Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis

References

• P. Assari, H. Adibi, and M. Dehghan, The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis, Appl. Numer. Math. 81 (2014), pp. 76–93. doi: 10.1016/j.apnum.2014.02.013
   Web of Science ®Google Scholar
• P. Baratella and A.P. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004), pp. 401–418. doi: 10.1016/j.cam.2003.08.047
   Web of Science ®Google Scholar
• H. Brunner, Implicitly linear collocation methods for nonlinear Volterra equations, Appl. Numer. Math. 9 (1992), pp. 235–247. doi: 10.1016/0168-9274(92)90018-9
   Web of Science ®Google Scholar
• C. Canuto, M.Y. Hussaini, A. Quarteroni, and A.T. Zang, Spectral Methods. Fundamentals in Single Domains, Springer, Berlin, 2006.
   Google Scholar
• Z. Chen, C.A. Micchelli, and Y. Xu, The Petro-Galerkin methods for second kind integral equations II: multiwavelet scheme, Adv. Comput. Math. 7 (1997), pp. 199–233. doi: 10.1023/A:1018994802659
   Web of Science ®Google Scholar
• M.K. EL-Daou and H.G. Khajah, Iterated solutions of linear operator equations with the Tau method, Math. Comput. 66(217) (1997), pp. 207–214. doi: 10.1090/S0025-5718-97-00803-X
   Web of Science ®Google Scholar
• F. Fakhar-Izadi and M. Dehghan, Space-time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains, Comput. Math. Appl. 67(10) (2014), pp. 1884–1904. doi: 10.1016/j.camwa.2014.03.016
   Web of Science ®Google Scholar
• J.-S. Guf and W.-S. Jiang, The Haar Wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996), pp. 623–628. doi: 10.1080/00207729608929258
   Web of Science ®Google Scholar
• S.M. Hosseini and S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation, Appl. Math. Comput. 136 (2003), pp. 559–570.
   Web of Science ®Google Scholar
• A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, 1999.
   Google Scholar
• S. Karimi Vanani and F. Soleymani, Tau approximate solution of weakly singular Volterra integral equations, Math. Comput. 57 (2013), pp. 494–502.
   Google Scholar
• S. Kumar and I.H. Sloan, A new collocation-type method for Hammerstein integral equations, J. Math. Comput. 48 (1987), pp. 123–129. doi: 10.1090/S0025-5718-1987-0878692-4
   Web of Science ®Google Scholar
• M. Lakestani, B. Nemati Saray, and M. Dehghan, Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets, J. Comput. Appl. Math. 235(11) (2011), pp. 3291–3303. doi: 10.1016/j.cam.2011.01.043
   Web of Science ®Google Scholar
• L.J. Lardy, A variation of Nystrom's method for Hammerstein equations, J. Int. Eqs. 3 (1981), pp. 43–60.
   Web of Science ®Google Scholar
• X. LI and T. Tang, Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind, Front. Math. China. 7 (2012), pp. 69–84. doi: 10.1007/s11464-012-0170-0
   Web of Science ®Google Scholar
• B.N. Mandal and A. Chakrabarti, Applied Singular Integral Equations, Science Publishers, Stevenson Ranch, CA, 2011.
   Google Scholar
• E.L. Ortiz and H. Samara, An operational approach to the Tau method for the numerical solution of non-linear differential equations, Computing 27 (1981), pp. 15–25. doi: 10.1007/BF02243435
   Web of Science ®Google Scholar
• R. Piessens, Computing integral transforms and solving integral equations using Chebyshev polynomial approximations, J. Comput. Appl. Math. 121 (2000), pp. 113–124. doi: 10.1016/S0377-0427(00)00349-6
   Web of Science ®Google Scholar
• M. Razzaghi and S. Yousefi, The Legendre Wavelets operational matrix of integration, Int. J. Syst. Sci. 32(4) (2001), pp. 495–502. doi: 10.1080/00207720120227
   Web of Science ®Google Scholar
• H.J.J.T. Riele, Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal. 2 (1982), pp. 437–449. doi: 10.1093/imanum/2.4.437
   Web of Science ®Google Scholar
• A. Saadatmandi and M. Dehghan, A collocation method for solving Abel's integral equations of first and second kinds, Z. Naturforsch. 63a(10) (2008), pp. 752–756.
   Google Scholar
• A. Saadatmandi and M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method, Numer. Meth. Part. D. E. 26(1) (2010), pp. 239–252. doi: 10.1002/num.20442
   Web of Science ®Google Scholar
• S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993.
   Google Scholar
• J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.
   Google Scholar
• A.N. Tikhonov and V.Y. Arsenin, Solution of Ill-Posed Problems, V.H. Winston and Sons, Washington, 1977.
   Google Scholar
• N. Zeilon, Sur quelques points de la theorie de equation integrale Abel, Arkiv. Mat. Astr. Fysik. 18 (1924), pp. 1–19.
   Google Scholar