An Iterative Method for Solving Two Dimensional Nonlinear Volterra Integral Equations

Document Type : Research Paper

Author

Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran

Abstract

In this paper, a numerical iterative algorithm based on combination of the successive approximations method and the quadrature formula for solving two-dimensional nonlinear Volterra integral equations is proposed. This algorithm uses a trapezoidal quadrature rule for Lipschitzian functions applied at each iterative step. The convergence analysis and error estimate of the method are proved. Finally, two numerical examples are presented to show the accuracy of the proposed method.

Keywords

References

  1. Qi. Tang and D. Waxman, An integral equation describing an asexual population in a changing environment. Nonlinear Analysis: Theory, Methods & Applications, 53(5), 683-699 (2003).
  2. R.M. Evans , U.N. Katugampola , D.A. Edwards, Applications of fractional calculus in solving Abel-type integral equations: surface-volume reaction problem. Comput. Math. Appl, 73 (6), 1346-1362 (2017).
  3. B. G. Pachpatte, Multidimensional integral equations and inequalities, volume 9. Springer Science & Business Media (2011).
  4. A. Jerri, Introduction to integral equations with applications. John Wiley & Sons (1999).
  5. A. G. Ramm, Dynamical systems method for solving operator equations. Communications in Nonlinear Science and Numerical Simulation, 9(4), 383-402 (2004).
  6. S. McKee, T. Tang, and T. Diogo, An Euler-type method for two-dimensional volterra integral equations of the first kind. IMA Journal of Numerical Analysis, 20(3), 423-440 (2000).
  7. R. J. Hanson and J. L. Phillips, Numerical solution of two-dimensional integral equations using linear elements. SIAM Journal on Numerical Analysis, 15(1), 113-121 (1978).
  8. G. Han and R. Wang, Richardson extrapolation of iterated discrete galerkin solution for two-dimensional fredholm integral equations. Journal of Computational and Applied Mathematics, 139(1), 49-63 (2002).
  9. H. Brunner, Collocation methods for Volterra integral and related functional differential equations, volume 15. Cambridge University Press (2004).
  10. K. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations. SIAMJ.Numer.Anal, 24, 1352-1373 (1987).
  11. H. Fatahi , J. Saberi-Nadjafi, E. Shivanian, A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional fredholm integral equations on general domains with error analysis. J. Comput. Appl. Math., 294, 196-209 (2016).
  12. S. Bazm, A. Hosseini, Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type. Computational and Applied Mathematics, 39(2), 49 (2020).
  13. H. Guoqiang , K. Itayami , K. Sugihara , W. Jiong, Extrapolation method of iterated collocation solution for two-dimensional nonlinear volterra integral equations. Appl. Math. Comput., 112 (1), 49-61 (2000) .
  14. Z. Avazzadeh , M. Heydari, Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind. Comput. Appl. Math. 31, 127-142 (2012).
  15. S. Fazeli, G. Hojjati, and H. Kheiri, A piecewise approximation for linear twodimensional volterra integral equation by chebyshev polynomials. International Journal of Nonlinear Science, 16(3), 255-261 (2013).
  16. I. Aziz, F. Khan, et al, A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations. Journal of Computational and Applied Mathematics, 272,70-80 (2014).
  17. F. Mirzaee , S. Alipour, Numerical solution of nonlinear partial quadratic integrodifferential equations of fractional order via hybrid of block-pulse and parabolic functions. Numer. Methods Partial Different Equ. ,35 (3), 1134-1151 (2019).
  18. S. Nemati , Y. Ordokhani, Solving nonlinear two-dimensional Volterra integral equations of the first-kind using the bivariate shifted Legendre functions. Int J Math Model Comput, 5(3),1–12 (2015).
  19. K. Maleknejad, N. Aghazadeh, and M. Rabbani, Numerical solution of second kind fredholm integral equations system by using a taylor-series expansion method. Applied Mathematics and Computation, 175(2), 1229-1234 (2006).
  20. A.A. Khajehnasiri, Numerical solution of nonlinear 2D Volterra–Fredholm integrodifferential equations by two-dimensional triangular function. Int J Appl Comput Math ,2(4), 575-591 (2016).
  21. Z. Avazzadeh , M. Heydari , G.B. Loghmani, A comparison between solving two dimensional integral equations by the traditional collocation method and radial basis functions. Appl. Math. Sciences. 5, 1145-1152 (2011).
  22. F. Mirzaee , N. Samadyar, Using radial basis functions to solve two dimensional linear stochastic integral equations on non-rectangular domains, Eng. Anal. Bound. Elem, 92, 180-195 (2018).
  23. S. Najafalizadeh , R. Ezzati, Numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using two-dimensional block pulse operational matrix, Appl. Math. Comput, 280, 46-56 (2016).
  24. E. Babolian, S. Bazm, and P. Lima, Numerical solution of nonlinear two-dimensional integral equations using rationalized haar functions. Communications in Nonlinear Science and Numerical Simulation, 16(3), 1164-1175 (2011).
  25. H. Guoqiang and W. Jiong, Extrapolation of nystrom solution for two-dimensional nonlinear fredholm integral equations. Journal of Computational and Applied Mathematics, 134(1), 259-268, (2001).
  26. F. Mirzaee , S. Alipour, Fractional order orthogonal bernstein polynomials for numerical solution of nonlinear fractional partial volterra integro-differ- ential equations. Math. Methods Appl. Sci. 42 (6), 1870-1893 (2019).
  27. F. Khan , M. Omar, Z. Ullah, Discretization method for the numerical solution of 2D Volterra integral equation based on two-dimensional Bernstein polynomial. AIP Advances, 8, 125209 (2018).
  28. Adhraa M. Muhammad, A. M. Ayal, Numerical Solution of Linear Volterra Integral Equation with Delay using Bernstein Polynomial. International Electronic Journal of Mathematics Education ,14(3), 735-740 (2019).
  29. A. Alturk, The regularization-homotopy method for the two-dimensional fredholm integral equations of the first kind. Mathematical and Computational Applications, 21(2), 9, (2016).
  30. E. Hetmaniok , I. Nowak , D. Sota, A. Zielonka, Homotopy approach for solving twodimensional integral equations of the second kind. Comput. Assist. Method. Eng. Sci. 23 (1), 19–41 (2017).
  31. M. Kazemi, H. M. Golshan, R. Ezzati, M. Sadatrasoul. New approach to solve twodimensional Fredholm integral equations, Journal of Computational and Applied Mathematics, 354, 66-79 (2019).
  32. M. Kazemi, R. Ezzati, Numerical solution of two-dimensional nonlinear integral equations via quadrature rules and iterative method, Advances in Differential Equations and Control Processes, 17, 27-56 (2016).
  33. A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear fredholm integral equations of the second kind. Journal of Computational and Applied Mathematics, 232(2), 449-454 (2009).
  34. A.-M. Wazwaz, Linear and nonlinear integral equations: methods and applications. Springer Science & Business Media (2011).
  35. A. M. Bica , M. Curila and S. Curila, About a numerical method of successive interpolations for functional Hammerstein integral equations. Journal of Computational and Applied Mathematics, 236(2), 2005-2024 (2012).
  36. F. Mirzaee, E. Hadadiyan, Application of two-dimensional hat functions for solving space-time integral equations. Journal of Applied Mathematics and Computing, 51(2), 453-486 (2016).
  37. M.A. Abdou , A.A. Badr , M.B. Soliman , On a method for solving a two-dimensional nonlinear integral equation of the second kind. J Comput Appl Math 235(12), 3589-3598 (2014).
  38. SA. Hosseini , S. Shahmorad , A. Tari , Existence of an Lp-solution for two dimensional integral equations of the Hammerstein type. Bull Iran Math Soc 40(4), 851-862 (2014).
  39. R. P. Agarwal, N. Hussain, and M.-A. Taoudi, Fixed point theorems in ordered banach spaces and applications to nonlinear integral equations. In Abstract and Applied Analysis, volume 2012. Hindawi Publishing Corporation (2012).
  40. M. Kazemi and R. Ezzati, Existence of Solutions for some Nonlinear Volterra Integral Equations via Petryshyn’s Fixed Point Theorem. Int. J. Nonlinear Anal. Appl. , 9, 1-12 (2018).
  41. M. Kazemi and R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness. Applied Mathematics and Computation, 275, 165-171 (2016).
  42. K. Maleknejad, R. Mollapourasl, K. Nouri, Study on existence of solutions for some nonlinear functional–integral equations. Nonlinear Anal, 69, 2582-2588 (2008).
Analytical and Numerical Solutions for Nonlinear Equations
Volume 6, Issue 1
August 2021
Pages 41-57

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History

  • Receive Date: 22 November 2020
  • Revise Date: 14 January 2021
  • Accept Date: 09 February 2021

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