In this study, a decomposition method for approximating the solutions of unsteady convection-diffusion problems is implemented. The approximate solution is calculated in the form of a convergent series with easily computable components. | Turk J Math 32 (2008) , 51 – 60. ¨ ITAK ˙ c TUB A Decomposition Method for Solving Unsteady Convection-Diffusion Problems Shaher Momani Abstract In this study, a decomposition method for approximating the solutions of unsteady convection-diffusion problems is implemented. The approximate solution is calculated in the form of a convergent series with easily computable components. The calculations are accelerated by using the noise terms phenomenon for nonhomogeneous problems. Numerical examples are investigated to illustrate the pertinent features of the proposed algorithm. Key Words: Convection-diffusion equation; Decomposition method; Noise terms. 1. Introduction Consider the following convection-diffusion equation: ∂u ∂t ∂u ∂2u ∂ 2u = f(t, x, y), + b1 (x, y) ∂u + b (y) − a + a 2 1 2 2 2 ∂x ∂y ∂x ∂y u(x, y, t) = g1 (x, t), u(x, y, 0) = g (x, y), 2 in Ω × J, on ∂Ω × J, () in Ω, where Ω = (0, 1) × (0, 1), J = (0, T ), b1 (x, y), b2 (y) are smooth functions and a1 , a2 are positive constants. This equation may be seen in computational hydraulics and fluid dynamics to model convection-diffusion of quantities such as mass, heat, energy, vorticity, 51 MOMANI etc. [1]. Several numerical methods have been proposed to solve convection-diffusion problems approximately. Among them are restrictive Taylor’s approximation [2], the alternating direction implicit (ADI) method [3], the upwind method [4], and the explicit predictor method [5]. Recently, the Adomian decomposition method (in short, ADM) [6, 7] has emerged as an alternative method for solving a wide ranges of problems whose mathematical models involve algebraic, differential, integro-differential, and partial differential equations. The decomposition method yields rapidly convergent series solutions for both linear and nonlinear deterministic and stochastic equations. The technique has many advantages over the classical techniques, mainly, it avoids discretization and .
