerical method for a singularly perturbed convection–diffusion problem with delay

• Published in: Applied mathematics and computation Vol. 216; no. 8; pp. 2351 - 2359
• Main Authors: Amiraliyev, Gabil M., Cimen, Erkan
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.06.2010; Elsevier
• Subjects: Boundary value problem; Convergence; Delay; Delay differential equation; Differential equations; Exact sciences and technology; Fitted difference method; Integrals; Mathematical analysis; Mathematical models; Mathematics; Norms; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Partial differential equations, boundary value problems; Perturbation methods; Sciences and techniques of general use; Sequences, series, summability; Singular perturbation; Special functions; Boundary value problem; Fitted difference method; Delay differential equation; Singular perturbation; Second order equation; Difference scheme; Grid pattern; Differential equation; Convection diffusion equation; Numerical method; Stochastic method; Delay equation; Exponential function; Partial differential equation; Convergence; Numerical approximation; Numerical analysis; Linear equation; Cubature; Applied mathematics; Numerical solution; Quadrature formula
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.03.080

This paper deals with the singularly perturbed boundary value problem for a linear second-order delay differential equation. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that one gets first order convergence in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.