Jacobi Wavelets Method for Numerical Solution of Weakly Singular Volterra Integral Equation

Document Type : Research Paper


1 Math Department, Tabriz Branch, Islamic Azad University

2 Math Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran

3 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran

4 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

5 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran


In this paper, the first and second kind weakly singular Volterra integral equations are approximated by using the Jacobi wavelets method. First, the operational matrices for fractional integration and product for Jacobi wavelets are computed with a new matrix approach, and then, it applied to solve numerically the first and second kind Volterra integral equations involving singularity. Illustrative numerical experiments with comparison are included to indicate the validity and practicability of the method.


  1. M. Rahman, Integral Equations and their Applications, Wit Press, Southampton, Boston, (2007).
  2. A. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, (1997).
  3. M. Metrovic, The modified decomposition method for eighth-order boundary value problems, Applied Mathematics and Computation, 188, 1437–1444 (2007).
  4. S. Siddiqi, G. Akram, Septic spline solutions of sixth-order boundary value problems, J. Comput. Appl. Math., 215, 288–301 (2008).
  5. Z. Chen, W. Jiang, An approximate solution for a mixed linear volterra-fredholm integral equation, Appl. Math. Lett., 25, 1131–1134 (2012).
  6. M. Noor, S. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Appl. Math. Comput., 189, 1929–1942 (2007).
  7. H. El-Hawary, T. El-Sheshtawy, Spectral method for solving the general form linear Fredholm-Volterra integro differential equations based on chebyshev polynomials, J. Mod. Met. Numer. Math., 1, 1–11 (2010).
  1. K. Maleknejad, N. Aghazadeh, Numerical solution of volterra integral equations of the second kind with convolution kernel by using taylor-series expansion method, Appl. Math. Comput., 161, 915–922 (2005).
  2. K. Maleknejad, R. Mollapourasl, M. Alizadeh, Numerical solution of the volterra type integral equation of the first kind with wavelet basis, Appl. Math. Comput., 194, 400–405 (2007).
  3. F. Usta, On new modification of bernstein operators: theory and applica- tions, Iran. J. Sci. Technol. Trans. Sci., 44, 1119–1124 (2020).
  4. F. Usta, Numerical analysis of fractional volterra integral equations via bernstein approximation method, J. Comput. Appl. Math., 384, 113198 (2021).
  5. R. Dehbozorgi, K. Nedaiaslb, Numerical solution of nonlinear weakly sin- gular volterra integral equations of the first kind: An hp-version collocation approach, Appl. Numer. Math., 161, 111–136 (2021).
  6. I. Podlubny, Fractional differential equations, Academic Press, San Diego, Calif, USA, (1999).
  7. K. Sadri, A. Amini, C. Cheng, A new operational method to solve abel’s and generalized abel’s integral equations, Applied Mathematics and Computation, 317, 49–67 (2018).
  8. A. Borhanifar, K. Sadri, A new operational approach for numerical solution of generalized functional integro-differential equations, Journal of Computational and Applied Mathematics, 279, 80–96 (2015).
  9. J. Guf, W. Jiang, The haar wavelet operational matrix of integratio, International Journal of Systems Science, 27, 623–628 (1996).
  10. E. Fathizadeh, R. Ezzati, K. Maleknejad, Cas wavelet function method for solving abel equations with error analysis, International Journal of Research in Industrial Engineering, 6, 350–364 (2017).
  11. S. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Engineering Journal., 57, 2591–2600 (2018).
  12. G. Phillips, Interpolation and Approximation by Polynomials, Springer, (2003).
  13. S. Micula, An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math., 339, 124–133 (2018).
  14. R. Plato, Fractional multistep methods for weakly singular volterra integral equations of the first kind with perturbed data, Numer. Funct. Anal. Optim., 26, 249–269 (2005).