Papers by Enes Duvnjakovic
We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion pr... more We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion problem, () () () y x f x y p x y y L , : 2 = − ′ ′ = ε ε , on () 1 , 0 () 0 0 = y ; () 0 1 = y , where f(x,y) is non-linear function. For this problem, using spline-metod with the natural choice of function, a difference scheme, on a non-uniform mesh, is given. Constructed non-linear difference scheme has uniform convergence in points of the uneven division segments..
We consider the singularly perturbed selfadjoint one-dimensional semilinear reaction-diffusion pr... more We consider the singularly perturbed selfadjoint one-dimensional semilinear reaction-diffusion problem () () 2 : , L y y x f x y ε ε ′′ = = , on () 1 , 0 () 0 0 = y ; () 0 1 = y , where f(x,y) is a non-linear function. For this problem, using the spline-method with the natural choice of functions, a new difference scheme is given on a non-uniform mesh. The constructed non-linear difference scheme has uniform convergence in points of uneven division segments. A numerical example is given.
For singularly perturbed selfadjoint one-dimenzional reaction-diffusion problem, using the Galerk... more For singularly perturbed selfadjoint one-dimenzional reaction-diffusion problem, using the Galerkin method with the natural choice of test function, a difference scheme on a non-uniform mesh is given, wich is second-order accurate at nodes for the fixed perturbation parameter. A numerical exaple, wich illustrating the theoretical results, is included.
We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion pr... more We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion problem, 2 L y: = ε y′ x − p y x = f x, y 0, 1 ( ) ( ) ( ) , on () ε y ( 0) = 0; ( 1) = 0 y, where f(x,y) is non-linear function. For this problem, using spline-metod with the natural choice of function, a difference scheme, on a non-uniform mesh, is given. Constructed non-linear difference scheme has uniform convergence in points of the uneven division segments.. Key words. Non-linear reaction-diffusion problem, difference scheme, singular perturbation problem 1.
We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion pr... more We consider the singularly perturbed selfadjoint one-dimenzional non-linear reaction-diffusion problem, 2 L y: = ε y′ x − p y x = f x, y 0, 1 ( ) ( ) ( ) , on () ε y ( 0) = 0; ( 1) = 0 y, where f(x,y) is non-linear function. For this problem, using spline-metod with the natural choice of function, a difference scheme, on a non-uniform mesh, is given. Constructed non-linear difference scheme has uniform convergence in points of the uneven division segments.. Key words. Non-linear reaction-diffusion problem, difference scheme, singular perturbation problem 1.
arXiv: Numerical Analysis, Dec 4, 2017
In this paper we consider two difference schemes for numerical solving of a one-dimensional singu... more 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 Shiskin mesh. Finally, we present four numerical experiments to confirm the theoretical results.
In this paper we consider two difference schemes for numerical solving of a one--dimensional sing... more 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.
In this paper we consider the semilinear singularly perturbed reaction--diffusion boundary value ... more 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.
Journal of Modern Methods in Numerical Mathematics, 2017
We consider an approximate solution for the one--dimensional semilinear singularly--perturbed bou... more 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 \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.
In this paper we consider the semilinear singularly perturbed reaction--diffusion boundary value ... more 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.
Journal of Modern Methods in Numerical Mathematics, 2017
We consider an approximate solution for the one--dimensional semilinear singularly--perturbed bou... more 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 \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.
Journal of Modern Methods in Numerical Mathematics, 2015
We are considering a semilinear singular perturbation reaction-diffusion boundary value problem w... more We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is ϵ-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.
Journal of Modern Methods in Numerical Mathematics, 2015
We are considering a semilinear singular perturbation reaction -diffusion boundary value problem ... more We are considering a semilinear singular perturbation reaction -diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is ϵ-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.
arXiv: Numerical Analysis, 2018
In this paper we consider the numerical solution of a singularly perturbed one-dimensional semili... more 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 $\\varepsilon$--uniform convergence of the second order $\\mathcal{O}\\left(1/N^2 \\right),$ 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.
For singularly perturbed selfadjoint one-dimensional reaction-diffusion problems, using the Galer... more For singularly perturbed selfadjoint one-dimensional reaction-diffusion problems, using the Galerkin method with an exponential test function, a class of difference schemes is given, which is second-order accurate at nodes for the fixed perturbation parameter. A numerical example is included.
We consider the singularly perturbed selfadjoint one-dimensional semilinear reaction-diffusion pr... more We consider the singularly perturbed selfadjoint one-dimensional semilinear reaction-diffusion problem () () 2 : , L y y x f x y ε ε ′′ = = , on () 1 , 0 () 0 0 = y ; () 0 1 = y , where f(x,y) is a non-linear function. For this problem, using the spline-method with the natural choice of functions, a new difference scheme is given on a non-uniform mesh. The constructed non-linear difference scheme has uniform convergence in points of uneven division segments. A numerical example is given.
Journal of Modern Methods in Numerical Mathematics, 2019
In this paper we consider the numerical solution of a singularly perturbed one-dimensional semili... more In this paper we consider the numerical solution of a singularly perturbed one-dimensional semilinear reaction-diffusion problem. We construct a class of finite-difference schemes to discretize the problem and we prove that the discrete system has a unique solution. The central result of the paper is second-order convergence uniform in the perturbation parameter, which we obtain for the discrete approximate solution on a modified Bakhvalov mesh. Numerical experiments with two representatives of the class of difference schemes show that our method is robust and confirm the theoretical results.
Advances in Mathematics: Scientific Journal, 2015
In this work we consider the singularly perturbed one-dimensional semi-linear reaction-diusion pr... more In this work we consider the singularly perturbed one-dimensional semi-linear reaction-diusion problem " y (x) = f (x; y); x 2 (0; 1) ; y(0) = 0; y(1) = 0; where f is a nonlinear function. Here the second-order derivative is multiplied by a small positive parameter and consequently, the solution of the problem has boundary layers. A new dierence scheme is constructed on a modied Shishkin mesh with O(N) points for this problem. We prove existence and uniqueness of a discrete solution on such a mesh and show that it is accurate to the order of N ln N in the discrete maximum norm. We present numerical results that verify this rate of convergence.
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Papers by Enes Duvnjakovic