SavedAdd your journal

Journals / Karadeniz Fen Bilimleri Dergisi / 2023 / Cilt: 13 Sayı: 4

Solution of Some Integral Equations by Point-Collocation Method

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

Birkan DURAK
Hasan Ömür ÖZER
Şule KAPKIN
Hüseyin YILDIZ
Pages
1894–1905
Publication Date
2023-12-15
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In several engineering or physics problems, particularly those involving electromagnetic theory, thermal and radiation effects, acoustics, elasticity, and some fluid mechanics, it is not always easy or possible to find the analytical solution of integral equations that describe them.
📊
Key Findings & Data:In this study, Point-collocation method was applied to linear and nonlinear, Volterra and Fredholm type integral equations and the performance and accuracy of the method was compared with several other methods that seem to be popular choices.
💡
Conclusion & Impact:The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Kollokasyon yöntemi · Doğrusal olmayan integral denklemler · Volterra denklemleri

Abstract

In several engineering or physics problems, particularly those involving electromagnetic theory, thermal and radiation effects, acoustics, elasticity, and some fluid mechanics, it is not always easy or possible to find the analytical solution of integral equations that describe them. For this reason, numerical techniques are used. In this study, Point-collocation method was applied to linear and nonlinear, Volterra and Fredholm type integral equations and the performance and accuracy of the method was compared with several other methods that seem to be popular choices. As the base functions, a suitably chosen family of polynomials were employed. The convergence of the method was verified by increasing the number of polynomial base functions. The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations. Nonlinear problems take longer to calculate approximate solution coefficients than linear problems. Furthermore, it is necessary to use the real and smallest coefficients found in order to obtain a suitable approximate solution to these problems.

Özet

Çeşitli mühendislik veya fizik problemlerinde, özellikle elektromanyetik teori, termal ve radyasyon etkileri, akustik, elastisite ve akışkanlar mekaniğinde, bunları tanımlayan integral denklemlerin analitik çözümünü bulmak her zaman kolay veya mümkün değildir. Bu yüzden sayısal teknikler kullanılır. Bu çalışmada temel bilimlerde ve mühendislikte karşılaşılan integral denklemlerin sayısal çözümleri için kullanılabilecek polinom temelli kollokasyon yöntemi sunulmuştur. Yöntem, doğrusal veya doğrusal olmayan Volterra ve Fredholm integral denklemlerine uygulanacak şekilde formüle edilmiştir. Doğrusal olmayan denklemlerin kollokasyon noktalarında cebirsel denklemlere indirgenmesi ve meydana gelen denklem sisteminin çözümü mümkün olmuştur. İncelenen örneklerin sayısal sonuçları, önerilen bu yöntemin iyi çalıştığını ve az sayıda kollokasyon noktası alındığında bile polinom seçiminin yaklaşık çözüm için uygun olduğunu göstermektedir. Ayrıca, yöntemin performansı farklı polinom mertebeleri için karşılaştırılmıştır. Doğrusal olmayan problemlerin yaklaşık çözüm katsayılarını hesaplamak doğrusal problemlere göre daha uzun sürmektedir. Ayrıca bu problemlere uygun yaklaşık çözüm elde edebilmek için bulunan gerçek ve en küçük katsayıların kullanılması gerekmektedir.

🔖

Cite This Article

DURAK, B., ÖZER, H. Ö., KAPKIN, Ş., YILDIZ, H. (2023). Solution of Some Integral Equations by Point-Collocation Method. Karadeniz Fen Bilimleri Dergisi, Cilt: 13 Sayı: 4, 1894–1905. https://ajindex.com/articles/1022682

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (12)

  1. Abbasbandy, S., (2006). Numerical solutions of the integral equations: homotopy perturbation method and adomian’s decomposition method. Applied Mathematics and Computation, 173(1), 493-500. https://doi.org/10.1016/j.amc.2005.04.077.
  2. Abbasbandy, S., and Shivanian, E.,(2011). A new analytical technique to solve Fredholm’s integral equations. Numerical Algorithms, 56, 27–43. https://doi.org/10.1007/s11075-010-9372-2
  3. Adawi, A., Awawdeh, F., and Jaradat, H., (2009). A numerical method for solving linear integral equations. Int. J. Contemp. Math. Sciences, 4(10), 485–496.
  4. Arikoglu, A., and Ozkol, I., (2008). Solutions of integral and integro-differential equation systems by using differential transform method. Computers & Mathematics with Applications, 56(9), 2411-2417. https://doi.org/10.1016/j.camwa.2008.05.017.
  5. Biazar, J., and Eslami, M., (2010). Modified hpm for solving systems of volterra integral equations of the second kind. Journal of King Saud University-Science, 23(1), 35-39. https://doi.org/10.1016/j.jksus.2010.06.004
  6. Brunner, H., Hairer, E., and Njersett, S. P.,(1982). Runge-Kutta theory for volterra integral equations of the second kind. Mathematics of Computation, 39, 147-163. https://doi.org/10.2307/2007625
  7. Chakraverty, S., Mahato, N.R., Karunakar, P., and Rao, T.D.,(2019). Advanced Numerical and Semi-Analytical Methods for Differential Equations. (1st ed.). USA: John Wiley & Sons, Inc. Daddi-Moussa-Ider, A., Kaoui, B., Löwen, H., (2019). Axisymmetric flow due to a stokeslet near a finite-sized elastic membrane. Journal of the Physical Society of Japan, 88, 054401, 1-15. https://doi.org/10.7566/JPSJ.88.054401
  8. Darania, P., Ebadian, A., and Oskoi, A. V., (2006). Linearization method for solving nonlinear integral equations. Mathematical Problems in Engineering, 073714, 1-10. https://doi.org/10.1155/MPE/2006/73714.
  9. Guo, P., (2020). Numerical simulation for fredholm integral equation of the second kind. Journal of Applied Mathematics and Physics, 8(11), 2438-2446. https://doi.org/10.4236/jamp.2020.811180
  10. Huang, Y., Peng, S., Habibi, M., and Moradi, Z., (2023). Buckling simulation of eccentrically rotating nanocomposite sector plates in thermal environment using the 2D Chebyshev collocation method. Thin-Walled Structures, 193(111203). https://doi.org/10.1016/j.tws.2023.111203
  11. Krasnov, M., Kiselev, A., and Makarenko, G., (1971). Problems and Exercises in Integral Equations. (1st ed.). Moscow, Mir Publishers.
  12. Matoog, R. T., Abdou, M. A., Abdel-Aty, M. A.,