Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle |
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Authors: | G I Shishkin L P Shishkina |
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Institution: | (1) Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia |
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Abstract: | The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?2, where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ?2ln2 N + N 0 ?1 , where \(N = \mathop {\min }\limits_s N_s \), N s + 1 and N 0 + 1 are the numbers of mesh points on the axes x s and t, respectively. |
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Keywords: | initial boundary value problem in a rectangle perturbation parameter ɛ system of parabolic reaction-diffusion equations finite difference approximation parabolic boundary layer a priori bounds on the solution and its derivatives ɛ -uniform convergence |
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