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Difference Scheme for Semilinear Reaction-Diffusion Problems on a Mesh of Bakhvalov Type
Abstract
The paper examines a semilinear singular reaction-diffusion problem. Using the col-
location method with naturally chosen splines of exponential type, a new difference scheme on a mesh of Bakhvalov type is constructed. A difference scheme generates the system of non-linear equations, and the theorem of existence and this system’s solution uniqueness is also provided. At the end, a numerical example, is given as well, which points to the convergence of the numerical solution to the exact one.
... Using the method of [1], we constructed new difference schemes in [3,4] and we carried out numerical experiments. ...
... A construction of a difference scheme, which will be used for calculation of the approximate solution of the problem (1)- (3) in the mesh points, is based on the representation of the exact solution on the interval [x i , x i+1 ], i = 0, . . . , N − 1 ...
... An answer to the question of existence and uniqueness will be given in the next theorem, however before that, it is necessary to define the operator (or discrete problem) F : R N+1 → R N+1 and a corresponding norm that is necessary in formulation of the theorem. Therefore, we will now use the difference scheme (8) in order to obtain a discrete problem of the problem (1)- (3). We have that ...
We consider an approximate solution for the one–dimensional semilinear singularly–perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green’s function. We present an "epsilon–uniform convergence of such gained the approximate solutions, in the maximum norm. After that, the constructed approximate solution is repaired and we obtain a solution,
which also has "epsilon–uniform" convergence. In the end a numerical
experiment is presented to confirm previously shown theoretical results.
... Many authors have analyzed and made a great contribution to the study of the problem (1.1)-(1. 3) with different assumptions about the function f ; and as well as more general nonlinear problems. ...
... Using the method of [2], authors constructed new difference schemes in papers [3] and [4] for the problem (1.1)-(1.3) and carried out numerical experiments. ...
... Many authors have analyzed and made a great contribution to the study of the problem (1.1)-(1. 3) with different assumptions about the function f ; and as well as more general nonlinear problems. ...
... Using the method of [2], authors constructed new difference schemes in papers [3] and [4] for the problem (1.1)-(1.3) and carried out numerical experiments. ...
In this paper we consider the numerical solution of a singularly perturbed one-dimensional semilinear reaction-diffusion problem. A class of differential schemes is constructed. There is a proof of the existence and uniqueness of the numerical solution for this constructed class of differential schemes. The central result of the paper is an --uniform convergence of the second order for the discrete approximate solution on the modified Bakhvalov mesh. At the end of the paper there are numerical experiments, two representatives of the class of differential schemes are tested and it is shown the robustness of the method and concurrence of theoretical and experimental results.
... was first developed by Boglaev [1], who constructed a difference scheme and showed convergence of order 1 on the modified Bakhvalov mesh. Using the method of [1], we constructed new difference schemes in [3,4] and we carried out numerical experiments. ...
We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an -uniform convergence of such gained the approximate solutions, in the maximum norm of the order on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has --uniform convergence, but now of order on [0,1]. In the end a numerical experiment is presented to confirm previously shown theoretical results.
... The author's results in the numerical solving of the problem (1a)-(1b) and others results can be seen in [2], [3], [5], [6], [7], [12], [4], [13], [16], [15], [14], [27], [26]. ...
A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified Bakhvalov meshes are included in the numerical experiments as well.
... [14]. Other results obtained in a similar fashion to the one presented in this paper are given in [2,3,4,9,10]. ...
In this paper we consider the semilinear singularly perturbed reaction--diffusion boundary value problem. In the first part of the paper a difference scheme is given for the considered problem. In the main part of the paper a cubic spline is constructed and we show that it represents a global approximate solution of the our problem. At the end of the paper numerical examples are given, which confirm the theoretical results.
... These difference schemes were constructed using the method first introduced by Boglaev [1], who constructed a difference scheme and showed convergence of order 1 on a modified Bakhvalov mesh. In our previous papers using the method [1], we constructed new difference schemes in [3,4,10,6,7,8,9,13] and performed numerical tests, in [5,11] we constructed new difference schemes and we proved the theorems on the uniqueness of the numerical solution and the ε-uniform convergence on the modified Shishkin mesh, and again performed the numerical test. In [12] we used the difference schemes from [11] and calculated the values of the approximate solutions of the problem (1.1)-(1.3) ...
... These difference schemes were constructed using the method first introduced by Boglaev [1], who constructed a difference scheme and showed convergence of order 1 on a modified Bakhvalov mesh. In our previous papers using the method [1], we constructed new difference schemes in [3,4,10,6,7,8,9,13] and performed numerical tests, in [5,11] we constructed new difference schemes and we proved the theorems on the uniqueness of the numerical solution and the ε-uniform convergence on the modified Shishkin mesh, and again performed the numerical test. In [12] we used the difference schemes from [11] and calculated the values of the approximate solutions of the problem (1.1)-(1.3) ...
In this paper we consider two difference schemes for numerical solving of a one-dimensional singularly perturbed boundary value problem. We proved an ε-uniform convergence for both difference schemes on a Shishkin mesh. Finally, we present four numerical experiments to confirm the theoretical results.
In this paper, the numerical solution of the singular-perturbation Cauchy problem by Runge-Kutta methods on the Shishkin grid is discussed. Numerical solutions of the observed problem were obtained using two explicit and one implicit Runge-Kutta method on the simplest layer-adaptive network. Finally, the obtained results were compared.
In the present paper we consider the numerical solving of a semilinear singular--perturbation reaction--diffusion boundary--value problem having boundary layers. A new difference scheme is constructed, the second order of convergence on a modified Shishkin mesh is shown. The numerical experiments are included in the paper, which confirm the theoretical results.
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