Abstract
AbstractThe initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a difference scheme which consists of the implicit-Euler method for the time derivative and a central difference method for the spatial derivative on uniform mesh. The time domain is discretized with a nonuniform grid generated by equidistributing a positive monitor function. The performance of the numerical scheme is tested which confirms the expected behavior of the method. The existing method is compared with other methods available in the recent literature.
Highlights
This paper concerns with the following singularly perturbed pseudo-parabolic initial boundary value problem (IBVP) in the domain D = Ω [, T], Ω = (, l), D = Ω (, T], Lu := εL ∂u ∂t − ∂ ∂x a(x, t) ∂u ∂x + b(x, t)+c(x, t)u = f (x, t), (x, t) ∈ D, (1.1) u(x,) = s(x), x ∈ Ω, u(
The initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a di erence scheme which consists of the implicit-Euler method for the time derivative and a central di erence method for the spatial derivative on uniform mesh
The proposed scheme consists of the central di erence scheme for the spatial derivative on an uniform mesh and implicit Euler scheme for the time derivative on an adaptively generated nonuniform mesh
Summary
The equation (1.1) is an example of Sobolev equation characterized by having mixed time and space derivatives appearing in the highest order terms. These type of problems were rst studied by Sobolev. Various types of numerical methods for a parameter-free version of IBVP(1.1) has been studied by many researchers. Very limited literature exists [1, 3] for the problem of type (1.1) with the presence of the parameter ε which makes it singularly perturbed in nature. Duru [10] analyzed Sobolev type equations involving a single space variable through a nite-di erence method to tackle the boundary layers. The nonuniform mesh is so generated that we hardly need any aprior information of the solution which
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