Numerical Study of the System of Nonlinear Volterra Integral Equations by Using Spline Method
DOI:
https://doi.org/10.29304/jqcm.2021.13.3.836Keywords:
Volterra integral equations, Non-polynomial spline, Exact solutionAbstract
The second order non-polynomial spline function for solving system of two nonlinear Volterra integral equations is proposed in this paper. An algorithm introduced as well to numerical examples to illustrate carry out of this method. Also, we compare the absolute error of quadratic non-polynomial spline method with absolute error of linear non-polynomial spline method and the exact solution.
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References
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[12] S. Kumar, J. Singh, D.Kumar, and S. Kapoor, “New homotopy analysis transform algorithm to solve Volterra integral equation,” Aim Shams Engineering Journal, 5, (2014), PP. 243-246.
[13] T. Lotfi, and K. Mahdiani, “Fuzzy galerkin method for solving Fredholm integral equations with error analysis,” Int. J. Industrial Mathematics, 3(4), (2011), PP. 237-249.
[14] Y. Mahmoudi, “Wavelet Galerkin method for numerical solution of nonlinear integral equation,” Applied Mathematics and Computation, 167, (2005), PP.1119–1129.
[15] K. Maleknejad, P. Torabi, and S. Sauter, “Numerical solution of a non-linear Volterra integral equation,” Vietnam Journal of Mathematics, 44(1), (2016), PP. 5-28.
[16] Z. Masouri, E. Babolian, and H. Varmazyar “An expansion–iterative method for numerically solving Volterra integral equation of the first kind,” Computers and Mathematics with Applications, 59, (2010), PP.1491–1499.
[17] D. A. Maturi, “Numerical solution of system of two nonlinear Volterra integral equations,” International Journal of Computers & Technology 12(10), (2014), PP.3967- 3975.
[18] A. M. Muhammad, “Numerical solution of Volterra integral equation with delay by using non polynomial spline function,” Misan Journal for Academic staduies, 16(32), (2017), PP. 133-142.
[19] M. M. Mustafa, and S. H. Harbi, “Solution of second kind Volterra integral equations using non-polynomial spline functions,” Baghdad Science Journal, 11(2), (2014), PP.207-210.
[20] Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, 48, (2008), PP. 1144–1149.
[21] J. Saberi-Nadjafi, and M. Haidari, “Solving nonlinear integral equations in the Urysohn form by Newton–Kantorovich–quadrature method,” Computers and Mathematics with Applications, 60 (2010), PP. 2058–2065.
[22] L. Saeedi, A. F. Tari, S. H. M. Masuleh, “Numerical solution of some nonlinear Volterra integral equations of the first kind,” Applications and Applied Mathematics, An International Journal (AAM), 8(1), (2013), PP. 214 – 226.
[23] B. A. Taha, and A. J. Kadim, “Non-polynomial spline finite difference method for solving second order boundary value problem,” Journal of AL-Qadisiyah for computer science and mathematics, 10(2), (2018), PP. 92-101.
[24] S. Yalcinbas, and k. Erdem, “Approximate solutions of nonlinear volterra integral equation systems,” International Journal of Modern Physics B, 24(32), (2010), PP. 6235–6258.
[2] M. Akbar, R. Nawaz, S. Ahsan, D. Baleanu, and K. S. Nisar, “Analytical solution of system of Volterra integral equations using OHAM,” Journal of Mathematics, Hindawi, 2020 (8845491), (2020), PP. 1-9.
[3] K.M.M. Al-Abrahemee, “Modification of high performance training algorithms for solve singularly perturbed Volterra integro-differential and integral equation,” Journal of AL-Qadisiyah for computer science and mathematics, 11(1), (2019), PP.36-44.
[4] A. M. Ayal, and A. M. Muhammad, “Cubic spline for solving Volterra integral equation with delay,” Misan Research Journal, 14(27), (2018), PP. 179–189.
[5] E. Babolian, and Z. Masouri, “Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions,” Journal of Computational and Applied Mathematics, 220, (2008), PP. 51 – 57.
[6] M. Sh. Bani issa, A. A. Hamoud, Giniswamy, and K. P. Ghadle, “Solving nonlinear volterra integral equations by using numerical techniques,” International Journal of Advances in Applied Mathematics and Mechanics, 6, (2019), PP: 50-45.
[7] J. Biazar, E. Babolian , and R. Islam, “ Solution of a system of Volterra integral equations of the first kind by Adomian method,” 139, (2003), PP.249-258.
[8] B. Cahlon, , and L.J. Nachman, “Numerical solutions of Volterra integral equations with a solution dependent delay,”Journal of Mathematical Analysis and Applicatins, 112, (1985), PP. 541-562.
[9] D. Costarelli, and R Spigler, “Solving Volterra integral equations of the second kind by sigmoidal functions approximation,” Journal of Integral Equations and Applications, 25(2), (2013), PP: 193-222.
[10] B. F. Jumaa, and A. H. Taqi, “Non-polynomial spline method for the solution of the system of two nonlinear Volterra integral equations,” Kirkuk University Journal /Scientific Studies (KUJSS), 11(3), (2016), PP.15-25.
[11] A. J., Kalee, H. H. Omran, and S. J. Majeed, “Modified mid-pointmethod method for solving linear Fredholm integral equations of the second kind,” Journal of Al-Qadisiyah for Computer Science and Mathematics, 3(1),(2011), PP. 28-46.
[12] S. Kumar, J. Singh, D.Kumar, and S. Kapoor, “New homotopy analysis transform algorithm to solve Volterra integral equation,” Aim Shams Engineering Journal, 5, (2014), PP. 243-246.
[13] T. Lotfi, and K. Mahdiani, “Fuzzy galerkin method for solving Fredholm integral equations with error analysis,” Int. J. Industrial Mathematics, 3(4), (2011), PP. 237-249.
[14] Y. Mahmoudi, “Wavelet Galerkin method for numerical solution of nonlinear integral equation,” Applied Mathematics and Computation, 167, (2005), PP.1119–1129.
[15] K. Maleknejad, P. Torabi, and S. Sauter, “Numerical solution of a non-linear Volterra integral equation,” Vietnam Journal of Mathematics, 44(1), (2016), PP. 5-28.
[16] Z. Masouri, E. Babolian, and H. Varmazyar “An expansion–iterative method for numerically solving Volterra integral equation of the first kind,” Computers and Mathematics with Applications, 59, (2010), PP.1491–1499.
[17] D. A. Maturi, “Numerical solution of system of two nonlinear Volterra integral equations,” International Journal of Computers & Technology 12(10), (2014), PP.3967- 3975.
[18] A. M. Muhammad, “Numerical solution of Volterra integral equation with delay by using non polynomial spline function,” Misan Journal for Academic staduies, 16(32), (2017), PP. 133-142.
[19] M. M. Mustafa, and S. H. Harbi, “Solution of second kind Volterra integral equations using non-polynomial spline functions,” Baghdad Science Journal, 11(2), (2014), PP.207-210.
[20] Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, 48, (2008), PP. 1144–1149.
[21] J. Saberi-Nadjafi, and M. Haidari, “Solving nonlinear integral equations in the Urysohn form by Newton–Kantorovich–quadrature method,” Computers and Mathematics with Applications, 60 (2010), PP. 2058–2065.
[22] L. Saeedi, A. F. Tari, S. H. M. Masuleh, “Numerical solution of some nonlinear Volterra integral equations of the first kind,” Applications and Applied Mathematics, An International Journal (AAM), 8(1), (2013), PP. 214 – 226.
[23] B. A. Taha, and A. J. Kadim, “Non-polynomial spline finite difference method for solving second order boundary value problem,” Journal of AL-Qadisiyah for computer science and mathematics, 10(2), (2018), PP. 92-101.
[24] S. Yalcinbas, and k. Erdem, “Approximate solutions of nonlinear volterra integral equation systems,” International Journal of Modern Physics B, 24(32), (2010), PP. 6235–6258.
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Published
2021-08-16
How to Cite
Khalaf, A. J., & Taha, B. A. (2021). Numerical Study of the System of Nonlinear Volterra Integral Equations by Using Spline Method. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(3), Math Page 34– 43. https://doi.org/10.29304/jqcm.2021.13.3.836
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Math Articles
