# A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY

## SOLVING SINGULARLY PERTURBED IBV SOBOLEV PROBLEMS WITH LARGE TIME DELAY USING S-MESH METHOD

### Abstract

The aim of this paper is to provide a numerical method for time delay singularly perturbed Sobolev-type equations. First, asymptotic estimates for the Sobolev problem solution with singular perturbation and delay parameters were obtained. This estimate showed that the solution depends on the initial data. On a special piecewise uniform mesh (Shishkin mesh), whose solution converges pointwise independently of the singular perturbation parameter, it is built and studied to solve this issue using the finite difference method. A discrete norm was used to investigate the stability of difference schemes. It is shown that the completely discrete scheme converges with order $%O\left( \tau ^{2}+N_{l}^{-2}\ln ^{2}N_{l}\right) $ in both space and time, independent of the perturbation parameter. Finally, with a test problem and numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.### References

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[2] G. M. Amiraliyev, 1990, Difference Method for the Solution of One Problem of the Theory of Dispersive Waves, Differential Equations, 26, 2146-2154. (Russian)

[3] G. M. Amiraliyev, 1987, Investigation of the Difference Schemes for the Quasi-Linear Sobolev Equations, Differential Equations, V.23, No. 8, 1453-1455. (Russian)

[4] A.R. Ansari, S.A. Bakr, G.I. Shishkin, 2007, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. of Comput. and App. Math., 205, No.1 , 552-566.

[5] A. Barati Chiyaneh, H. Duru, 2019, ON Adaptive Mesh for the Initial-Boundary Value Singularly Perturbed Delay Sobolev Problems, Numerical Methods for Partial Differential Equations, DOI: 10.1002/num.22417.

[6] A. Barati Chiyaneh, H. Duru, 2019, Uniform Difference Method for Singularly Perturbed Delay Sobolev Problems on a Piecewise Uniform Mesh, Quaestiones Mathematicae, DOI:10.2989/16073606.2019.1653395.

[7] R. K. Bullough and P. J. Caudrey, 1980, Solitons, Springer-Verlag, New York, 1-13.

[8] G. V. Demidenko and S. V. Uspenskii, 1998, Equations and systems unsolved with respect to the highest derivative, Nauchnaya Kniga, Novosibirsk (Russian).

[9] E. P. Doolan, J. J. Miller, W. H. A. Schilders, 1980, Uniform Numerical Methods for

Problems with Initial and Boundary Layers, Boole Press. Dublin.

[10] H. Duru, 2004, Difference schemes for the singularly perturbed Sobolev periodic

boundary problem, App. Math. And Comput. 149: 187–201.

[11] I. E. Egorov, S. G. Pyatkov, and S. V. Popov, 2000, Non-classical differential-operator

equations, Nauka, Novosibirsk (Russian).

[12] R. E. Ewing, 1978, Time Stepping Galerkin Methods for Nonlinear Sobolev Partial Differential Equations, SIAM J. Numer Anal. 15, 1125-1150.

[13] A. Favini and A. Yagi, 1999, Degenerate differential equations in Banach spaces, Marcel

Dekker, New York.

[14] W. E. Ford, T. W. Ting, 1974, Uniform Error Estimates for Difference Approximations to Nonlinear Pseudo-Parabolic Partial Differential Equations, SIAM J. Numer. Anal. 11, 155-169.

[15] H. Gajewski, K. Groeger, and K. Zacharias, 1974, Nichtlineare Operatorgleichungen und Operator differentialgleichungen, Mathematische Lehrbu ̈cher und Monographien, Band 38, Akademie-Verlag, Berlin; Russian transl., Mir, Moscow 1978.

[16] H. Ikezi, 1978, Experiments on Solitons in Plasmas, Solitons in Action, Ed. K. Lonngren and A. Scott, Academic Press, 152-170.

[17] M. K. Kadalbajoo and Y. N. Reddy, 1989, Asymptotic and Numerical Analysis of Singular Perturbation Problems, A survey, App. Math. And Comput. 30:223-259.

[18] J. L. Langnese, 1972, General Boundary-Value Problems for Differential Equations of Sobolev Type, SIAM J. Math. Anal., v.3, 105-119.

[19] V. I. Lebedev, 1957, The Method of Difference for the Equations of Sobolev Type, Dokl. Acad. Sci. USSR, V.114, No. 6, 1166-1169.

[20] K. E. Lonngren, 1978, Observation of Solitons on Nonlinear Dispersive Transmission Lines, Soliton in Action, Academic Press. 127-148.

[21] A. A. Samarskii, 1983, Theory of Difference Schemes, 2ndEd., Nauka, Moscow.

[22] C. L. Sobolev, 1954, On a New Problems of Mathematical Physics, Izv. Acad. Sci. USSR, Mathematics, 18, No.1, 3-50.

[2] G. M. Amiraliyev, 1990, Difference Method for the Solution of One Problem of the Theory of Dispersive Waves, Differential Equations, 26, 2146-2154. (Russian)

[3] G. M. Amiraliyev, 1987, Investigation of the Difference Schemes for the Quasi-Linear Sobolev Equations, Differential Equations, V.23, No. 8, 1453-1455. (Russian)

[4] A.R. Ansari, S.A. Bakr, G.I. Shishkin, 2007, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. of Comput. and App. Math., 205, No.1 , 552-566.

[5] A. Barati Chiyaneh, H. Duru, 2019, ON Adaptive Mesh for the Initial-Boundary Value Singularly Perturbed Delay Sobolev Problems, Numerical Methods for Partial Differential Equations, DOI: 10.1002/num.22417.

[6] A. Barati Chiyaneh, H. Duru, 2019, Uniform Difference Method for Singularly Perturbed Delay Sobolev Problems on a Piecewise Uniform Mesh, Quaestiones Mathematicae, DOI:10.2989/16073606.2019.1653395.

[7] R. K. Bullough and P. J. Caudrey, 1980, Solitons, Springer-Verlag, New York, 1-13.

[8] G. V. Demidenko and S. V. Uspenskii, 1998, Equations and systems unsolved with respect to the highest derivative, Nauchnaya Kniga, Novosibirsk (Russian).

[9] E. P. Doolan, J. J. Miller, W. H. A. Schilders, 1980, Uniform Numerical Methods for

Problems with Initial and Boundary Layers, Boole Press. Dublin.

[10] H. Duru, 2004, Difference schemes for the singularly perturbed Sobolev periodic

boundary problem, App. Math. And Comput. 149: 187–201.

[11] I. E. Egorov, S. G. Pyatkov, and S. V. Popov, 2000, Non-classical differential-operator

equations, Nauka, Novosibirsk (Russian).

[12] R. E. Ewing, 1978, Time Stepping Galerkin Methods for Nonlinear Sobolev Partial Differential Equations, SIAM J. Numer Anal. 15, 1125-1150.

[13] A. Favini and A. Yagi, 1999, Degenerate differential equations in Banach spaces, Marcel

Dekker, New York.

[14] W. E. Ford, T. W. Ting, 1974, Uniform Error Estimates for Difference Approximations to Nonlinear Pseudo-Parabolic Partial Differential Equations, SIAM J. Numer. Anal. 11, 155-169.

[15] H. Gajewski, K. Groeger, and K. Zacharias, 1974, Nichtlineare Operatorgleichungen und Operator differentialgleichungen, Mathematische Lehrbu ̈cher und Monographien, Band 38, Akademie-Verlag, Berlin; Russian transl., Mir, Moscow 1978.

[16] H. Ikezi, 1978, Experiments on Solitons in Plasmas, Solitons in Action, Ed. K. Lonngren and A. Scott, Academic Press, 152-170.

[17] M. K. Kadalbajoo and Y. N. Reddy, 1989, Asymptotic and Numerical Analysis of Singular Perturbation Problems, A survey, App. Math. And Comput. 30:223-259.

[18] J. L. Langnese, 1972, General Boundary-Value Problems for Differential Equations of Sobolev Type, SIAM J. Math. Anal., v.3, 105-119.

[19] V. I. Lebedev, 1957, The Method of Difference for the Equations of Sobolev Type, Dokl. Acad. Sci. USSR, V.114, No. 6, 1166-1169.

[20] K. E. Lonngren, 1978, Observation of Solitons on Nonlinear Dispersive Transmission Lines, Soliton in Action, Academic Press. 127-148.

[21] A. A. Samarskii, 1983, Theory of Difference Schemes, 2ndEd., Nauka, Moscow.

[22] C. L. Sobolev, 1954, On a New Problems of Mathematical Physics, Izv. Acad. Sci. USSR, Mathematics, 18, No.1, 3-50.

How to Cite

CHIYANEH, A. B.; DURU, H..
A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 117, n. 1, apr. 2023. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1245>. Date accessed: 08 june 2023. doi: https://doi.org/10.26577/JMMCS.2023.v117.i1.08.
Section

Applied Mathematics

Keywords
Delayed partial differential equation, Finite difference method, Shishkin mesh, Singular perturbation, Sobolev problem