International Journal of Computer Mathematics, 1989
In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear ... more In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear two-point boundary-value problems. First, Newton's iteration is used to linearize such problems, reducing these to a sequence of linear singularly perturbed two-point boundary-value problems. Then,a difference scheme is applied to solve the linear systems. The difference scheme is accurate up to O(h 2). Test examples are included to demonstrate the efficiency of the method.
Turkish journal of mathematics & computer science, Jun 30, 2022
In this paper, singularly perturbed quasilinear boundary value problems are taken into account. W... more In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With this purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules are used to establish the numerical scheme. Then, an error estimate is derived. A numerical experiment is demonstrated to verify the theory.
Journal of Computational and Applied Mathematics, Oct 1, 2005
We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order sin... more We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.
This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobo... more This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobolev boundary value problems with Robin type boundary condition. Firstly, the asymptotic behavior of the solution is analyzed. By using interpolating quadrature rules and basis functions, a completely exponentially fitted tree-level difference scheme is constructed on the uniform mesh. Then an error estimation is investigated in a discrete energy norm. Two numerical examples are solved and the computational results are tabulated.
The generalized with respect to ðb; cÞ-distance Riesz potential defined on Sobolev space W l p ðR... more The generalized with respect to ðb; cÞ-distance Riesz potential defined on Sobolev space W l p ðR n Þ is constructed and for this potential the theorem of Hardy-Littlewood-Sobolev type has been established.
Journal of Applied Mathematics and Computational Mechanics, Dec 1, 2020
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptot... more In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
We study uniform finite-difference method for solving first-order singularly perturbed boundary v... more We study uniform finite-difference method for solving first-order singularly perturbed boundary value problem (BVP) depending on a parameter. Uniform error estimates in the discrete maximum norm are obtained for the numerical solution. Numerical results support the theoretical analysis. ε > 0 is a small parameter and {u(x), λ} is a solution. For ε 1, the function u(x) has a boundary layer of thickness O(ε) near x = 0. Under the above conditions, there exists a unique solution to problem (1.1), (1.2) (see [7, 12]). An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [6
We consider a uniform finite difference method on a B-mesh is applied to solve a singularly pertu... more We consider a uniform finite difference method on a B-mesh is applied to solve a singularly perturbed quasilinear boundary value problem (BVP) depending on a parameter. We give a uniform first-order error estimates in a discrete maximum norm. Numerical results are presented that demonstrate the sharpness of our theoretical analysis.
A finite-difference method is suggested and analyzed for evolution equations of Sobolev type in a... more A finite-difference method is suggested and analyzed for evolution equations of Sobolev type in a single space variable and with boundary layers. The convergence and error estimates for an exponentially fitted difference scheme in an equidistant mesh are obtained. The numerical methods discussed here are extremely robust, in the sense that they have good convergence properties not only for small values of the perturbation parameter, but also for moderate and large values.
Numerical Methods for Partial Differential Equations, Aug 1, 2019
A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value pro... more A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.
Journal of Difference Equations and Applications, Mar 2, 2022
We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the as... more We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete maximum norm. Finally, three numerical examples are solved to validate the theory.
Erzincan University Journal of Science and Technology, Mar 24, 2019
Bu çalışmada singüler pertürbe özellikli lineer olmayan reaksiyon-difüzyon sınır değer problemi e... more Bu çalışmada singüler pertürbe özellikli lineer olmayan reaksiyon-difüzyon sınır değer problemi ele alınmıştır. Kalan terimi integral biçiminde olan ve baz fonksiyonu içeren interpolasyon kuadratür kuralları kullanılarak parçalı düzgün şebeke üzerinde fark şeması kurulmuştur. Sunulan metodun kararlı olduğu gösterilmiş ve yakınsaklık analizi yapılmıştır. Kurulan metodun yaklaşık çözüme düzgün yakınsadığı gösterilmiştir. Nümerik sonuçların teorik sonuçları desteklediği örnek üzerinde gösterilmiştir.
The present study is concerned with the numerical solution, using finite difference method of a o... more The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.
Yüzüncü yıl üniversitesi fen bilimleri enstitüsü dergisi, May 1, 2022
In this study, the numerical solution of the singularly perturbed semilinear differential equatio... more In this study, the numerical solution of the singularly perturbed semilinear differential equations with constant delay is investigated by the method of integral identities with use of linear basis functions and interpolating quadrature formulas. The finite difference scheme is established on Boglaev-Bakhvalov type mesh. The error approximations are obtained in the discrete maximum norm. A numerical example is solved to clarify the theoretical analysis.
Journal of Applied Mathematics and Computational Mechanics, 2020
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptot... more In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic e... more In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.
Journal of Computational and Applied Mathematics, 2005
We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order sin... more We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.
International Journal of Computer Mathematics, 1989
In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear ... more In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear two-point boundary-value problems. First, Newton's iteration is used to linearize such problems, reducing these to a sequence of linear singularly perturbed two-point boundary-value problems. Then,a difference scheme is applied to solve the linear systems. The difference scheme is accurate up to O(h 2). Test examples are included to demonstrate the efficiency of the method.
Turkish journal of mathematics & computer science, Jun 30, 2022
In this paper, singularly perturbed quasilinear boundary value problems are taken into account. W... more In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With this purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules are used to establish the numerical scheme. Then, an error estimate is derived. A numerical experiment is demonstrated to verify the theory.
Journal of Computational and Applied Mathematics, Oct 1, 2005
We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order sin... more We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.
This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobo... more This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobolev boundary value problems with Robin type boundary condition. Firstly, the asymptotic behavior of the solution is analyzed. By using interpolating quadrature rules and basis functions, a completely exponentially fitted tree-level difference scheme is constructed on the uniform mesh. Then an error estimation is investigated in a discrete energy norm. Two numerical examples are solved and the computational results are tabulated.
The generalized with respect to ðb; cÞ-distance Riesz potential defined on Sobolev space W l p ðR... more The generalized with respect to ðb; cÞ-distance Riesz potential defined on Sobolev space W l p ðR n Þ is constructed and for this potential the theorem of Hardy-Littlewood-Sobolev type has been established.
Journal of Applied Mathematics and Computational Mechanics, Dec 1, 2020
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptot... more In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
We study uniform finite-difference method for solving first-order singularly perturbed boundary v... more We study uniform finite-difference method for solving first-order singularly perturbed boundary value problem (BVP) depending on a parameter. Uniform error estimates in the discrete maximum norm are obtained for the numerical solution. Numerical results support the theoretical analysis. ε > 0 is a small parameter and {u(x), λ} is a solution. For ε 1, the function u(x) has a boundary layer of thickness O(ε) near x = 0. Under the above conditions, there exists a unique solution to problem (1.1), (1.2) (see [7, 12]). An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [6
We consider a uniform finite difference method on a B-mesh is applied to solve a singularly pertu... more We consider a uniform finite difference method on a B-mesh is applied to solve a singularly perturbed quasilinear boundary value problem (BVP) depending on a parameter. We give a uniform first-order error estimates in a discrete maximum norm. Numerical results are presented that demonstrate the sharpness of our theoretical analysis.
A finite-difference method is suggested and analyzed for evolution equations of Sobolev type in a... more A finite-difference method is suggested and analyzed for evolution equations of Sobolev type in a single space variable and with boundary layers. The convergence and error estimates for an exponentially fitted difference scheme in an equidistant mesh are obtained. The numerical methods discussed here are extremely robust, in the sense that they have good convergence properties not only for small values of the perturbation parameter, but also for moderate and large values.
Numerical Methods for Partial Differential Equations, Aug 1, 2019
A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value pro... more A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.
Journal of Difference Equations and Applications, Mar 2, 2022
We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the as... more We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete maximum norm. Finally, three numerical examples are solved to validate the theory.
Erzincan University Journal of Science and Technology, Mar 24, 2019
Bu çalışmada singüler pertürbe özellikli lineer olmayan reaksiyon-difüzyon sınır değer problemi e... more Bu çalışmada singüler pertürbe özellikli lineer olmayan reaksiyon-difüzyon sınır değer problemi ele alınmıştır. Kalan terimi integral biçiminde olan ve baz fonksiyonu içeren interpolasyon kuadratür kuralları kullanılarak parçalı düzgün şebeke üzerinde fark şeması kurulmuştur. Sunulan metodun kararlı olduğu gösterilmiş ve yakınsaklık analizi yapılmıştır. Kurulan metodun yaklaşık çözüme düzgün yakınsadığı gösterilmiştir. Nümerik sonuçların teorik sonuçları desteklediği örnek üzerinde gösterilmiştir.
The present study is concerned with the numerical solution, using finite difference method of a o... more The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.
Yüzüncü yıl üniversitesi fen bilimleri enstitüsü dergisi, May 1, 2022
In this study, the numerical solution of the singularly perturbed semilinear differential equatio... more In this study, the numerical solution of the singularly perturbed semilinear differential equations with constant delay is investigated by the method of integral identities with use of linear basis functions and interpolating quadrature formulas. The finite difference scheme is established on Boglaev-Bakhvalov type mesh. The error approximations are obtained in the discrete maximum norm. A numerical example is solved to clarify the theoretical analysis.
Journal of Applied Mathematics and Computational Mechanics, 2020
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptot... more In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic e... more In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.
Journal of Computational and Applied Mathematics, 2005
We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order sin... more We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.
Uploads
Papers by hakkı Duru