Singüler Pertürbe Özellikli Fredholm İntegro Diferansiyel Denkleminin Katman Davranışının İncelenmesi

Yıl 2024, , 1301 - 1309, 01.09.2024

Muhammet Enes Durmaz

https://doi.org/10.21597/jist.1483651

Öz

Çalışma, ikinci mertebeden lineer singüler pertürbe özellikli Fredholm integro diferansiyel denklemini ele almaktadır. Bu tür problemlerin niteliksel analizi, çözümün sınır katmanlarındaki davranışının hızlı değişmesi nedeniyle oldukça zordur. Bu çalışmada sınır katmanlı Fredholm integro diferansiyel denkleminin çözümü ve çözümün birinci ve ikinci türevleri için asimptotik değerlendirmeler sunulmuştur. Elde edilen değerlendirmeler, matematiksel modelleme ve analizde uygun yaklaşık yöntemlerin geliştirilmesine ve değerlendirilmesine katkı sağlaması açısından önem taşımaktadır. Ayrıca sunulan örnek, teorik sonuçların geçerliliğine ve değerlendirmelerin doğruluğuna destek sağlamaktadır.

Anahtar Kelimeler

Asimptotik değerlendirme, Sınır katmanı, Singüler pertürbasyon

Survey of the Layer Behaviour of the Singularly Perturbed Fredholm Integro-Differential Equation

Öz

The work handles a second order linear singularly perturbed Fredholm integro differential equation. The qualitative analysis of such problems is quite difficult due to the rapid change in behavior of the solution within the boundary layer. In this study, asymptotic estimates for the solution and its first and second derivatives of the Fredholm integro differential equation with a boundary layer have been presented. The obtained estimates have significance in their contribution to the development and evaluation of appropriate approximate methods in mathematical modeling and analysis. Furthermore, the presented example provides support for the validity of the theoretical results and the accuracy of the estimates.

Anahtar Kelimeler

Asymptotic estimate, Boundary layer, Singular perturbation

Kaynakça

• Abdulghani, M., Hamoud, A., & Ghandle, K., (2019). The effective modification of some analytical techniques for Fredholm integro-differential equations. Bulletin of the International Mathematical Virtual Institute, 9, 345-353.
• Abdullah, J. T., (2021). Numerical solution for linear Fredholm integro-differential equation using Touchard polynomials. Baghdad Science Journal, 18(2), 330-337.
• Amirali, I., Durmaz, M. E., Acar, H., & Amiraliyev G. M., (2023). First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition. Journal of Physics: Conference Series, 2514, 012003.
• Amiraliyev, G. M., Durmaz, M. E., & Kudu, M., (2020). Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation. Bulletin of the Belgian Mathematical Society - Simon Stevin, 271, 71-88.
• Amiraliyev, G. M., Durmaz, M. E., & Kudu, M., (2021). A numerical method for a second order singularly perturbed Fredholm integro-differential equation. Miskolc Mathematical Notes, 221, 37-48.
• Arqub, O. A., Al-Smadi, M., & Shawagfeh, N., (2013). Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method. Applied Mathematics and Computation, 219(17), 8938-8948.
• Cakir, M., Ekinci, Y., & Cimen, E., (2022), A numerical approach for solving nonlinear Fredholm integro-differential equation with boundary layer. Computational and Applied Mathematics, 41, 259.
• Cakir, M., & Cimen, E., (2023), A Novel Uniform Numerical Approach to Solve a Singularly Perturbed Volterra Integro-Differential Equation. Computational Mathematics and Mathematical Physics, 63, 1800-1816.
• Chen, J., He, M., & Huang, Y., (2020), A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions. Journal of Computational and Applied Mathematics, 364, 112352.
• Cimen, E., & Cakir, M., (2021), A uniform numerical method for solving singularly perturbed Fredholm integro-differential problem. Computational and Applied Mathematics, 40, 42.
• Dag, H. G., & Bicer, K. E., (2020). Boole collocation method based on residual correction for solving linear Fredholm integro-differential equation. Journal of Science and Arts, 3(52), 597-610.
• Durmaz, M. E., Amiraliyev, G., & Kudu, M., (2022). Numerical solution of a singularly perturbed Fredholm integro differential equation with Robin boundary condition. Turkish Journal of Mathematics, 46(1), 207-224.
• Durmaz, M. E., Çakır, M., & Amirali, G., (2022). Parameter uniform second-order numerical approximation for the integro-differential equations involving boundary layers. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 954-967.
• Dzhumabaev, D. S., Nazarova, K. Z., & Uteshova, R. E., (2020). A modification of the parameterization method for a linear boundary value problem for a Fredholm integro-differential equation. Lobachevskii Journal of Mathematics, 41, 1791-1800.
• El-Zahar, E. R., (2020). Approximate analytical solution of singularly perturbed boundary value problems in MAPLE. AIMS Mathematics, 53, 2272-2284.
• Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O’Riordan, E., & Shishkin, G. I., (2000). Robust computational techniques for boundary layers. Chapman Hall/CRC, New York.
• Hosseini, S. M., & Shahmorad, S., (2003). Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Applied Mathematical Modelling, 27(2), 145-154.
• Hamoud, A. A., & Ghadle, K. P., (2019). Usage of the variational iteration technique for solving Fredholm integro-differential equations. Journal of Applied and Computational Mechanics, 50(2), 303-307.
• Kadalbajoo, M. K., & Gupta, V., (2010). A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217, 3641-3716.
• Kevorkian, J., & Cole, J. D., (1981). Perturbation methods in applied mathematics. Springer, New York.
• Lin, X., Liu, J., & Wang, C., (2020). The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems. Boundary Value Problems, 14.
• Miller, J. J., O’Riordan, H. E., & Shishkin, G. I., (2012). Fitted numerical methods for singular perturbation problems. Rev. Ed., World Scientific, Singapore.
• 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, 89, Springer-Verlag, New York.
• Panda, A., Mohapatra, J., Amirali, I., Durmaz, M. E., Amiraliyev, G. M., (2024). A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations. Mathematics and Computers in Simulation, 220, 618-629.
• Reddy, Y. N., & Chakravarthy, P. P., (2004). An initial-value approach for solving singularly perturbed two-point boundary value problems. Applied Mathematics and Computation, 1551, 95-110.
• Roos, H. G., Stynes, M., & Tobiska, L., (2008). Robust numerical methods for singularly perturbed differential equations. Springer-Verlag, Berlin Heidelberg.
• Schmisser, C., & Weiss, R., (1986). Asymptotic analysis of singularly perturbed boundary value problems. SIAM Journal on Mathematical Analysis, 17, 560-579.
• Tair, B., Guebbai, H., Segni, S., & Ghiat, M., (2022). An approximation solution of linear Fredholm integro-differential equation using Collocation and Kantorovich methods. Journal of Applied Mathematics and Computing, 68, 3505-3525.
• Tair, B., Guebbai, H., Segni, S., & Ghiat, M., (2021). Solving linear Fredholm integro-differential equation by Nyström method. Journal of Applied Mathematics and Computational Mechanics, 20(3), 53-64.
• Vougalter, V., & Volpert, V., (2018). On the existence in the sense of sequences of stationary solutions for some systems of non-Fredholm integro-differential equations. Mediterranean Journal of Mathematics, 15(5), 205.
• Yalcinbas, S., Sezer, M., & Sorkun, H. H., (2009). Legendre polynomial solutions of high-order linear Fredholm integro-differential equations. Applied Mathematics and Computation, 210(2), 334-349.