Applied Mathematics and Computation 274 (2016) 208–219 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Quartic and quintic B-spline methods for advection–diffusion equation Alper Korkmaz a,∗ , Idris Da ˘ g b a Mathematics Department, Çankırı Karatekin University, Çankırı, Turkey b Department of Mathematics and Computer Science, Eski ¸ sehir Osmangazi University, Eski ¸ sehir, Turkey article info Keywords: B-spline Differential quadrature method Advection–diffusion equation abstract Differential quadrature methods based on B-spline functions of degree four and five have been introduced to solve advection–diffusion equation numerically. Two initial-boundary value problems modeling the transportation of a concentration and distribution of an initial pulse are simulated using both methods. The errors of the numerical results obtained by both meth- ods have been computed. Stability analysis for both methods is also studied by the use of matrix stability. © 2015 Elsevier Inc. All rights reserved. 1. Introduction In industrialized world, a wide variety of contaminants are released to the environment everyday from industrial sources and plants. Modeling the contamination rate or transportation of contamination physically may probably the first step to solve an environment problem. The advection-diffusion equation (ADE) is a mathematical model for transport, dispersion, diffusion or intrusion in various media. Consider one-dimensional form of the ADE given by ∂U ∂ t + α ∂U ∂ x − β ∂ 2 U ∂ x 2 = 0, 0 ≤ x ≤ L (1) with the initial condition U (x, 0) = U 0 (x), 0 ≤ x ≤ L (2) and the boundary conditions U (0, t ) = f (t ), U (L, t ) = g(t ) (3) in a finite domain [0, L] where α and β are parameters, ∂U ∂ x and ∂ 2 U ∂ x 2 are advection and diffusion terms, respectively [1]. In many environment problems, U(x, t) represents concentration of the pollutant or contaminant material at point x at the time t. Some- times, the solutions refer to mass, heat, water or energy transportation in various media containing draining film or soil [2–4]. In some studies, the ADE models many engineering and chemistry problems covering dispersion in porous media, the intrusion of fluids of different densities, the absorption of chemicals, dispersion of contaminants in rivers, lakes, embouchures and coasts, flow of a solute material through a tube, the transportation of pollutants in atmosphere, cooling problems in generators, the ther- mal pollution in water systems [5–13] etc. The ADE was solved numerically as a model in some financial forecasting problems [14]. ∗ Corresponding author. Tel.: +90 545 6497717. E-mail address: alperkorkmaz7@gmail.com (A. Korkmaz). http://dx.doi.org/10.1016/j.amc.2015.11.004 0096-3003/© 2015 Elsevier Inc. All rights reserved.