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Nümerik İntegrasyon Metodu ile Singüler Pertürbe Problemlerin Yaklaşık Çözümü

Year 2022, Volume: 27 Issue: 3, 612 - 618, 25.12.2022
https://doi.org/10.53433/yyufbed.1094184

Abstract

Bu çalışmada, singüler pertürbe Volterra integro-diferansiyel denklemlerin yaklaşık çözümü için nümerik integrasyon yöntemi uygulanır. İlk olarak düzgün bir şebeke üzerinde sonlu fark metodu ile başlanır daha sonra integraller için trapez metodu kullanılır. Buradan elde edilen denklem sistemi Thomas algoritması ile çözülür. Önerilen yöntemin doğruluğunu ve ekonomikliğini ortaya koyan bir örnek sunulur.

References

  • Amiraliyev, G. M., & Amirali, I. (2018). Nümerik Analiz Teori ve Uygulamalarla. Ankara, Türkiye: Seçkin Yayıncılık.
  • Andargie, A. & Reddy, Y. N. (2008). Numerical integration method for singular perturbation problems with mixed boundary conditions. Journal of Applied Mathematics & Informatics, 26(5-6), 1273-1287.
  • Arslan, D. (2020). A numerical solution for singularly perturbed multi-point boundary value problems with the numerical integration method. BEU Journal of Science, 9(1), 157-167. doi: 10.17798/bitlisfen.662732
  • Burton, T. A. (2005). Volterra Integral and Differential Equations. 2nd Ed. Amsterdam, Netherland: Elsevier.
  • Cimen E. (2018). A computational method for Volterra integro-differential equation. Erzincan University Journal of Science and Technology, 11(3), 347-352. doi: 10.18185/erzifbed.435331
  • Celik, E. & Tabatabaei, K. (2013). Solving a class of Volterra integral equation systems by the differential transform method. International Journal of Nonlinear Science, 16(1), 87-91.
  • De Gaetano, A. & Arino, O. (2000). Mathematical modelling of the intravenous glucose tolerance test. Journal of Mathematical Biology, 40, 136-168. doi: 10.1007/s002850050007
  • Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O'Riordan E., & Shishkin, G. I. (2000). Robust Computational Techniques for Boundary Layers. New York, USA: Chapman-Hall/CRC.
  • Jerri, A. (1999). Introduction to Integral Equations with Applications. New York, USA: Wiley.
  • Kauthen, J. P. (1997). A survey on singularly perturbed Volterra equations. Applied Numerical Mathematics, 24, 95-114. doi: 10.1016/S0168-9274(97)00014-7
  • Kythe, P. K., & Puri, P. (2002). Computational Methods for Linear Integral Equations. Boston, USA: Birkhauser.
  • Lodge, A. S., McLeod, J. B., & Nohel, J. A. A. (1978). A nonlinear singularly perturbed Volterra integro differential equation occurring in polymer rheology. 80, 99-137. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, doi: 10.1017/S0308210500010167
  • Mbroh, N. A., Noutchie, S. C. O., & Massoukou, R. Y. M. (2007). A second order finite difference scheme for singularly perturbed Volterra integro-differential equation. Alexandria Engineering Journal, 59, 2441-2447. doi: 10.1016/j.aej.2020.03.007
  • Miller, J. J. H., O'Riordan, E., & Shishkin, G. I. (1996). Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. Singapore: World Scientific.
  • Nefedov, N. N., Nikitin, A. G., & Urazgil'dina, T. A. (2006). The Cauchy problem for a singularly perturbed Volterra integro-differential equation. Computational Mathematics and Mathematical Physics, 46, 768-775. doi: 10.1134/S0965542506050046
  • Ramos, J. I. (2007). Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations. Applied Mathematics and Computation, 188, 1221-1233. doi: 10.1016/j.amc.2006.10.076
  • Ramos, J. I. (2008). Exponential techniques and implicit Runge-Kutta method for singularly perturbed Volterra integro differential equations. Neural, Parallel and Scientific Computations, 16, 387-404.
  • Ranjan, R., & Prasad, H. S. (2018). An efficient method of numerical integration for a class of singularly perturbed two point boundary value problems. Mathematics, 17, 265-273.
  • Reddy, Y. N. (1990). A Numerical integration method for solving singular perturbation problems. Applied Mathematics and Computation, 37, 83-95. doi: 10.1016/0096-3003(90)90037-4
  • Ross, H. G., Stynes, M., & Tobiska, L. (1996). Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems. Berlin, Germany: Springer Verlag.
  • Salama, A. A., & Bakr, S. A. (2007). Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems. Applied Mathematical Modelling, 31, 866-879. doi: 10.1016/j.apm.2006.02.007
  • Sevgin, S. (2014). Numerical solution of a singularly perturbed Volterra integro-differential equation. Advances in Difference Equation, 2014, 1-15. doi: 10.1186/1687-1847-2014-171
  • Soujanya, G., & Phnaeendra, K. (2015). Numerical intergration method for singular-singularly perturbed two- point boundary value problems. Procedia Engineering, 127, 545-552. doi: 10.1016/j.proeng.2015.11.343
  • Tao, X., & Zhang, Y. (2019). The coupled method for singularly perturbed Volterra integro-differential equations. Advances in Continuous and Discrete Models, 217, 1-16. doi: 10.1186/s13662-019-2139-8
  • Yapman, O., & Amiraliyev, G. M. (2020). A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation. International Journal of Computer Mathematics, 97, 1293-1302. doi: 10.1080/00207160.2019.1614565

Approximate Solution of Singularly Perturbed Problems with Numerical Integration Method

Year 2022, Volume: 27 Issue: 3, 612 - 618, 25.12.2022
https://doi.org/10.53433/yyufbed.1094184

Abstract

In this study, the numerical integration method is performed for approximate solution of the singularly perturbed Volterra integro-differential equations. Firstly, it starts with the finite difference method on the uniform mesh points, then the trapezoidal method is used for integrals. The system of equations obtained here is solved with the Thomas algorithm. An example is presented that demonstrates the accuracy and economy of the proposed method.

References

  • Amiraliyev, G. M., & Amirali, I. (2018). Nümerik Analiz Teori ve Uygulamalarla. Ankara, Türkiye: Seçkin Yayıncılık.
  • Andargie, A. & Reddy, Y. N. (2008). Numerical integration method for singular perturbation problems with mixed boundary conditions. Journal of Applied Mathematics & Informatics, 26(5-6), 1273-1287.
  • Arslan, D. (2020). A numerical solution for singularly perturbed multi-point boundary value problems with the numerical integration method. BEU Journal of Science, 9(1), 157-167. doi: 10.17798/bitlisfen.662732
  • Burton, T. A. (2005). Volterra Integral and Differential Equations. 2nd Ed. Amsterdam, Netherland: Elsevier.
  • Cimen E. (2018). A computational method for Volterra integro-differential equation. Erzincan University Journal of Science and Technology, 11(3), 347-352. doi: 10.18185/erzifbed.435331
  • Celik, E. & Tabatabaei, K. (2013). Solving a class of Volterra integral equation systems by the differential transform method. International Journal of Nonlinear Science, 16(1), 87-91.
  • De Gaetano, A. & Arino, O. (2000). Mathematical modelling of the intravenous glucose tolerance test. Journal of Mathematical Biology, 40, 136-168. doi: 10.1007/s002850050007
  • Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O'Riordan E., & Shishkin, G. I. (2000). Robust Computational Techniques for Boundary Layers. New York, USA: Chapman-Hall/CRC.
  • Jerri, A. (1999). Introduction to Integral Equations with Applications. New York, USA: Wiley.
  • Kauthen, J. P. (1997). A survey on singularly perturbed Volterra equations. Applied Numerical Mathematics, 24, 95-114. doi: 10.1016/S0168-9274(97)00014-7
  • Kythe, P. K., & Puri, P. (2002). Computational Methods for Linear Integral Equations. Boston, USA: Birkhauser.
  • Lodge, A. S., McLeod, J. B., & Nohel, J. A. A. (1978). A nonlinear singularly perturbed Volterra integro differential equation occurring in polymer rheology. 80, 99-137. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, doi: 10.1017/S0308210500010167
  • Mbroh, N. A., Noutchie, S. C. O., & Massoukou, R. Y. M. (2007). A second order finite difference scheme for singularly perturbed Volterra integro-differential equation. Alexandria Engineering Journal, 59, 2441-2447. doi: 10.1016/j.aej.2020.03.007
  • Miller, J. J. H., O'Riordan, E., & Shishkin, G. I. (1996). Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. Singapore: World Scientific.
  • Nefedov, N. N., Nikitin, A. G., & Urazgil'dina, T. A. (2006). The Cauchy problem for a singularly perturbed Volterra integro-differential equation. Computational Mathematics and Mathematical Physics, 46, 768-775. doi: 10.1134/S0965542506050046
  • Ramos, J. I. (2007). Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations. Applied Mathematics and Computation, 188, 1221-1233. doi: 10.1016/j.amc.2006.10.076
  • Ramos, J. I. (2008). Exponential techniques and implicit Runge-Kutta method for singularly perturbed Volterra integro differential equations. Neural, Parallel and Scientific Computations, 16, 387-404.
  • Ranjan, R., & Prasad, H. S. (2018). An efficient method of numerical integration for a class of singularly perturbed two point boundary value problems. Mathematics, 17, 265-273.
  • Reddy, Y. N. (1990). A Numerical integration method for solving singular perturbation problems. Applied Mathematics and Computation, 37, 83-95. doi: 10.1016/0096-3003(90)90037-4
  • Ross, H. G., Stynes, M., & Tobiska, L. (1996). Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems. Berlin, Germany: Springer Verlag.
  • Salama, A. A., & Bakr, S. A. (2007). Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems. Applied Mathematical Modelling, 31, 866-879. doi: 10.1016/j.apm.2006.02.007
  • Sevgin, S. (2014). Numerical solution of a singularly perturbed Volterra integro-differential equation. Advances in Difference Equation, 2014, 1-15. doi: 10.1186/1687-1847-2014-171
  • Soujanya, G., & Phnaeendra, K. (2015). Numerical intergration method for singular-singularly perturbed two- point boundary value problems. Procedia Engineering, 127, 545-552. doi: 10.1016/j.proeng.2015.11.343
  • Tao, X., & Zhang, Y. (2019). The coupled method for singularly perturbed Volterra integro-differential equations. Advances in Continuous and Discrete Models, 217, 1-16. doi: 10.1186/s13662-019-2139-8
  • Yapman, O., & Amiraliyev, G. M. (2020). A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation. International Journal of Computer Mathematics, 97, 1293-1302. doi: 10.1080/00207160.2019.1614565
There are 25 citations in total.

Details

Primary Language Turkish
Journal Section Articles
Authors

Derya Arslan 0000-0001-6138-0607

Early Pub Date December 25, 2022
Publication Date December 25, 2022
Submission Date March 27, 2022
Published in Issue Year 2022 Volume: 27 Issue: 3

Cite

APA Arslan, D. (2022). Nümerik İntegrasyon Metodu ile Singüler Pertürbe Problemlerin Yaklaşık Çözümü. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27(3), 612-618. https://doi.org/10.53433/yyufbed.1094184