This study deals with a numerical solution of a nonlinear Volterra integral equation of the first kind. The method of this research is based on a new kind of orthogonal wavelets, called the Chebyshev cardinal wavelets. These wavelets known as new basis functions contain numerous beneficial features like orthogonality, spectral accuracy, and cardinality. In addition, we assume an expansion of the terms of Chebyshev cardinal wavelets within unknown coefficients as a substitute for an unknown solution. Relatively, considering the mentioned expansion and the cardinality feature within the generated operational matrix of the introduced wavelets, a system of nonlinear algebraic equations is extracted for the stated problem. Finally, by solving the yielded system, the estimated solution results.
[1] T. Abdeljawad, R. P. Agarwal, E. Karapnar, and P. S. Kumari, Solutions of the nonlinear integral equation and fractional diﬀerential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry, 11 (2019), 686-704.
[2] M. A. Alzhrani, H. O. Bakodah, and M. Al-mazmumy, A 3/8 Simpson’s Numerical Scheme for the Classes of Volterra Integral Equations of First Kind, Nonlinear Anal. Diﬀer. Equ., 1 (2019), 99-113.
[3] K. E. Atkinson and J. Flores, The collocation method for nonlinear integral equations, IMA J. Numer. Anal., 13 (1993), 195-213.
[4] E. Babolian and L. M. Delves, An augmented Galerkin method for first kind Fredholm equations, IMA J. Appl. Math., 24 (1979), 157-174.
[5] E. Babolian, K. Maleknejad, M. Mordad, and B. Rahimi, A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix, J. Comput. Appl. Math., 235 (2011), 39653971.
[6] E. Babolian and Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, J. Comput. Appl. Math., 220 (2008), 5157.
[7] E. Babolian and A. S. Shamloo, Numerical solution of Volterra integral and integro-diﬀerential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J. Comput. Appl. Math., 214 (2008), 495508.
[8] P. Baratella, A Nystrom interpolant for some weakly singular linear Volterra integral equations, Comput. Appl. Math., 231 (2009), 725734.
[9] M. A. Bartoshevich, On one heat conduction problem, Inz-Fiz Zh, 28 (1975), 340345.
[10] J. Biazar, E. Babolian, and R. Islam, Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. Math. Comput., 139 (2003), 249-258.
[11] J. Biazar, M. Eslami, and H. Aminikhah, Application of homotopy perturbation method for systems of Volterra integral equations of the first kind, Chaos Solitons Fractals, 42 (2009), 3020-3026.
[12] L. K. Bieniasz, Modelling electroanalytical experiments by the integral equation method, Berlin Heidelberg, Springer, 2015.
[13] H. Brunner, Collocation methods for Volterra integral and related functional diﬀerential equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2004.
[14] C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988.
[15] L. M. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge 1988.
[16] H. J. Ding, H. M. Wang, and W. Q. Chen, Analytical solution for the electrostatic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere, Arch. Appl. Mech., 73 (2003), 4962.
[17] G. Gripenberg, S.-O. Londen, and O. Staﬀans, Volterra integral and functional equations, vol. 34 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.
[18] R. C. Guru Sekar and K. Murugesan, STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind, Int. J. Comput. Math., 94 (2017), 18671878.
[19] M. H. Heydari, Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations, Appl. Numer. Math. 144 (2019), 190-203.
[20] M. H. Heydari, M. R. Mahmoudi, A. Shakiba, and Z. Avazzadeh, Chebyshev cardinal wavelets and their application in solving nonlinear stochastic diﬀerential equations with fractional Brownian motion, Commun. Nonlinear Sci. Numer. Simul. 64 (2018), 98-121.
[21] M. H. Heydari and M. Razzaghi, Extended Chebyshev cardinal wavelets for nonlinear fractional delay optimal control problems, Internat. J. Syst. Sci. 53 (2022), 190-203.
[22] M. H. Heydari, R. Tavakoli, and M. Razzaghi, Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative, Internat. J. Syst. Sci. 64 (2022), 98-121.
[23] N. Khan, M. S. Hashmi, S. Iqbal, and T. Mahmood, Optimal homotopy asymptotic method for solving Volterra integral equation of first kind, Alex. Eng. J., 53 (2014), 751755.
[24] S. Kumar, Modifications of Linz methods for nonlinear second kind Volterra integral equations with singular or periodic kernels, Journal of Mathematical and Physical Sciences, 26 (1992), 591-597.
[25] P. K. Lamm, Approximation of ill-posed Volterra problems via predictor-corrector regularization methods, SIAM J. Appl. Math., 56 (1996), 524-541.
[26] P. K. Lamm, Solution of ill-posed Volterra equations via variable-smoothing Tikhonov regularization, In Inverse Problems in Geophysical Applications, Philadelphia, 1997.
[27] P. K. Lamm and L. Eldn, Numerical solution of first kind Volterra equations by sequential Tikhonov regularization, SIAM J. Numer. Anal., 34 (1997), 1432-1450.
[28] Y. Ma, J. Huang, C. Wang, and H. Li, Sinc Nystro¨m method for a class of nonlinear Volterra integral equations of the first kind. Adv. Diﬀer. Equ., 151 (2016), 1-15.
[29] K. Maleknejad and R. Dehbozorgi, Adaptive numerical approach based upon Chebyshev operational vector for nonlinear Volterra integral equations and its convergence analysis, J. Comput. Appl. Math. 344 (2018), 356366.
[30] K. Maleknejad, E. Hashemizadeh, and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernsteins approximation, Commun. Nonlin. Sci. Numer. Simulat., 16 (2011), 647655.
[31] K. Maleknejad, R. Mollapourasl, and M. Alizadeh, Numerical solution of Volterra type integral equation of the first kind with wavelet basis, Appl. Math. Comput., 194 (2007), 400-405.
[32] K. Maleknejad, R. Mollapourasl, and K. Nouri, Convergence of numerical solution of the Fredholm integral equation of the first kind with degenerate kernel, Appl. Math. Comput. 181 (2006), 1000-1007.
[33] K. Maleknejad, K. Nouri, and L. Torkzadeh, Comparison projection method with Adomian’s decomposition method for solving system of integral equations, Bull. Malays. Math. Sci. Soc. 34 (2011), 379388.
[34] K. Maleknejad and B. Rahimi, Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind, Commun. Nonlin. Sci. Numer. Simulat., 16 (2011), 24692477.
[35] K. Maleknejad, S. Sohrabi, and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, J. Appl. Math. Comput., 188 (2007), 123128.
[36] Z. Masouri, E. Babolian, and S. Hatamzadeh-Varmazyar, An expansion-iterative method for numerically solving Volterra integral equation of the first kind, Comput. Math. Appl., 59 (2010), 14911499.
[37] F. Mirzaee, Numerical solution of optimal control problem of the non-linear Volterra integral equations via gen- eralized hat functions, IMA J. Math. Control Inf., 34 (2017), 889-904.
[38] F. Mirzaee, Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials, Comput. Methods Diﬀer. Equ., 5 (2017), 88-102.
[39] F. Mirzaee and E. Hadadiyan, Using operational matrix for solving nonlinear class of mixed Volterra-Fredholm integral equations, Math. Methods Appl. Sci., 40 (2016), 3433-3444.
[40] F. Mirzaee and S. F. Hoseini, Hybrid functions of Bernstein polynomials and block-pulse functions for solving optimal control of the nonlinear Volterra integral equations, Indag. Math., 27 (2016), 835-849.
[41] F. Mirzaee and N. Samadyar, Explicit representation of orthonormal Bernoulli polynomials and its application for solving Volterra-Fredholm-Hammerstein integral equations, SeMA Journal, 77 (2020), 8196.
[42] F. Mirzaee and N. Samadyar, Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed VolterraFredholm integral equations, Trans. A. Razmadze Math. Inst., 172 (2018), 631-64.
[43] M. Mohamadi, E. Babolian, and S. A. Yousefi, A Solution For Volterra Integral Equations of the First Kind Based on Bernstein Polynomials, Int. J. Industrial Mathematics, 10 (2018), 1-9.
[44] K. Nedaiasl, R. Dehbozorghi, and K. Maleknejad, hp-version collocation method for a class of nonlinear Volterra integral equations of the first kind, Appl. Numer. Math., 150 (2020), 452-477.
[45] N. Negarchi and K. Nouri, A new direct method for solving optimal control problem of nonlinear VolterraFredholm integral equation via the MntzLegendre polynomials, Bull. Iran. Math. Soc., 45 (2019), 917-934.
[46] N. Negarchi and K. Nouri, Numerical solution of VolterraFredholm integral equations using the collocation method based on a special form of the MntzLegendre polynomials, J. Comput. Appl. Math., 344 (2018), 15-24.
[47] S. K. Panda, A. Tassaddiq, and R. P. Agarwal, A new approach to the solution of nonlinear integral equations via various FBe -contractions, Symmetry, 11 (2019), 206-226.
[48] B. Salehi, K. Nouri, and L. Torkzadeh, An approximate method for solving optimal control problems with Chebyshev cardinal wavelets, Iranian Journal of Operations Research, 12 (2021), 20-33.
[49] I. Singh and S. Kumar, Haar wavelet method for some nonlinear Volterra integral equations of the first kind, J. Comput. Appl. Math., 292 (2016), 541552.
[50] A. N. Tikhonov and V. Y. Arsenin, Solutions of ill-posed problems, Winston, Washington DC, 1977.
[51] S. A. Yousefi, Numerical solution of Abels integral equation by using Legendre wavelets, Appl. Math. Comput., 175 (2006), 574580.
[52] T. Zhang and H. Liang, Multistep collocation approximations to solutions of first-kind Volterra integral equations, Appl. Numer. Math., 130 (2018), 171183.
Salehi, B., Nouri, K., & Torkzadeh, L. (2023). An efficient numerical approach for solving nonlinear Volterra integral equations. Computational Methods for Differential Equations, 11(3), 615-629. doi: 10.22034/cmde.2022.52804.2226
MLA
Behnam Salehi; Kazem Nouri; Leila Torkzadeh. "An efficient numerical approach for solving nonlinear Volterra integral equations". Computational Methods for Differential Equations, 11, 3, 2023, 615-629. doi: 10.22034/cmde.2022.52804.2226
HARVARD
Salehi, B., Nouri, K., Torkzadeh, L. (2023). 'An efficient numerical approach for solving nonlinear Volterra integral equations', Computational Methods for Differential Equations, 11(3), pp. 615-629. doi: 10.22034/cmde.2022.52804.2226
VANCOUVER
Salehi, B., Nouri, K., Torkzadeh, L. An efficient numerical approach for solving nonlinear Volterra integral equations. Computational Methods for Differential Equations, 2023; 11(3): 615-629. doi: 10.22034/cmde.2022.52804.2226