# Newton-Krylov generalized minimal residual algorithm in solving nonlinear Volterra-Fredholm-Hammerstein integral equations

Document Type : Original Article

Author

Malek Ashtar University of Technology, Tehran, Iran.

Abstract

In this paper, Galerkin and collocation methods based on shifted Legendre polynomials and spectral methods have been applied on nonlinear Volterra-Fredholm-Hammerstein (VFH) integral equations, these methods transfer the finding solution of a nonlinear integral equation to finding the solution of nonlinear algebraic equations, in order to solve these nonlinear algebraic equations we use Newton method composed by generalized minimal residual (NGMRes) method, the iteration number and running time for implementation of NGMRes method are important parameters that have been considered to solve this type of integral equations. These methods are applied on several various nonlinear VFH integral equations that confirm accuracy and efficiency of the methods.

Keywords 20.1001.1.27172708.2021.2.1.1.6

#### References

 M.A. Abdou, Khamis I. Mohamed, A.S. Ismail, On the numerical solutions of FredholmVolterra integral equation, Applied mathematics and computation, 146(2-3) 2003, 713-728.
 H. Asgharzadeh, I. Borazjani, A NewtonKrylov method with an approximate analytical Jacobian for implicit solution of NavierStokes equations on staggered overset-curvilinear grids with immersed boundaries, Journal of computational physics, 331 2017, 227-256.
 J. Boersma and E. Danick, On the solution of an integral equation arising in potential problems for circular and elliptic disks, SIAM Journal on Applied Mathematics, 53(4) 1993, 931-941.
 J.P. Boyd, Chebyshev and Fourier spectral methods, 2nd ed, New York Dover, 2000.
 R.L. Burden, J.D. Faires, Numerical Analysis, Youngstown State University, Youngstown, 2001.
 Y. Chen, C. Shen, A Jacobian-free Newton-GMRES (m) method with adaptive preconditioner and its application for power flow calculations, IEEE Transactions on Power Systems, 21(3) 2006, 1096-1103.
 J. Chen, C. Vuik, Globalization technique for projected NewtonKrylov methods, International Journal for Numerical Methods in Engineering, 110(7) 2017, 661-674.
 D. Fadrani, V. Rostami, K. Maleknejad, Fast iterative methods for solving of boundary nonlinear integral equations with singularity, Journal of Computational Analysis and Applications, 1(2) 1999, 219-234.
 D. Fadrani, V. Rostami, K. Maleknejad, Preconditioners for solving stochastic boundary integral equations with weakly singular kernels, Computing, 63(1) 1999, 47-67.
 Z. Gouyandeh, T. Allahviranloo, A. Armand, Numerical solution of nonlinear Volterra Fredholm Hammerstein integral equations via Tau-collocation method with convergence analysis, Journal of Computational and Applied Mathematics 100(308) 2016, 435-446.
 M. Hadizadeh, R. Azizi, A reliable computational approach for approximate solution of Hammerstein integral equations of mixed type, International Journal of Computer Mathematics 81(7) 2004, 889-900.
 M. Hadizadeh, M. Mohamadsohi, Numerical solvability of a class of Volterra-Hammerstein integral equations with noncompact kernels, Journal of Applied Mathematics, 2005(2) 2005, 171-181.
 S. Hatamzadeh-Varmazyar, M. Naser-Moghadasi, E. Babolian, Z. Masouri, Numerical approach to survey the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions, Progress In Electromagnetics Research, 81 2008, 393-412.
 G. Han, Asymptotic error expansion variation of a collocation method for Volterra Hammerstein equations, Appl. Numer. Math, 13 1993, 357-369.
 D.A. Knoll, D.E. Keyes, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, Journal of Computational Physics, 193(2) 2004, 357-397.
 E.V. Kovalenko, Some approximate methods of solving integral equations of mixed problems, Journal of Applied Mathematics and Mechanics, 53(1) 1989, 85-92.
 F. Li, Y. Li, Z. Liang, Existence of solutions to nonlinear Hammerstein integral equations and applications, Journal of Mathematical Analysis and Applications, 323(1) 2006, 209-227.
 M. Lakestani, M. Razzaghi, M. Dehghan, Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets, Mathematical problems in engineering, 2005(1) 2005,
113-121.
 L.J. Lardy, A Variation of Nystrm's Method for Hammerstein Equations, The Journal of Integral Equations, 1981, 43-60.
 A.V. Manzhirov, A mixed integral equation of mechanics and a generalized projection method of its solution, Doklady Physics, 61(10) 2016.
 H.R. Marzban, H.R. Tabrizidooz, M. Razzaghi, A composite collocation method for the nonlinear mixed VolterraFredholmHammerstein integral equations, Communications in Nonlinear Science and Numerical Simulation, 16(3) 2011, 1186-1194.
 M.V. Mirkin and A.J. Bard, Multidimensional integral equations: a new approach to solving micro electrode diffusion problems: Part 2. Applications to microband electrodes and the scanning electrochemical microscope, Journal of Electroanalytical Chemistry, 323(1-2) 1992, 29-51.
 Y. Ordokhani, Solution of nonlinear VolterraFredholmHammerstein integral equations via rationalized Haar functions, Applied Mathematics and Computation, 180(2) 2006, 436-443.
 Y. Ordokhani, Solution of FredholmHammerstein integral equations with WalshHybrid functions, International Mathematical Forum, 4(20) 2009.
 K. Parand, A. Bahramnezhad, H. Farahani, A numerical method based on rational Gegenbauer functions for solving boundary layer flow of a PowellEyring non-Newtonian fluid, Computational and Applied Mathematics, 37(5) 2018, 6053-6075
 K. Parand, M. Delkhosh, Operational matrices to solve nonlinear volterra-fredholm integro-differential equations of multi-arbitrary order, Gazi University Journal of Science, 29(4) 2016, 895-907.
 K. Parand, M. Delkhosh, Solving Volterras population growth model of arbitrary order using the generalized fractional-order of the Chebyshev functions Authors, Ricerche di Matematica, 65(1) 2016, 307-328.
 K. Parand, J.A. Rad, Numerical solution of nonlinear VolterraFredholmHammerstein integral equations via collocation method based on radial basis functions, Applied Mathematics and Computation, 218(9) 2012, 5292-5309.
 K. Parand, J.A. Rad, M. Nikarya, A new numerical algorithm based on the first kind of Modi ed Bessel function to solve population growth in a closed system, International Journal of Computer Mathematics, 91(6) 2014, 1239-1254.
 K. Paranda, M. Nikarya, A numerical method to solve the 1D and the 2D reaction diffusion equation based on Bessel functions and Jacobian free Newton-Krylov subspace methods, The European Physical Journal Plus 132(11) 2017, 1-18.
 K. Parand, S. Lati , M.M. Moayeri, M. Delkhosh, Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method For Solving Linear and Nonlinear Fokker-Planck equations, Communications in Theoretical Physics, 69(5) 2018, 519.
 J. Radlow, A two-dimensional singular integral equation of diffraction theory, Bulletin of the American Mathematical Society, (70)4 1964, 596-599.
 Y. Saad, M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM Journal on scientific and statistical computing, 7(3) 1986, 856-869.
 A. Soulaimani, N.B. Salah, Y. Saad, Enhanced GMRES acceleration techniques for some CFD problems, International Journal of Computational Fluid Dynamics, 16(1) 2002, 1-20.
 J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Publishing Company, Incorporated, 2011.
 S.C. Shiralashetti, R.A. Mundewadi, S.S. Naregal, B. Veeresh, Haar Wavelet Collocation Method for the Numerical Solution of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations, Global Journal of Pure and Applied Mathematics, 13(2) 2017, 463-474.
 M.S. Tong, A Stable Integral Equation Solver for Electromagnetic Scattering by Large Scatterers with Concave Surface, Progress In Electromagnetics Research, 74 2007, 113-130.
 E. Voltchkova, Integro-Differential Equations for Option Prices in Exponential Lvy Models, Finance and Stochastics, 9(3) 2005, 299-325.
 P. Wolfe, Eigenfunctions of the Integral Equation for the Potential of the Charged Disk, Journal of Mathematical Physics, 12(7) 1971, 1215-1218.
 S. Youse, M. Razzaghi, Legendre wavelets method for the nonlinear VolterraFredholm integral equations, Mathematics and computers in simulation, 70(1) 2005, 1-8.

### History

• Receive Date: 25 November 2020
• Revise Date: 04 January 2021
• Accept Date: 15 January 2021