An effective and simple scheme for solving nonlinear Fredholm integral equations
Abstract
In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.
Keyword : Fredholm integral equation, Lagrange polynomials, Gauss-Legendre integration, interpolation, convergence and stability
How to Cite
Shahsavaran, A., & Fotros, F. (2022). An effective and simple scheme for solving nonlinear Fredholm integral equations. Mathematical Modelling and Analysis, 27(2), 215–231. https://doi.org/10.3846/mma.2022.14194
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Molabahrami. A modified degenerate kernel method for the system of Fredholm integral equations of the second kind. Iranian Journal of Mathematical Sciences and Informatics, 14(1):43–53, 2019.
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S. Panda, S.C. Martha and A. Chakrabarti. A modified approach to numerical solution of Fredholm integral equations of the second kind. Applied Mathematics and Computation, 271:102–112, 2015. https://doi.org/10.1016/j.amc.2015.08.111
B.L. Panigrahi, M. Mandal and G. Nelakanti. Legendre multi-Galerkin methods for Fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem. Journal of Computational and Applied Mathematics, 346:224–236, 2019. https://doi.org/10.1016/j.cam.2018.07.010
A. Rahmoune. Spectral collocation method for solving Fredholm integral equations on the half-line. Applied Mathematics and Computation, 219(17):9254– 9260, 2013. https://doi.org/10.1016/j.amc.2013.03.043
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M.H.A. Sathar, A.F.N. Rasedee, A.A. Ahmedov and N. Bachok. Numerical solution of nonlinear Fredholm and Volterra integrals by NewtonKantorovich and haar wavelets methods. Symmetry, 12(12), 2020. https://doi.org/10.3390/sym12122034
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J. Stoer and R. Bulirsch. Introduction to numerical analysis. Springer-Verlag, 1991.
P.N. Swarztrauber. On computing the points and weights for Gauss-Legendre quadrature. SIAM Journal on Scientific Computing, 24(8):945–954, 2003. https://doi.org/10.1137/S1064827500379690
A.M. Wazwaz. Linear and nonlinear integral equations: methods and applications. Higher education, Springer, 2011.
M.A. Zaky and A.S. Hendy. Convergence analysis of a Legendre spectral collocation method for nonlinear Fredholm integral equations in multidimensions. Mathematical Methods in the Applied Sciences, 2020. https://doi.org/10.1002/mma.6443
H. Almasieh and M. Roodaki. Triangular functions method for the solution of Fredholm integral equations system. Ain Shams Engineering Journal, 3(4):411– 416, 2012. https://doi.org/10.1016/j.asej.2012.04.006
P. Assari. A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique. Journal of Applied Analysis & Computation, 9(1):75–104, 2019. https://doi.org/10.11948/2019.75
P. Assari. The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions. Filomat, 33(3):667–682, 2019. https://doi.org/10.2298/FIL1903667A
E. Babolian and A. Shahsavaran. Numerical solution of nonlinear Fredholm integral equations of the second kind using haar wavelets. Journal of Computational and Applied Mathematics, 1(225):87–95, 2009. https://doi.org/10.1016/j.cam.2008.07.003
S. Bazm. Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations. Journal of Computational and Applied Mathematics, 275:44–60, 2015. https://doi.org/10.1016/j.cam.2014.07.018
H. Beyrami, T. Lotfi and K. Mahdiani. A new efficient method with error analysis for solving the second kind Fredholm integral equations with Cauchy kernel. Journal of Computational and Applied Mathematics, 300(C):385–399, 2016. https://doi.org/10.1016/j.cam.2016.01.011
M.C. Bounaya, S. Lemita, M. Ghiat and M.Z. Aissaoui. On a nonlinear integro-differential equation of Fredholm type. International Journal of Computing Science and Mathematics, 13(2), 2021. https://doi.org/10.1504/IJCSM.2021.114188
V. Cristini, J. Lowengrub and Q. Nie. Nonlinear simulation of tumor growth. Journal of Mathematical Biology, 46:191–224, 2003. https://doi.org/10.1007/s00285-002-0174-6
A. Daddi-Moussa-Ider, B. Kaoui and H. Lo¨wen. Axisymmetric flow due to a Stokeslet near a finite-sized elastic membrane. Journal of the Physical Society of Japan, 88(5), 2019. https://doi.org/10.7566/JPSJ.88.054401
L.M. Delves and J.L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, 1985. https://doi.org/10.1017/CBO9780511569609
M. Derakhshan and M. Zarebnia. On the numerical treatment and analysis of Hammerstein integral equation. Computational Methods for Differential Equations, 9(2):493–510, 2021. https://doi.org/10.22034/cmde.2019.29825.1435
S.A. Edalatpanah. Legendre wavelets collocation method for systems of linear and nonlinear Fredholm integral equations. Aloy Journal of Soft Computing and Applications, 2(1):33–44, 2014.
D. Elliott. A Chebyshev series method for the numerical solution of Fredholm integral equations. The Computer Journal, 6(1):102–112, 1963. https://doi.org/10.1093/comjnl/6.1.102
R. Ezzati and S. Ziari. Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method. Applied Mathematics and Computation, 225:33–42, 2013. https://doi.org/10.1016/j.amc.2013.09.020
F. Fattahzadeh. Numerical solution of general nonlinear Fredholm-Volterra integral equations using Chebyshev approximation. International Journal Industrial Mathematics, 8(1):81–86, 2016.
L. Fermo. A Nystro¨m method for a class of Fredholm integral equations of the third kind on unbounded domains. Applied Numerical Mathematics, 59(12):2970–2989, 2009. https://doi.org/10.1016/j.apnum.2009.07.002
G. He, S. Xiang and Z. Xu. A Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels. Journal of Computational and Applied Mathematics, 300:354–368, 2016. https://doi.org/10.1016/j.cam.2015.12.027
M.A. Hernández-Verón and E. Martínez. On nonlinear Fredholm integral equations with non-differentiable Nemystkii operator. Mathematical Methods in the Applied Sciences, 43(14):7961–7976, 2020. https://doi.org/10.1002/mma.5801
K.G.T. Hollands. The simplified-Fredholm integral equation solver and its use in thermal radiation. ASME Journal of Heat Transfer, 132(2):1–6, 2010. https://doi.org/10.1115/1.4000183
N. Karamollahi, M. Heydari and G.B. Loghmani. Approximate solution of nonlinear Fredholm integral equations of the second kind using a class of Hermite interpolation polynomials. Mathematics and Computers in Simulation, 187:414– 432, 2021. https://doi.org/10.1016/j.matcom.2021.03.015
R. Katani. Numerical solution of the Fredholm integral equations with a quadrature method. SeMA Journal, 76(2):271–276, 2019. https://doi.org/10.1007/s40324-018-0175-z
W.V. Lovitt. Linear Integral Equations. Dover, New York, 1950.
Y. Lu, L. Shen and Y. Xu. Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Problems, 26(4):1–32, 2010. https://doi.org/10.1088/0266-5611/26/4/045006
N. Madbouly. Solutions of Hammerstein integral equations arising from chemical reactor theory (Ph.D. thesis). University of Strathclyde, 1996.
K. Maleknejad, J. Rashidinia and H. Jalilian. Quintic spline functions and Fredholm integral equation. Computational Methods for Differential Equations, 9(1):211–224, 2021. https://doi.org/10.22034/cmde.2019.31983.1492
A. Molabahrami. A modified degenerate kernel method for the system of Fredholm integral equations of the second kind. Iranian Journal of Mathematical Sciences and Informatics, 14(1):43–53, 2019.
H. Saberi Najafi, S.A. Edalatpanah and H. Aminikhah. An algorithmic approach for solution of nonlinear Fredholm-Hammerstein integral equations. Iranian Journal Science and Technology, 39(3):399–406, 2015. https://doi.org/10.22099/IJSTS.2015.3263
S. Panda, S.C. Martha and A. Chakrabarti. A modified approach to numerical solution of Fredholm integral equations of the second kind. Applied Mathematics and Computation, 271:102–112, 2015. https://doi.org/10.1016/j.amc.2015.08.111
B.L. Panigrahi, M. Mandal and G. Nelakanti. Legendre multi-Galerkin methods for Fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem. Journal of Computational and Applied Mathematics, 346:224–236, 2019. https://doi.org/10.1016/j.cam.2018.07.010
A. Rahmoune. Spectral collocation method for solving Fredholm integral equations on the half-line. Applied Mathematics and Computation, 219(17):9254– 9260, 2013. https://doi.org/10.1016/j.amc.2013.03.043
P.K. Sahu, A.K. Ranjan and S.S. Ray. B-spline wavelet method for solving Fredholm Hammerstein integral equation arising from chemical reactor theory. Nonlinear Engineering, 7(3):163–169, 2018. https://doi.org/10.1515/nleng-2017-0116
M.H.A. Sathar, A.F.N. Rasedee, A.A. Ahmedov and N. Bachok. Numerical solution of nonlinear Fredholm and Volterra integrals by NewtonKantorovich and haar wavelets methods. Symmetry, 12(12), 2020. https://doi.org/10.3390/sym12122034
G. Spiga, R.L. Bowden and V.C. Boffi. On the solution to a class of nonlinear integral equations arising in transport theory. Journal of Mathematical Physics, 25(12):3444–3450, 1984. https://doi.org/10.1063/1.526099
J. Stoer and R. Bulirsch. Introduction to numerical analysis. Springer-Verlag, 1991.
P.N. Swarztrauber. On computing the points and weights for Gauss-Legendre quadrature. SIAM Journal on Scientific Computing, 24(8):945–954, 2003. https://doi.org/10.1137/S1064827500379690
A.M. Wazwaz. Linear and nonlinear integral equations: methods and applications. Higher education, Springer, 2011.
M.A. Zaky and A.S. Hendy. Convergence analysis of a Legendre spectral collocation method for nonlinear Fredholm integral equations in multidimensions. Mathematical Methods in the Applied Sciences, 2020. https://doi.org/10.1002/mma.6443