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Solution of Some Integral Equations by Point-Collocation Method

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

Thanks

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

References

  • 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.
  • 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
  • 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.
  • 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.
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • Krasnov, M., Kiselev, A., and Makarenko, G., (1971). Problems and Exercises in Integral Equations. (1st ed.). Moscow, Mir Publishers.
  • Matoog, R. T., Abdou, M. A., Abdel-Aty, M. A., (2023). New algorithms for solving nonlinear mixed integral equations. AIMS Mathematics, 8(11), 27488-27512. https://doi: 10.3934/math.20231406
  • Maturi, D.A.,(2019). The successive approximation method for solving nonlinear Fredholm integral equation of the second kind using maple. Advances in Pure Mathematics, 9(10), 832-843. https://doi.org/10.4236/apm.2019.910040
  • Prajapati, R., Mohan, R., and Kumar, P., (2012). Numerical solution of generalized Abel’s integral equation by variational iteration method. American Journal of Computational Mathematics, 2(4), 312-315. https://doi.org/10.4236/ajcm.2012.24042.
  • Shakeri, S., Saadati, R., Vaezpaur, S.M., and Vahidi, J., (2009). Variational iteration method for solving integral equations. Journal of Applied Sciences, 9(4), 799-800. https://doi.org/10.3923/jas.2009.799.800
  • Tannhäuser, K., Serrao P.H., and Kozinov, S., (2024). A three-dimensional collocation finite element method for higher-order electromechanical coupling. Computers & Structures, 291. https://doi.org/10.1016/j.compstruc.2023.107219
  • Wazwaz, A.M., and Khuri, S.A., (1996). Two methods for solving integral equations. Applied Mathematics and Computation, 77(1), 79-89. https://doi.org/10.1016/0096-3003(95)00189-1.
  • Wazwaz, A.M., (2011). Linear and nonlinear integral equations methods and applications. (1st ed.). Beijing:Higher Education Press.
  • Xi, Q., Fu, Z., Zou, M., and Zhang, C., (2024). An efficient hybrid collocation scheme for vibro-acoustic analysis of the underwater functionally graded structures in the shallow ocean. Computer Methods in Applied Mechanics and Engineering, 418(116537). https://doi.org/10.1016/j.cma.2023.116537
  • Yang, X., Wu, L., and Zhang, H., (2023). A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Applied Mathematics and Computation, 457. https://doi.org/10.1016/j.amc.2023.128192.
  • Yuzbasi, S., and Yildirim, G., (2023). A Pell–Lucas Collocation Approach for an SIR Model on the Spread of the Novel Coronavirus (SARS CoV-2) Pandemic: The Case of Turkey. Mathematics, 11(3), 697. https://doi.org/10.3390/math11030697

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

Year 2023, Volume: 13 Issue: 4, 1894 - 1905, 15.12.2023
https://doi.org/10.31466/kfbd.1372548

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.

References

  • 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.
  • 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
  • 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.
  • 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.
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • Krasnov, M., Kiselev, A., and Makarenko, G., (1971). Problems and Exercises in Integral Equations. (1st ed.). Moscow, Mir Publishers.
  • Matoog, R. T., Abdou, M. A., Abdel-Aty, M. A., (2023). New algorithms for solving nonlinear mixed integral equations. AIMS Mathematics, 8(11), 27488-27512. https://doi: 10.3934/math.20231406
  • Maturi, D.A.,(2019). The successive approximation method for solving nonlinear Fredholm integral equation of the second kind using maple. Advances in Pure Mathematics, 9(10), 832-843. https://doi.org/10.4236/apm.2019.910040
  • Prajapati, R., Mohan, R., and Kumar, P., (2012). Numerical solution of generalized Abel’s integral equation by variational iteration method. American Journal of Computational Mathematics, 2(4), 312-315. https://doi.org/10.4236/ajcm.2012.24042.
  • Shakeri, S., Saadati, R., Vaezpaur, S.M., and Vahidi, J., (2009). Variational iteration method for solving integral equations. Journal of Applied Sciences, 9(4), 799-800. https://doi.org/10.3923/jas.2009.799.800
  • Tannhäuser, K., Serrao P.H., and Kozinov, S., (2024). A three-dimensional collocation finite element method for higher-order electromechanical coupling. Computers & Structures, 291. https://doi.org/10.1016/j.compstruc.2023.107219
  • Wazwaz, A.M., and Khuri, S.A., (1996). Two methods for solving integral equations. Applied Mathematics and Computation, 77(1), 79-89. https://doi.org/10.1016/0096-3003(95)00189-1.
  • Wazwaz, A.M., (2011). Linear and nonlinear integral equations methods and applications. (1st ed.). Beijing:Higher Education Press.
  • Xi, Q., Fu, Z., Zou, M., and Zhang, C., (2024). An efficient hybrid collocation scheme for vibro-acoustic analysis of the underwater functionally graded structures in the shallow ocean. Computer Methods in Applied Mechanics and Engineering, 418(116537). https://doi.org/10.1016/j.cma.2023.116537
  • Yang, X., Wu, L., and Zhang, H., (2023). A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Applied Mathematics and Computation, 457. https://doi.org/10.1016/j.amc.2023.128192.
  • Yuzbasi, S., and Yildirim, G., (2023). A Pell–Lucas Collocation Approach for an SIR Model on the Spread of the Novel Coronavirus (SARS CoV-2) Pandemic: The Case of Turkey. Mathematics, 11(3), 697. https://doi.org/10.3390/math11030697
There are 22 citations in total.

Details

Primary Language English
Subjects Classical Physics (Other)
Journal Section Articles
Authors

Birkan Durak 0000-0002-8196-5407

Hasan Ömür Özer 0000-0002-6388-4638

Şule Kapkın 0000-0003-4951-7089

Hüseyin Yıldız 0000-0002-0575-3904

Early Pub Date December 18, 2023
Publication Date December 15, 2023
Published in Issue Year 2023 Volume: 13 Issue: 4

Cite

APA Durak, B., Özer, H. Ö., Kapkın, Ş., Yıldız, H. (2023). Solution of Some Integral Equations by Point-Collocation Method. Karadeniz Fen Bilimleri Dergisi, 13(4), 1894-1905. https://doi.org/10.31466/kfbd.1372548