Matrix method with Legendre wavelets for solving some nonlinear integral equations of the first kind with error analysis and existence of the solution

[1] Akhiezer, N.I.i. and Glazman, I.M., 2013. Theory of linear operators in Hilbert space. Courier Corporation.
[2] Baboliana, E. and Salimi Shamloo, A. 2008. Numerical solution of Volterra integral and integrodifferential equations of convolution type by using operational matrices of piecewise constant orthogonal functions. Journal of Computational and Applied Mathematics, 214(2), pp. 495–508. doi.org/10.1016/j.cam.2007.03.007
[3] Bazm, S., 2015. Bernoulli polynomials for the numerical solution of some classes of linear and
nonlinear integral equations. Journal of Computational and Applied Mathematics, 275(c), pp.44­60. doi:10.1016/j.cam.2014.07.018
[4] Conway, J.B., 2013. A course in functional analysis. Springer Science & Busi­ness Media.
[5] Erfanian, M., Gachpazan, M. and Beiglo, H., 2015. Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis. Applied Mathematics and Computation, 256(4), pp.304­312. doi:10.1016/j.amc.2015.05.010
[6] Fazeli S., 2020. Numerical methods for solving nonlinear Volterra integro­differential equations based on Hermite­Birkhoff interpolation. Iranian Journal of Numerical Analysis and Optimization, 10(2), pp.131­153. doi:10.22067/IJNAO.V10I2.85756
[7] Ghomanjani, F., Farahi, M.H. and Pariz, N., 2017. A new approach for numer­ical solution of a
linear system with distributed delays, Volterra delay­integro­differential equations, and nonlinear
Volterra­Fredholm integral equation by Bezier curves. Computational and applied mathematics, 36(3), pp.1349­1365. doi:10.1007/s40314­015­0296­2
[8] Hesameddini, E. and Riahi, M., 2019. Bernoulli Galerkin matrix method and its convergence analysis for solving system of Volterra Fredholm integro­differential equations. Iranian Journal of Science and Technology, Transactions A: Science, 43(4), pp.1203­1214. doi:10.1007/s40995­018­0584­y

[9] Hsiao, G.C., Kopp, P. and Wendland, W.L., 1980. A Galerkin collocation method for some integral equations of the first kind. Computing, 25(2), pp. 89–130. doi.org/10.1007/BF02259638

[10] Lepik, U., 2008. Solving differential and integral equations by the Haar wavelet method; revisited. International Journal of Mathematics and Computation, 1(8), pp.43­52.
[11] Maleknejad, K. and Saeedipoor, E., 2017. An efficient method based on hybrid functions for Fredholm integral equation of the first kind with convergence analysis. Applied Mathematics and Computation, 304(c), pp.93­102. doi:10.1016/j.amc.2017.01.013
[12] Maleknejad, K., Mollapourasl, R. and Alizadeh, M.,2007. Numerical solution of Volterra type integral equation of the first kind with wavelet basis. Applied Mathematics and Computation, 194(2), pp.400­405. doi:10.1016/j.amc.2007.04.031
[13] Maleknejad, K., Nosrati, M. and Najafi, E., 2012. Wavelet Galerkin method for solving singular
integral equations. Computational & Applied Mathematics, 31(2), pp.373­390. doi:10.1590/S1807­
03022012000200009

[14] Miller, K.S. and Ross, B., 1993. An introduction to the fractional calculus and fractional differential equations. Wiley. DRID:1130282270123129216
[15] Myint­U, T. and Debnath, L., 2007. Linear partial differential equations for scientists and engineers. Springer Science & Business Media.
[16] Nemati, S., 2015. Numerical solution of Volterra­Fredholm integral equations using Legendre collocation method. Journal of Computational and Applied Mathematics, 278(c), pp.29­36.
doi:10.1016/j.cam.2014.09.030
[17] Parand, K. and Rad, J., 2012. Numerical solution of nonlinear Volterra­Fredholm­Hammerstein integral equations via collocation method based on radial basis functions. Applied Mathematics and Computation, 218(9), pp.5292­5309. doi:10.1016/j.amc.2011.11.013
[18] Patil, D.P., Shinde, P.D. and Tile, G.K., 202. Volterra integral equations of first kind by using Anuj transform. International Journal of Advances in Engineering and Management, 4(5), pp.917­920.doi:10.35629/5252­0405917920
[19] Patil, D., Suryawanshi, Y. and Nehete, M., 2022. Application of Soham tranform for solving Volterra integral equations of first kind. International Advanced Research Journal in Science, Engineering and Technology, 9(4), pp.668­674. doi:10.17148/IARJSET.2022.94108
[20] Phillips, M.G., 2003. Interpolation and Approximation by Polynomials. Springer, NY.
[21] Quarteroni, A., Sacco, R. and Saleri, F. 2007. Numerical Mathematics. Springer, NY.
[22] Rivlin, T.J. 2003. An Introduction to the Approximation of Functions. Dover Publications, NY.
[23] Sahu, P. and Saha Ray, S., 2017. A new Bernoulli Wavelet method for numerical solutions of nonlinear weakly singular Volterra integro­differential equations. International Journal of Computational Methods, 14(3), pp.1750022. doi:10.1142/S0219876217500220