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

• Autorzy: Nemer Ahlem , Kaboul Hanane , Mokhtari Zouhir

Identyfikatory

DOI

10.1515/jaa-2021-2055

• Języki publikacji: EN

Abstrakty

EN

In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion

Słowa kluczowe

EN

Volterra integral equation; product integration method; linear equation

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2021
• Tom: Vol. 27, nr 2
• Strony: 289--297
• Opis fizyczny: Bibliogr. 50 poz.

Twórcy

autor

Nemer Ahlem

• Laboratoire de Mathématiques Appliquées, University of Biskra, BP 145 RP, Biskra 07000, Algeria

autor

Kaboul Hanane

• Laboratoire de Mathématiques Appliquées, University of Biskra, BP 145 RP, Biskra 07000, Algeria

autor

Mokhtari Zouhir

• Laboratoire de Mathématiques Appliquées, University of Biskra, BP 145 RP, Biskra 07000, Algeria

Bibliografia

• [1] J. M. Anderson, Mathematics for Quantum Chemistry, Courier Corporation, North Chelmsford, 2012.
• [2] S. András, Weakly singular Volterra and Fredholm-Volterra integral equations, Studia Univ. Babeş-Bolyai Math. 48 (2003), no. 3, 147-155.
• [3] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monogr. Appl. Comput. Math. 4, Cambridge University, Cambridge, 1997.
• [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.
• [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.
• [6] V. Bilet, O. Dovgoshey and J. Prestin, Boundedness of Lebesgue constants and interpolating Faber bases, preprint (2016), https://arxiv.org/abs/1610.05026.
• [7] R. C. Blattberg, B.-D. Kim and S. A. Neslin, Database Marketing: Analyzing and Managing Customers, Springer, New York, 2008.
• [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.
• [9] H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monogr. Appl. Comput. Math. 15, Cambridge University, Cambridge, 2004.
• [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.
• [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.
• [12] L. J. Corwin and R. H. Szczarba, Multivariable Calculus, Monogr. Textb. Pure Appl. Math. 64, Marcel Dekker, New York, 1982.
• [13] F. Cottet-Emard, Analyse 2: Calcul différentiel, intégrales multiples, séries de Fourier, De Boeck Supérieur, Bruxelles, 2006.
• [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.
• [15] G. Debeaumarché, F. Dorra and M. Hochart, Mathématiques PSI-PSI*: Cours complet avec tests, exercices et problèmes corrigés, Pearson Education, Paris, 2010.
• [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.
• [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.
• [18] B. S. Gan, An Isogeometric Approach to Beam Structures: Bridging the Classical to Modern Technique, Springer, Cham, 2018.
• [19] L. Grammont, M. Ahues and H. Kaboul, An extension of the product integration method to L1 with applications in astrophysics, Math. Model. Anal. 21 (2016), no. 6, 774-793.
• [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.
• [21] L. Grammont, H. Kaboul and M. Ahues, A product integration type method for solving nonlinear integral equations in L1, Appl. Numer. Math. 114 (2017), 38-46.
• [22] D. Guinin and B. Joppin, Mathématiques Analyse MP, Bréal, Paris, 2004.
• [23] M. Hazewinkel, Encyclopaedia of Mathematics. Vol. 1: A-Integral-Coordinates, Springer, New York, 2013.
• [24] K. B. Howell, Principles of Fourier Analysis, CRC Press, Boca Raton, 2001.
• [25] B. A. Ibrahimoglu, Lebesgue functions and Lebesgue constants in polynomial interpolation, J. Inequal. Appl. 2016 (2016), Paper No. 93.
• [26] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Courier Corporation, North Chelmsford, 2012.
• [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.
• [28] N. Kovvali, Theory and Applications of Gaussian Quadrature Methods, Morgan & Claypool, Williston, 2011.
• [29] P. K. Kythe and M. R. Schäferkotter, Handbook of Computational Methods for Integration, CRC Press, Boca Raton, 2004.
• [30] G. Mastroianni and G. V. Milovanović, Interpolation Processes: Basic Theory and Applications, Springer, Berlin, 2008.
• [31] P. R. Mercer, More Calculus of a Single Variable, Springer, New York, 2014.
• [32] M. Moskowitz and F. Paliogiannis, Functions of Several Real Variables, World Scientific, Hackensack, 2011.
• [33] F. N. Najm, Circuit Simulation, John Wiley & Sons, New York, 2010.
• [34] Z. Nitecki, Calculus in 3D: Geometry, Vectors, and Multivariate Calculus, American Mathematical Society, Providence, 2018.
• [35] K. M. O’Connor, Calculus: Labs for Matlab, Jones & Bartlett Learning, Burlington, 2005.
• [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.
• [37] M. J. D. Powell, Approximation Theory and Methods, Cambridge University, Cambridge, 1981.
• [38] T. J. Rivlin, An Introduction to the Approximation of Functions, Courier Corporation, North Chelmsford, 2003.
• [39] K. H. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, 1999.
• [40] I. M. Roussos, Improper Riemann Integrals, CRC Press, Boca Raton, 2016.
• [41] S. Sarfati and M. Fegyvères, Mathématiques: Méthodes, savoir-faire et astuces, Editions Bréal, Paris, 1997.
• [42] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Oxford University, Oxford, 2002.
• [43] S. J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform. 33 (2006), 109-123.
• [44] G. W. Stewart, Afternotes Goes to Graduate School: Lectures on Advanced Numerical Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998.
• [45] S. M. Stewart, How to Integrate it: A Practical Guide to Finding Elementary Integrals, Cambridge University, Cambridge, 2017.
• [46] G. Strang and K. Borre, Linear algebra, geodesy and GPS, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1997.
• [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.
• [48] S. S. M. Wong, Computational Methods in Physics and Engineering, 2nd ed., World Scientific, Singapore, 1997.
• [49] W. Y. Yang, W. Cao, T.-S. Chung and J. Morris, Applied Numerical Methods Using MATLAB, John Wiley & Sons, Hoboken, 2005.
• [50] V. A. Zorich, Mathematical Analysis. I, Springer, Berlin, 2004.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021)