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In this paper we consider two difference schemes for numerical solving of a one--dimensional singularly perturbed boundary value problem. We proved an $\varepsilon$--uniform convergence for both difference schemes on a Shiskin mesh. Finally, we present four numerical experiments to confirm the theoretical results.
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Citations
... 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]. ...
... where y = (y 0 , y 1 , . . . , y N ) T the numerical solution of the problem (1a)-(1b), obtained by using the difference scheme (15). Now, we can state and prove the theorem of stability. ...
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]. ...
... Comparing these results to the ones obtained in [9] related to the discrete approximate solution, we can conclude that the rate of convergence of the cubic spline global approximate solution on the Shishkin mesh is of order O(ln 2 N/N 2 ) within the boundary layer, while outside of the layer it is of order O(1/N). ...
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.
... In [9,18] authors constructed new difference schemes and proved the uniqueness of the numerical solution and an ε-uniform convergence on a modified Shishkin mesh, and at the end presented numerical experiments, others results are in [19,20] and [5,6,7,8,17]. ...
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.
... 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.
The book was written on the basis of materials that we presented at several faculties, either as lectures or as part of auditory exercises. Aware that there are more books and textbooks in the area in which the topics covered by this book are covered, we tried, based on the mentioned experience, to write a book oriented towards students.
