Araştırma Makalesi
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Analysis of the Layer Behavior to the Parameterized Problem with Integral Boundary Condition

Yıl 2017, Cilt: 10 Sayı: 2, 296 - 302, 22.12.2017

Öz

A parameterized
singularly perturbed first order quasilinear boundary value problem
with integral
boundary conditions
is considered.
Asymptotic estimates for the solution and its first derivative have been
established. Given an example supports
these theoretical results and indicate that the estimates are sharp.
The estimates are obtained with the use of a mathematical technique that can
also be applied in appropriate grid computations.

Kaynakça

  • Ahmad, B., Khan, R.A. and Sivasundaram, S. 2005. Generalized quasilinearization method for a first order differential equation with integral boundary condition. Dynam. Contin. Discrete Impuls. Systems, Ser. A, Math. Anal., 12: 289-296.
  • Amiraliyev, G.M., Amiraliyeva, I.G., Kudu, M. 2007. A numerical treatment for singularly perturbed differential equations with integral boundary condition. Applied Mathematics and Computations, 185:574-582.
  • Amiraliyev, G.M., Duru, H. 2005. A note on a parameterized singular perturbation problem. Journal of Computational and Applied Mathematics 182 (2005) 233–242.
  • Ashyralyev, A., Sharifov Y. A. 2013. Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions, Advances in Difference Equations 2013(173), DOI: 10.1186/1687-1847-2013-173.
  • Benchohra, M., Hamani S., Nieto J.J. 2010. The method of upper and lower solutions for second order differential inclusions with integral boundary conditions, Rocky Mountain J. Math., 40(1):13-26.
  • Benchohra, M., Nieto J.J., Quahab, A. 2011. Second-order boundary value problem with integral boundary conditions. Boundary Value Problems, 2011, Article ID 260309.
  • Cakir, M., Amiraliyev, G.M. 2007. Numerical solution of a singularly perturbed three-point boundary value problem. International Journal of Computer Mathematics. 84:1465-1481.
  • Jankowski, T. 2002. Application of the numerical-analytical method to systems of differential equations with a parameter. Ukrain. Math. J., 54(4): 671-683.
  • Jankowski, T. 2003. Existensions of quasilinearization method for differential equations with integral boundary conditions. Math. Comput. Model., 37(2003), 155-165.
  • Kevorkian, J. and Cole, J.D. 1981. Perturbation Methods in Applied Mathematics. Springer, New York, http://dx.doi.org/10.1007/978-1-4757-4213-8.,
  • Khan, R.A. 2003. The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions. JQTDE (http://www.math.u.szeged.hu/ejqtde/), 10: 1-9.
  • Kudu, M. and Amirali, I. 2016. A priori estimates of solution of parametrized singularly perturbed problem. Journal of Applied Mathematics and Physics. 4: 73-78. doi: 10.4236/jamp.2016.41011.
  • Kudu, M. and Amiraliyev, G.M 2015. Finite difference method for a singularly perturbed differential equations with ıntegral boundary condition. International Journal of Mathematics and Computation. 26(3): 72-79.
  • Kudu, M., Amirali, I., Amiraliyev, Gabil M. 2016. A layer analysıs of parameterızed singularly perturbed boundary value problems. International Journal of Applied Mathematics, 29(4): 439-449.
  • Kudu, M. 2014. Asymptotic estimates for second-order parameterized singularly perturbed promlem. Applied Mathematics, 5:1988-1992. doi: 10.4236/am.2014.513191.
  • Liu, X., Mcare F.A. 2001. A Monotone iterative methods for boundary value problems of parameteric differential equation. J. Appl. Math. Stoch. Anal. 14:183-187.
  • Miller, J. J., O'Riordan, H. E., Shishkin G. I. 2012. Fitted Numerical Methods for Singular Perturbation Problems, Rev. Ed. World Scientific, Singapore.
  • Na, T.Y. 1979. Computational Methods in Engineering Boundary Value Problems, Academic Press, New York.
  • Nayfeh, A. H. 1993. Introduction to Perturbation Techniques. Wiley, New York.
  • O’Malley, R. E. 1991. Singular Perturbations Methods for Ordinary Differential Equations, Applied Mathemetical Sciences, vol. 89, Springer-Verlag, New York.
  • Pomentale, T. A. 1976. Constructive theorem of existence and uniqueness for the problem. Zeitschrift für Angewandte Mathematik und Mechanik. 56:387-388.
  • Roos, H.G., Stynes M., Tobiska L. 2008. Robust Numerical Methods for Singularly Perturbed Differential Equations Convection-Diffusion-Reaction and Flow Problems. Springer-Verlag, Berlin Heidelberg.
  • Samoilenko, A.M., Ronto N.I., Martynyuk, S.V. 1991. On the numerical–analytic method for problems with integral boundary conditions, Dokl. Akad. Nauk Ukrain. SSR 4: 34–37 (in Russian).

Analysis of the Layer Behavior to the Parameterized Problem with Integral Boundary Condition

Yıl 2017, Cilt: 10 Sayı: 2, 296 - 302, 22.12.2017

Öz

Bu çalışmada, integral
sınır şartlı parametreye bağlı singüler pertürbe özellikli kuazi-lineer
sınır-değer problemi ele alınmıştır. Problemin çözümü ve birinci türevleri için
asimptotik değerlendirmeler elde edilmiştir. Bu teorik sonuçları destekleyen ve
değerlendirmelerin kesin olduğunu gösteren bir örnek verilmiştir. Asimptotik
değerlendirmelerin elde edilmesinde kullanılan yöntem uygun nümerik çözümlerin
incelenmesinde kullanılabilir.

Kaynakça

  • Ahmad, B., Khan, R.A. and Sivasundaram, S. 2005. Generalized quasilinearization method for a first order differential equation with integral boundary condition. Dynam. Contin. Discrete Impuls. Systems, Ser. A, Math. Anal., 12: 289-296.
  • Amiraliyev, G.M., Amiraliyeva, I.G., Kudu, M. 2007. A numerical treatment for singularly perturbed differential equations with integral boundary condition. Applied Mathematics and Computations, 185:574-582.
  • Amiraliyev, G.M., Duru, H. 2005. A note on a parameterized singular perturbation problem. Journal of Computational and Applied Mathematics 182 (2005) 233–242.
  • Ashyralyev, A., Sharifov Y. A. 2013. Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions, Advances in Difference Equations 2013(173), DOI: 10.1186/1687-1847-2013-173.
  • Benchohra, M., Hamani S., Nieto J.J. 2010. The method of upper and lower solutions for second order differential inclusions with integral boundary conditions, Rocky Mountain J. Math., 40(1):13-26.
  • Benchohra, M., Nieto J.J., Quahab, A. 2011. Second-order boundary value problem with integral boundary conditions. Boundary Value Problems, 2011, Article ID 260309.
  • Cakir, M., Amiraliyev, G.M. 2007. Numerical solution of a singularly perturbed three-point boundary value problem. International Journal of Computer Mathematics. 84:1465-1481.
  • Jankowski, T. 2002. Application of the numerical-analytical method to systems of differential equations with a parameter. Ukrain. Math. J., 54(4): 671-683.
  • Jankowski, T. 2003. Existensions of quasilinearization method for differential equations with integral boundary conditions. Math. Comput. Model., 37(2003), 155-165.
  • Kevorkian, J. and Cole, J.D. 1981. Perturbation Methods in Applied Mathematics. Springer, New York, http://dx.doi.org/10.1007/978-1-4757-4213-8.,
  • Khan, R.A. 2003. The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions. JQTDE (http://www.math.u.szeged.hu/ejqtde/), 10: 1-9.
  • Kudu, M. and Amirali, I. 2016. A priori estimates of solution of parametrized singularly perturbed problem. Journal of Applied Mathematics and Physics. 4: 73-78. doi: 10.4236/jamp.2016.41011.
  • Kudu, M. and Amiraliyev, G.M 2015. Finite difference method for a singularly perturbed differential equations with ıntegral boundary condition. International Journal of Mathematics and Computation. 26(3): 72-79.
  • Kudu, M., Amirali, I., Amiraliyev, Gabil M. 2016. A layer analysıs of parameterızed singularly perturbed boundary value problems. International Journal of Applied Mathematics, 29(4): 439-449.
  • Kudu, M. 2014. Asymptotic estimates for second-order parameterized singularly perturbed promlem. Applied Mathematics, 5:1988-1992. doi: 10.4236/am.2014.513191.
  • Liu, X., Mcare F.A. 2001. A Monotone iterative methods for boundary value problems of parameteric differential equation. J. Appl. Math. Stoch. Anal. 14:183-187.
  • Miller, J. J., O'Riordan, H. E., Shishkin G. I. 2012. Fitted Numerical Methods for Singular Perturbation Problems, Rev. Ed. World Scientific, Singapore.
  • Na, T.Y. 1979. Computational Methods in Engineering Boundary Value Problems, Academic Press, New York.
  • Nayfeh, A. H. 1993. Introduction to Perturbation Techniques. Wiley, New York.
  • O’Malley, R. E. 1991. Singular Perturbations Methods for Ordinary Differential Equations, Applied Mathemetical Sciences, vol. 89, Springer-Verlag, New York.
  • Pomentale, T. A. 1976. Constructive theorem of existence and uniqueness for the problem. Zeitschrift für Angewandte Mathematik und Mechanik. 56:387-388.
  • Roos, H.G., Stynes M., Tobiska L. 2008. Robust Numerical Methods for Singularly Perturbed Differential Equations Convection-Diffusion-Reaction and Flow Problems. Springer-Verlag, Berlin Heidelberg.
  • Samoilenko, A.M., Ronto N.I., Martynyuk, S.V. 1991. On the numerical–analytic method for problems with integral boundary conditions, Dokl. Akad. Nauk Ukrain. SSR 4: 34–37 (in Russian).
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Mustafa Kudu

Yayımlanma Tarihi 22 Aralık 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 10 Sayı: 2

Kaynak Göster

APA Kudu, M. (2017). Analysis of the Layer Behavior to the Parameterized Problem with Integral Boundary Condition. Erzincan University Journal of Science and Technology, 10(2), 296-302. https://doi.org/10.18185/erzifbed.322346