An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel

References

[1] J. M. Anderson, Mathematics for Quantum Chemistry, Courier Corporation, North Chelmsford, 2012. Search in Google Scholar

[2] S. András, Weakly singular Volterra and Fredholm–Volterra integral equations, Studia Univ. Babeş-Bolyai Math. 48 (2003), no. 3, 147–155. Search in Google Scholar

[3] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monogr. Appl. Comput. Math. 4, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511626340Search in Google Scholar

[4] 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), no. 2, 401–418. 10.1016/j.cam.2003.08.047Search in Google Scholar

[5] B. Bertram, On the product integration method for solving singular integral equations in scattering theory, J. Comput. Appl. Math. 25 (1989), no. 1, 79–92. 10.1016/0377-0427(89)90077-0Search in Google Scholar

[6] V. Bilet, O. Dovgoshey and J. Prestin, Boundedness of Lebesgue constants and interpolating Faber bases, preprint (2016), https://arxiv.org/abs/1610.05026. 10.20535/1810-0546.2017.4.108444Search in Google Scholar

[7] R. C. Blattberg, B.-D. Kim and S. A. Neslin, Database Marketing: Analyzing and Managing Customers, Springer, New York, 2008. 10.1007/978-0-387-72579-6Search in Google Scholar

[8] H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp. 45 (1985), no. 172, 417–437. 10.1090/S0025-5718-1985-0804933-3Search in Google Scholar

[9] H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monogr. Appl. Comput. Math. 15, Cambridge University, Cambridge, 2004. 10.1017/CBO9780511543234Search in Google Scholar

[10] Y. Chen and T. Tang, Convergence analysis for the Chebyshev collocation methods to Volterra integral equations with a weakly singular kernel, SIAM J. Numer. Anal. 233 (2009), 938–950. 10.1016/j.cam.2009.08.057Search in Google Scholar

[11] Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp. 79 (2010), no. 269, 147–167. 10.1090/S0025-5718-09-02269-8Search in Google Scholar

[12] L. J. Corwin and R. H. Szczarba, Multivariable Calculus, Monogr. Textb. Pure Appl. Math. 64, Marcel Dekker, New York, 1982. Search in Google Scholar

[13] F. Cottet-Emard, Analyse 2: Calcul différentiel, intégrales multiples, séries de Fourier, De Boeck Supérieur, Bruxelles, 2006. Search in Google Scholar

[14] F. d’Almeida, O. Titaud and P. B. Vasconcelos, A numerical study of iterative refinement schemes for weakly singular integral equations, Appl. Math. Lett. 18 (2005), no. 5, 571–576. 10.1016/j.aml.2004.03.020Search in Google Scholar

[15] G. Debeaumarché, F. Dorra and M. Hochart, Mathématiques PSI-PSI*: Cours complet avec tests, exercices et problèmes corrigés, Pearson Education, Paris, 2010. Search in Google Scholar

[16] T. Diogo, N. B. Franco and P. Lima, High order product integration methods for a Volterra integral equation with logarithmic singular kernel, Commun. Pure Appl. Anal. 3 (2004), no. 2, 217–235. 10.3934/cpaa.2004.3.217Search in Google Scholar

[17] J. Douchet and B. Zwahlen, Calcul différentiel et intégral: fonctions réelles d’une ou de plusieurs variables réelles, PPUR Presses Polytechniques et Universitaires Romandes, Lausanne, 2006. Search in Google Scholar

[18] B. S. Gan, An Isogeometric Approach to Beam Structures: Bridging the Classical to Modern Technique, Springer, Cham, 2018. 10.1007/978-3-319-56493-7Search in Google Scholar

[19] L. Grammont, M. Ahues and H. Kaboul, An extension of the product integration method to L 1 with applications in astrophysics, Math. Model. Anal. 21 (2016), no. 6, 774–793. 10.3846/13926292.2016.1243590Search in Google Scholar

[20] L. Grammont and H. Kaboul, An improvement of the product integration method for a weakly singular Hammerstein equation, preprint (2016), https://hal.archives-ouvertes.fr/hal-01295843. Search in Google Scholar

[21] L. Grammont, H. Kaboul and M. Ahues, A product integration type method for solving nonlinear integral equations in L 1 , Appl. Numer. Math. 114 (2017), 38–46. 10.1016/j.apnum.2016.11.006Search in Google Scholar

[22] D. Guinin and B. Joppin, Mathématiques Analyse MP, Bréal, Paris, 2004. Search in Google Scholar

[23] M. Hazewinkel, Encyclopaedia of Mathematics. Vol. 1: A-Integral-Coordinates, Springer, New York, 2013. Search in Google Scholar

[24] K. B. Howell, Principles of Fourier Analysis, CRC Press, Boca Raton, 2001. 10.1201/9781420036909Search in Google Scholar

[25] B. A. Ibrahimoglu, Lebesgue functions and Lebesgue constants in polynomial interpolation, J. Inequal. Appl. 2016 (2016), Paper No. 93. 10.1186/s13660-016-1030-3Search in Google Scholar

[26] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Courier Corporation, North Chelmsford, 2012. Search in Google Scholar

[27] B. Jumarhon and S. McKee, Product integration methods for solving a system of nonlinear Volterra integral equations, J. Comput. Appl. Math. 69 (1996), no. 2, 285–301. 10.1016/0377-0427(95)00041-0Search in Google Scholar

[28] N. Kovvali, Theory and Applications of Gaussian Quadrature Methods, Morgan & Claypool, Williston, 2011. 10.2200/S00366ED1V01Y201105ASE008Search in Google Scholar

[29] P. K. Kythe and M. R. Schäferkotter, Handbook of Computational Methods for Integration, CRC Press, Boca Raton, 2004. 10.1201/9780203490303Search in Google Scholar

[30] G. Mastroianni and G. V. Milovanović, Interpolation Processes: Basic Theory and Applications, Springer, Berlin, 2008. 10.1007/978-3-540-68349-0Search in Google Scholar

[31] P. R. Mercer, More Calculus of a Single Variable, Springer, New York, 2014. 10.1007/978-1-4939-1926-0Search in Google Scholar

[32] M. Moskowitz and F. Paliogiannis, Functions of Several Real Variables, World Scientific, Hackensack, 2011. 10.1142/7672Search in Google Scholar

[33] F. N. Najm, Circuit Simulation, John Wiley & Sons, New York, 2010. 10.1002/9780470561218Search in Google Scholar

[34] Z. Nitecki, Calculus in 3D: Geometry, Vectors, and Multivariate Calculus, American Mathematical Society, Providence, 2018. Search in Google Scholar

[35] K. M. O’Connor, Calculus: Labs for Matlab, Jones & Bartlett Learning, Burlington, 2005. Search in Google Scholar

[36] A. Palamara Orsi, Product integration for Volterra integral equations of the second kind with weakly singular kernels, Math. Comp. 65 (1996), no. 215, 1201–1212. 10.1090/S0025-5718-96-00736-3Search in Google Scholar

[37] M. J. D. Powell, Approximation Theory and Methods, Cambridge University, Cambridge, 1981. 10.1017/CBO9781139171502Search in Google Scholar

[38] T. J. Rivlin, An Introduction to the Approximation of Functions, Courier Corporation, North Chelmsford, 2003. Search in Google Scholar

[39] K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, 1999. 10.1201/9781439832905Search in Google Scholar

[40] I. M. Roussos, Improper Riemann Integrals, CRC Press, Boca Raton, 2016. 10.1201/b16296Search in Google Scholar

[41] S. Sarfati and M. Fegyvères, Mathématiques: Méthodes, savoir-faire et astuces, Editions Bréal, Paris, 1997. Search in Google Scholar

[42] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Oxford University, Oxford, 2002. 10.1093/oso/9780198502791.001.0001Search in Google Scholar

[43] S. J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform. 33 (2006), 109–123. Search in Google Scholar

[44] G. W. Stewart, Afternotes Goes to Graduate School: Lectures on Advanced Numerical Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998. 10.1137/1.9781611971422Search in Google Scholar

[45] S. M. Stewart, How to Integrate it: A Practical Guide to Finding Elementary Integrals, Cambridge University, Cambridge, 2017. 10.1017/9781108291507Search in Google Scholar

[46] G. Strang and K. Borre, Linear algebra, geodesy and GPS, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1997. Search in Google Scholar

[47] T. Tang, X. Xu and J. Cheng, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math. 26 (2008), no. 6, 825–837. Search in Google Scholar

[48] S. S. M. Wong, Computational Methods in Physics and Engineering, 2nd ed., World Scientific, Singapore, 1997. 10.1142/3365Search in Google Scholar

[49] W. Y. Yang, W. Cao, T.-S. Chung and J. Morris, Applied Numerical Methods Using MATLAB, John Wiley & Sons, Hoboken, 2005. 10.1002/0471705195Search in Google Scholar

[50] V. A. Zorich, Mathematical Analysis. I, Springer, Berlin, 2004. Search in Google Scholar