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Construction of the Difference Scheme for Semilinear Reaction-Diffusion Problem on a Bakhvalov Type Mesh
... 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.
... 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. ...
... Using the method of [2], authors constructed new difference schemes in papers [4] and [3] 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.
... 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.
... The exact solution of this problem is given by 6 8.1585e − 04 2.00 2.8932e − 03 2.02 2.5827e − 02 2.05 2 7 2.7762e − 04 2.00 9.7397e − 04 2.01 8.5547e − 03 1.96 2 8 9.0650e − 05 2.00 3.1625e − 04 2.00 2.8566e − 03 1.99 2 9 3.5410e − 05 2.00 1.2353e − 04 2.00 1.2111e − 03 2.00 2 10 1.5738e − 05 2.00 5.4904e − 05 2.00 4.9827e − 04 2.00 2 11 ...
In this paper we consider two difference schemes for numerical solving of a one--dimensional singularly perturbed boundary value problem. We proved an --uniform convergence for both difference schemes on a Shiskin 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.
Diferentna schema za nelinearni singularno-perturbacioni problem
- Enes Duvnjaković
enes duvnjaković, nermin okičić, Diferentna schema za nelinearni singularno-perturbacioni problem, Godišnjak PMF
1, 53-60 (2004)
Difference Scheme for Semilinear Reaction-Diffusion Problem, 14 th International Research/Expert Conference "Trends in the Development of Machinery and Associated Technology
enes duvnjaković, nermin okičić, samir karasuljić, Difference Scheme for Semilinear Reaction-Diffusion Problem,
14 th International Research/Expert Conference "Trends in the Development of Machinery and Associated
Technology"TMT 2010, Mediterranean Cruise (2010)