Solution of Some Integral Equations by Point-Collocation Method

Year 2023, Volume: 13 Issue: 4, 1894 - 1905, 15.12.2023

Birkan Durak , Hasan Ömür Özer , Şule Kapkın , Hüseyin Yıldız

https://doi.org/10.31466/kfbd.1372548

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.

Keywords

Collocation method, Nonlinear integral equations, Volterra equations, Fredholm equations, Approximate solution method

Thanks

The authors would like to thank Prof. Dr. Erol UZAL for his valuable suggestions and contributions.

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

Abstract

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.

Keywords

Kollokasyon yöntemi, Doğrusal olmayan integral denklemler, Volterra denklemleri, Fredholm denklemleri, Yaklaşık çözüm yöntemi

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