Numerical Method for a Singularly Perturbed Differential System

Communicated by the former editorial board e se™ondEorder —™™ur—te di'eren™e s™heme is developed to solve foussinesq sysE temF por the time integr—tionD — gr—nkExi™olson type s™heme is usedF „he error estim—tes for the numeri™—l solution —re o˜t—inedF xumeri™—l results —re —lso preE sentedF AMS 2010 Subject Classication: TSwHTD TSwIPD TSwISF Key words: foussinesq systemD di'eren™e s™hemeD error estim—tesF IF INTRODUCTION We shall be concerned with developing dierence schemes for approximating the solution {u(x, t), v(x, t)} , arising in the long water waves theory. L 1 [u, v] := ∂u ∂t − ∂ 3 u ∂t∂x 2 + αu ∂u ∂x + a 0 (x, t)u + a 1 (x, t)v + a 2 (x, t) ∂v ∂x = f 1 (x, t), (1.1) (x, t) ∈ Q, L 2 [u, v] := ∂v ∂t − ∂ 3 v ∂t∂x 2 + β ∂ ∂x (uv) + b 0 (x, t)v + b 1 (x, t)u + b 2 (x, t) ∂u ∂x = f 2 (x, t), (1.2)