scholarly journals Numerical solutions of a system of singularly perturbed reaction-diffusion problems

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4889-4905
Author(s):  
Ali Barati ◽  
Ali Atabaigi
Keyword(s):  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Interior Layers ◽  
Boundary And Interior Layers ◽  
Diffusion Problems

This paper addresses the numerical approximation of solutions to a coupled system of singularly perturbed reaction-diffusion equations. The components of the solution exhibit overlapping boundary and interior layers. Sinc procedure can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. Also the obtained results show that the proposed method is applicable even for small perturbation parameter as ? = 2-30. The convergence analysis of proposed technique is discussed, it is shown that the approximate solutions converge to the exact solutions at an exponential rate. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the method.

A FEM FOR A SYSTEM OF SINGULARLY PERTURBED REACTION-DIFFUSION EQUATIONS WITH DISCONTINUOUS SOURCE TERM

2013 ◽  
Vol 10 (05) ◽  
pp. 1350057
Author(s):  
A. RAMESH BABU ◽  
N. RAMANUJAM
Keyword(s):  
Reaction Diffusion ◽  
Coupled System ◽  
Perturbation Parameter ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Shishkin Meshes ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Discontinuous Source Term ◽  
Interior Layers

In this paper, we consider a weakly coupled system of two reaction-diffusion equations with discontinuous source terms. When a parameter multiplying the second order derivatives in the equations is small, their solutions exhibit boundary layers as well as interior layers. A numerical method based on finite element and Shishkin and Bakhvalov–Shishkin meshes is presented. We derive an error estimate of order O(N-1ln N) in the energy norm with respect to the perturbation parameter. Numerical experiments are also presented to support our theoritical results.

scholarly journals Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

2001 ◽  
Vol 1 (3) ◽  
pp. 298-315 ◽  
Author(s):  
Grigorii Shishkin
Keyword(s):  
Reaction Diffusion ◽  
Boundary Function ◽  
Perturbation Parameter ◽  
Continuity Condition ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Type ◽  
Local Accuracy ◽  
Time Variables ◽  
The Right

AbstractIn this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.

scholarly journals Solving Systems of Singularly Perturbed Convection Diffusion Problems via Initial Value Method

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Wondwosen Gebeyaw Melesse ◽  
Awoke Andargie Tiruneh ◽  
Getachew Adamu Derese
Keyword(s):  
Approximate Solution ◽  
Coupled System ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Convection Diffusion ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
The Given ◽  
Diffusion Problems

In this paper, an initial value method for solving a weakly coupled system of two second-order singularly perturbed Convection–diffusion problems exhibiting a boundary layer at one end is proposed. In this approach, the approximate solution for the given problem is obtained by solving, a coupled system of initial value problem (namely, the reduced system), and two decoupled initial value problems (namely, the layer correction problems), which are easily deduced from the given system of equations. Both the reduced system and the layer correction problems are independent of perturbation parameter, ε. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the problem. Further, error estimates are derived and examples are provided to illustrate the method.

scholarly journals CONSERVATIVE NUMERICAL METHOD FOR A SYSTEM OF SEMILINEAR SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS

2009 ◽  
Vol 14 (2) ◽  
pp. 211-228 ◽  
Author(s):  
Lidia Shishkina ◽  
Grigorii Shishkin
Keyword(s):  
Unit Length ◽  
Reaction Diffusion ◽  
Perturbation Parameter ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Finite Difference Schemes ◽  
Vertical Strip ◽  
Parabolic Boundary ◽  
Conservative Numerical Method

On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction‐diffusion equations connected only by terms that do not involve derivatives. The highest‐order derivatives in the equations, having divergent form, are multiplied by the perturbation parameter e ; ϵ ∈ (0,1]. When ϵ → 0, the parabolic boundary layer appears in a neighbourhood of the strip boundary. Using the integro‐interpolational method, conservative nonlinear and linearized finite difference schemes are constructed on piecewise‐uniform meshes in the x 1 ‐axis (orthogonal to the boundary) whose solutions converge ϵ‐uniformly at the rate O (N1−2 ln2 N 1 + N 2 −2 + N 0 −1). Here N 1 + 1 and N 0 + 1 denote the number of nodes on the x 1‐axis and t‐axis, respectively, and N 2 + 1 is the number of nodes in the x 2‐axis on per unit length.

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