In the current study, a fast, accurate, and reliable numerical scheme for approximating second-kind nonlinear Fredholm, Volterra, and Fredholm-Volterra integral equations with a weakly singular kernel and invertible nonlinearity is presented. The computational approach is based upon function, especially the hybrid one. Hybrid functions give us the opportunity to achieve an appropriate solution by adjusting a suitable order for polynomials’ degrees and block-pulse functions. The basic idea of this method is based on using the invertibility of the nonlinear function as a benefit to reduce the total error and simplify the procedure. The scheme reduces these types of equations to nonlinear systems of algebraic equations. Convergence analysis of the method under the infinity norm is wellstudied. Numerical results indicate the superiority of the present method compared with another existing method in the literature
[1] H. Adibi and P. Assari, On the numerical solution of weakly singular Fredholm integral equations of the second kind using Legendre wavelets, Journal of Vibration and Control, 17 (2011), 689–698.
[2] M. N. Ahmadabadi and H. L. Dastjerdi, Tau approximation method for the weakly singular Volterra-Ham-merstein integral equations, Appl. Math. Comput., 285 (2016), 241–247.
[3] M. Ahmadian, H. Afshari, and M. Heydari, Numerical solution of It-Volterra integral equation by least squares method, Numerical Algorithms, 84 (2020), 591–602.
[4] A. Aghajani, J. Bana´s, and Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl., 62 (2011), 1215–1227.
[5] P. Assari and M. Dehghan, The numerical solution of nonlinear weakly singular Fredholm integral equations based on the dual-Chebyshev wavelets, Applied and Computational Mathematics, 19 (2020), 3–19.
[6] K. E. Atkinson, An existence theorem for Abel integral equations, SIAM J. Math. Anal., 5 (1974), 729–736.
[7] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge university press, 1997.
[8] Z. Avazzadeh, M. Heydari, and G. Brid Loghmani, Chebyshev cardinal functions for solving Volterra-Fredholm integrodifferential equations using operational matrices, Iranian Journal of Science and Technology (Sciences), 36 (2012), 13–24.
[9] Z. Avazzadeh, M. Heydari, G. Wenchen, and G. Brid Loghmani, Smooth solution of partial integro-differential equations using radial basis functions, The Journal of Applied Analysis and Computation, 4 (2014), 115–127.
[10] P. Baratella and A. P. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equa- tions, J. Comput. Appl. Math., 163 (2004), 401–418.
[11] H. Brunner, A. Pedas, and G. Vainikko, The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations, Math Comp., 68 (1999), 1079–1095.
[12] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge M. A., 2004.
[13] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods on Fluid Dynamics, Springer-Verlag, 1988.
[14] M. Colangeli, Small scale hydrodynamics in applications of chaos and nonlinear dynamics in science and engi- neering, Springer International Publishing, 4 (2015), 65–104.
[15] M. A. Darwish, On quadratic integral equation of fractional orders, J Math Anal Appl., 311 (2005), 112–119.
[16] K. B. Datta and B. M. Mohan, Orthogonal Functions in Systems and Control, World Sci. Publishing Co., 1995.
[17] R. Dehbozorgi and K. Nedaiasl, Numerical solution of nonlinear weakly singular Volterra integral equations of the first kind: An hp-version collocation approach, Applied Numerical Mathematics, 161 (2021), 111–136.
[18] R. Dehbozorgi and K. Maleknejad, Direct Operational Vector Scheme for First-Kind Nonlinear Volterra Integral Equations and Its Convergence Analysis, Mediterranean Journal of Mathematics, 18 (2021), 1–22.
[19] J. Eshaghi, H. Adibi, and S. Kazem, Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudo-spectral method, Math Methods Appl. Sci., 39 (2016), 3411–3425.
[20] M. Gholamian, J. Saberi-Nadjafi, and A. R. Soheili, Cubic B-splines collocation method for solving a partial integro differential equation with a weakly singular kernel, Computational Methods Differential Weakly Singular for Equations, 7 (2019), 497–510.
[21] W. Hackbusch, Stability for Discretization of Integral Equations. In The Concept of Stability in Numerical Math- ematics, Springer, Berlin, Heidelberg, 2014, 167–184.
[22] M. Heydari, G. Brid Loghmani, S. M. Hosseini, and S. M. Karbassi, Application of hybrid functions for solving duffing-harmonic oscillator, Journal of Difference Equations, 2014 (2014), 1–9.
[23] H. Kaneko, R. Noren, and Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel, Integral Equations and Operator Theory, 13 (1990), 660–670.
[24] N. Karamollahi, M. Heydari, and G. Brid Loghmani, Approximate solution of nonlinear Fredholm integral equa- tions of the second kind using a class of Hermite interpolation polynomials, Mathematics and Computers in Simulation, 187 (2021), 414–432.
[25] N. Karamollahi, M. Heydari, and G. Brid Loghmani, An interpolation-based method for solving Volterra integral equations, Journal of Applied Mathematics and Computing, 2021, 1–32.
[26] E. G. Ladopoulos, Singular integral equations: linear and non-linear theory and its applications in science and engineering, Springer Science & Business Media, 2013.
[27] U. Lepik and E. Tamme, Solution of nonlinear Fredholm integral equations via the Haar wavelet method, Proc Estonian Acad Sci Phys Math., 56 (2007), 17–27.
[28] X. Li, T. Tang, and C. Xu, Numerical solutions for weakly singular Volterra integral equations using Chebyshev and Legendre pseudo-spectral Galerkin methods, Journal of Scientific Computing, 67 (2016), 43–64.
[29] C. Lubich, Rung-Kutta theory for Volterra and Abel integral equations of the second kind, Math Comput., 41 (1983), 87–102.
[30] K. Maleknejad and A. Ebrahimzadeh, The use of rationalized Haar wavelet collocation method for solving optimal control of Volterra integral equations, J. Vib. Control, 21 (2015), 1958–1967.
[31] K. Maleknejad and H. S. Kalalagh, Approximate solution of some nonlinear classes of Abel integral equations using hybrid expansion, Applied Numerical Mathematics, 159 (2021), 61–72.
[32] K. Maleknejad, R. Mollapourasl, and A. Ostadi, Convergence analysis of Sinc-collocation methods for nonlinear Fredholm integral equations with a weakly singular kernel, J. Comput. Appl. Math., 278 (2015), 1–11.
[33] A. S. Mohamed, Shifted Jacobi collocation method for Volterra-Fredholm integral equation, Computational Meth- ods for Differential Equations, Accepted manuscript available online from 01 May 2021.
[34] D. O´regan, R. P. Agarwal, and K. Perera, Nonlinear integral equations singular in the dependent variable, Appl. Math. Lett., 20 (2007), 1137–1141.
[35] S. Paul, M. M. Panja, and B. N. Mandal, Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis, J Comput Appl Math., 300 (2016), 275–289.
[36] M. Saffarzadeh, G. Brid Loghmani, and M. Heydari, An iterative technique for the numerical solution of nonlinear stochastic It Volterra integral equations, Journal of Computational and Applied Mathematics, 333 (2017), 74–86.
[37] M. Saffarzadeh, M. Heydari, and G. Brid Loghmani, Convergence analysis of an iterative numerical algorithm for solving nonlinear stochastic It-Volterra integral equations with m-dimensional Brownian motion, Applied Nu- merical Mathematics, 146 (2014), 182–198.
[38] M. Saffarzadeh, M. Heydari, and G. Barid Loghmani, Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic ItVolterra integral equations, Mathematical Methods in the Applied Sciences, 43 (2020), 5212–5233.
[39] M. N. Sahlan and H. Feyzollahzadeh, Operational matrices of Chebyshev polynomials for solving singular Volterra integral equations, Mathematical Sciences, 11 (2017), 165–171.
[40] L. Zhu and Y. Wang, Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method, Appl Math Comput., 260 (2015), 63–70.
Mahmoodi Darani, N. (2023). Hybrid collocation method for some classes of second-kind nonlinear weakly singular integral equations. Computational Methods for Differential Equations, 11(1), 183-196. doi: 10.22034/cmde.2022.47657.1992
MLA
Narges Mahmoodi Darani. "Hybrid collocation method for some classes of second-kind nonlinear weakly singular integral equations". Computational Methods for Differential Equations, 11, 1, 2023, 183-196. doi: 10.22034/cmde.2022.47657.1992
HARVARD
Mahmoodi Darani, N. (2023). 'Hybrid collocation method for some classes of second-kind nonlinear weakly singular integral equations', Computational Methods for Differential Equations, 11(1), pp. 183-196. doi: 10.22034/cmde.2022.47657.1992
VANCOUVER
Mahmoodi Darani, N. Hybrid collocation method for some classes of second-kind nonlinear weakly singular integral equations. Computational Methods for Differential Equations, 2023; 11(1): 183-196. doi: 10.22034/cmde.2022.47657.1992
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