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Bernstein polynomials based iterative method for solving fractional integral equations

  • Zoltan Satmari EMAIL logo and Alexandru Mihai Bica
Published/Copyright: December 4, 2022
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Abstract

A novel iterative numerical method is constructed for solving second kind Volterra fractional integral equations. The method uses at each iterative step a Bernstein spline interpolation procedure combined with the corresponding quadrature formula. In this way, based on the nice approximation and shape preserving properties of the Bernstein polynomials, we propose an alternative to the classical product integration technique that uses trapezoidal, Simpson, Gauss type and other well-known quadrature formulas. The convergence of the method is proved with the error estimate expressed in terms of the Lipschitz constants and the accuracy is illustrated on some numerical experiments.

MSC 2010: Primary 65R20; 65R20
  1. (Communicated by Michal Fečkan )

References

[1] Abbas, S.—Benchohra M.: Fractional order Riemann-Liouville integral equations with multiple time delays, Appl. Math. E-Notes 12 (2012), 79–87.10.1007/978-1-4614-4036-9_8Search in Google Scholar

[2] Abbas, S.—Benchohra M.: Fractional order integral equations of two independent variables, Appl. Math. Comput. 227 (2014), 755–761.10.1016/j.amc.2013.10.086Search in Google Scholar

[3] Agarwal, R.—Jain, S.—Agarwal, R. P.: Solution of fractional Volterra integral equation and non-homogeneous time fractional heat equation using integral transform of pathway type, Progr. Fract. Differ. Appl. 1(3) (2015), 145–155.Search in Google Scholar

[4] Agarwal, R. P.—Benchohra, M.—Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), 973–1033.10.1007/s10440-008-9356-6Search in Google Scholar

[5] Agheli, B.—Adabitabar Firozja, M.: A fuzzy transform method for numerical solution of fractional Volterra integral equations, Int. J. Appl. Comput. Math. 6 (2020), Art. No. 5.10.1007/s40819-019-0758-0Search in Google Scholar

[6] Alkahtani, B. S. T.—Atangana, A.: Analysis of non-homogeneous heat model with new trend of derivative with fractional order, Chaos Solitons Fractals 89 (2016), 566–571.10.1016/j.chaos.2016.03.027Search in Google Scholar

[7] Alkahtani, B. S. T.: Modeling the transmission dynamics of flagellated protozoan parasite with Atangana–Baleanu derivative: Application of 3/8 Simpson and Boole’s numerical rules for fractional integral, Chaos Solitons Fractals 115 (2018), 212–223.10.1016/j.chaos.2018.07.036Search in Google Scholar

[8] Alqudah, M. A.—Mohammed, P. O.—Abdeljawad, T.: Solution of singular integral equations via Riemann–Liouville fractional integrals, Math. Probl. Eng. 2020 (2020), Art. ID 1250970.10.1155/2020/1250970Search in Google Scholar

[9] Amin, R.—Shah, K.—Asif, M.—Khan, I.—Ullah, F.: An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math. 381 (2021), Art. ID 113028.10.1016/j.cam.2020.113028Search in Google Scholar

[10] András, S.: Weakly singular Volterra and Fredholm-Volterra integral equations, Studia Univ. Babeş-Bolyai Math. 48(3) (2003), 147–155.Search in Google Scholar

[11] Atangana, A.—Bildik, N.: Existence and numerical solution of the Volterra fractional integral equations of the second kind, Math. Probl. Eng. 2013 (2013), 1–12.10.1155/2013/981526Search in Google Scholar

[12] Atangana, A.—Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Sci. 20(2) (2016), 763–769.10.2298/TSCI160111018ASearch in Google Scholar

[13] Atkinson, K. E.: An Introduction to Numerical Analysis, 2nd ed., John Wiley & Sons, New York, 1989.Search in Google Scholar

[14] Atkinson, K. E.: The numerical solution of an Abel integral equation by a product trapezoidal method, SIAM J. Numer. Anal. 11(1) (1974), 97–101.10.1137/0711011Search in Google Scholar

[15] Baleanu, D.—Diethelm, K.—Scalas, E.—Trujillo, J. J.: Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, vol. 3, World Scientific Publishers, Co., N. Jersey, London, Singapore, 2012.10.1142/8180Search in Google Scholar

[16] Bachher, M.—Sarkar, N.—Lahiri, A.: Fractional order thermoelastic interactions in an infinite porous material due to distributed timedependent heat sources, Meccanica 50 (2015), 2167–2178.10.1007/s11012-015-0152-xSearch in Google Scholar

[17] Bagley, R. L.—Calico, R. A.: Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Contr. Dynam. 14 (1991), 304–311.10.2514/6.1989-1213Search in Google Scholar

[18] Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp. 45(172) (1985), 417–437.10.1090/S0025-5718-1985-0804933-3Search in Google Scholar

[19] Brunner, H.—Pedas, A.—Vainikko, G: The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations, Math. Comp. 68(227) (1999), 1079–1095.10.1090/S0025-5718-99-01073-XSearch in Google Scholar

[20] Cai, M.—Li, C.: Numerical approaches to fractional integrals and derivatives: a review, Mathematics 8(1) (2020) 43.10.3390/math8010043Search in Google Scholar

[21] Cakana, U.—Ozdemirb, I.: Existence of nondecreasing solutions of some nonlinear integral equations of fractional order, J. Nonlinear Sci. Appl. 8(6) (2015), 1112–1126.10.22436/jnsa.008.06.20Search in Google Scholar

[22] Darwish, M. A.: On existence and asymptotic behavior of solutions of a fractional integral equation, Appl. Anal. 88(2) (2009), 169–181.10.1080/00036810802713800Search in Google Scholar

[23] De Angelis, P.—De Marchis, R.—Martire, A. L.—Oliva, I: A mean-value approach to solve fractional differential and integral equations, Chaos Solitons Fractals 138 (2020), Art. ID 109895.10.1016/j.chaos.2020.109895Search in Google Scholar

[24] Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Math. 2004, Springer-Verlag Berlin Heidelberg, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[25] Diogo, T: Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math. 229 (2009), 363–372.10.1016/j.cam.2008.04.002Search in Google Scholar

[26] Du, M.—Wang, Z.—Hu, H.: Measuring memory with order of fractional derivative, Scientific Reports 3 (2013), Art. No. 3431.10.1038/srep03431Search in Google Scholar PubMed PubMed Central

[27] Dubey, V. P.—Kumar, R.—Kumar, D.: Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration, Phys. A 521 (2019), 762–772.10.1016/j.physa.2019.01.094Search in Google Scholar

[28] Ezzat, M. A.—Sabbah, A. S.—El-Bary, A. A.—Ezzat, S. M.: Thermoelectric viscoelastic fluid with fractional integral and derivative heat transfer, Adv. Appl. Math. Mech. 7 (2015), 528–548.10.4208/aamm.2013.m333Search in Google Scholar

[29] Gorenflo, R.—Vessella, S.: Abel Integral Equations: Analysis and Applications. Lecture Notes in Math. 1461, Springer Verlag, Berlin, 1991.10.1007/BFb0084665Search in Google Scholar

[30] Goswami, A.—Sushila, J.—Singh, J.—Kumar, D.: Numerical computation of fractional Kersten-Krasil’shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math. 5(3) (2020), 2346–2368.10.3934/math.2020155Search in Google Scholar

[31] Hamdan, S.—Qatanani, N.—Daraghmeh, A.: Numerical techniques for solving linear Volterra fractional integral equation, J. Appl. Math. 2019 (2019), Art. ID 5678103.10.1155/2019/5678103Search in Google Scholar

[32] Ibrahim, R. W.— Momani, S.: Upper and lower bounds of solutions for fractional integral equations, Surv. Math. Appl. 2 (2007), 145–156.Search in Google Scholar

[33] Jahanshahi, S.—Babolian, E.—Torres, D. F. M.—Vahidi, A. R.: A fractional Gauss–Jacobi quadrature rule for approximating fractional integrals and derivatives, Chaos Solitons Fractals 102 (2017), 295–304.10.1016/j.chaos.2017.04.034Search in Google Scholar

[34] Kilbas, A. A.—Srivastava, H. M.—Trujillo, J. J.: Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.Search in Google Scholar

[35] Kumar, S.—Kumar, A.—Kumar, D.—Singh, J.—Singh, A.: Analytical solution of Abel integral equation arising in astrophysics via Laplace transform, J. Egyptian Math. Soc. 23 (2015), 102–107.10.1016/j.joems.2014.02.004Search in Google Scholar

[36] Kumar, D.—Singh, J.—Baleanu, D.—Sushila: Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffer type kernel, Phys. A 492 (2018), 155–167.10.1016/j.physa.2017.10.002Search in Google Scholar

[37] Labora, D. C.: Fractional integral equations tell us how to impose initial values in fractional differential equations, Mathematics 8 (2020), Art. ID 1093.10.3390/math8071093Search in Google Scholar

[38] Lakshmikantham, V.—Leela, S.—Vasundhara, J.: Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.Search in Google Scholar

[39] Lubich, C.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp. 41(163) (1983), 87–102.10.1090/S0025-5718-1983-0701626-6Search in Google Scholar

[40] Micula, S.: An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math. 339 (2018) 124–133.10.1016/j.cam.2017.12.006Search in Google Scholar

[41] Miller, K. S.—Ross, B.: An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.Search in Google Scholar

[42] Mohammad, M.—Trounev, A.: Implicit Riesz wavelets based-method for solving singular fractional integro-differential equations with applications to hematopoietic stem cell modeling, Chaos Solitons Fractals 138 (2020), Art. ID 109991.10.1016/j.chaos.2020.109991Search in Google Scholar PubMed PubMed Central

[43] Muskhelishvili, N. I.—Radok, J. R. M.: Singular Integral Equations: Boundary Problems of Function Theory and their Application to Mathematical Physics, Courier Corporation, Chelmsford, 2008.Search in Google Scholar

[44] Plato, R.: Fractional multistep methods for weakly singular Volterra integral equations of the first kind with perturbed data, Numer. Funct. Anal. Optim. 26(2) (2005), 249–269.10.1081/NFA-200064396Search in Google Scholar

[45] Podlubny, I.: Fractional Differential Equation, Academic Press, San Diego, 1999.Search in Google Scholar

[46] Te Riele, H. J. J.: Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal. 2 (1982), 437–449.10.1093/imanum/2.4.437Search in Google Scholar

[47] Lorentz, G. G.: Bernstein Polynomials, Univ. of Toronto Press, Toronto, 1953.Search in Google Scholar

[48] Saeedi, H.—Mollahasani, N.—Moghadam, M. M.—Chuev, G. N.: An operational Haar wavelet method for solving fractional Volterra integral equations, Int. J. Appl. Math. Comput. Sci. 21(3) (2011), 535–547.10.2478/v10006-011-0042-xSearch in Google Scholar

[49] Sajjadi, S. A.—Pishbin, S.: Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels, Numerical Algorithms 86 (2021), 25–54.10.1007/s11075-020-00877-xSearch in Google Scholar

[50] Schneider, C.: Product integration for weakly singular integral equations, Math. Comp. 36(153) (1981), 207–213.10.1090/S0025-5718-1981-0595053-0Search in Google Scholar

[51] Shikrani, R.—Hashmi, M. S.—Khan, N.—Ghaffar, A.—Nisar, K. S.—Singh, J.—Kumar, D.: An efficient numerical approach for space fractional partial differential equations, Alexandria Engineering J. 59 (2020), 2911–2919.10.1016/j.aej.2020.02.036Search in Google Scholar

[52] Singh, J.—Kumar, D.—Qurashi, M. A.—Baleanu, D.: A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs and spraying, Adv. Differ. Equ. 2019 (2019), Art. No. 278.10.1186/s13662-019-2199-9Search in Google Scholar

[53] Srivastava, H.—Dubey, V. P.—Kumar, R.—Singh, J.—Kumar, D.—Baleanu, D.: An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Solitons Fractals 138 (2020), Art. ID 109880.10.1016/j.chaos.2020.109880Search in Google Scholar

[54] Sulaiman, T. A.—Yavuz, M.—Bulut, H.—Baskonus, H. M.: Investigation of the fractional coupled viscous Burgers equation involving Mittag-Leffler kernel, Phys. A 527 (2019), 121–126.10.1016/j.physa.2019.121126Search in Google Scholar

[55] Sweilam, N. H.—AL-Mekhlaf, S. M.—Almutairi, A.—Baleanu, D.: A hybrid fractional COVID-19 model with general population mask use: Numerical treatments, Alexandria Engineering J. 60 (2021), 3219–3232.10.1016/j.aej.2021.01.057Search in Google Scholar

[56] Talaei, Y.—Shahmorad, S.—Mokhtary, P.: A new recursive formulation of the Taumethod for solving linear Abel–Volterra integral equations and its application to fractional differential equations, Calcolo 56 (2019), Art. No. 50.10.1007/s10092-019-0347-ySearch in Google Scholar

[57] Torvik, P. J.—Bagley, R. L.: On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51 (1984), 294–298.10.1115/1.3167615Search in Google Scholar

[58] Usta, F.: Numerical analysis of fractional Volterra integral equations via Bernstein approximation method, J. Comput. Appl. Math. 384 (2021), Art. ID 113198.10.1016/j.cam.2020.113198Search in Google Scholar

[59] Verma, V.—Prakash, V.—Kumar, D.—Singh, J.: Numerical study of fractional model of multi-dimensional dispersive partial differential equation, J. Ocean Engin. Science 4 (2019), 338–351.10.1016/j.joes.2019.06.001Search in Google Scholar

[60] Wazwaz, A. M.: A First Course in Integral Equations, World Scientific, Singapore, 1997.10.1142/3444Search in Google Scholar

[61] Wu, G. C.—Baleanu, D.: Variational iteration method for fractional calculus - a universal approach by Laplace transform, Adv. Differential Equations 2013(18) (2013), 1–9.10.1186/1687-1847-2013-18Search in Google Scholar

[62] Yavuz, M.—Ozdemir, N.: European vanilla option pricing model of fractional order without singular kernel, Fractal Fractional 2(1) (2018), 3.10.3390/fractalfract2010003Search in Google Scholar

[63] Yousefi, S. A.: Numerical solution of Abel’s integral equation by using Legendre wavelets, Appl. Math. Comput. 175 (2006), 574–580.10.1016/j.amc.2005.07.032Search in Google Scholar

[64] Yousefi, A.—Javadi, S.—Babolian, E.: A computational approach for solving fractional integral equations based on Legendre collocation method, Math. Sciences 13 (2019), 231–240.10.1007/s40096-019-0292-6Search in Google Scholar

[65] Zheng, B.: Explicit bounds derived by some new inequalities and applications in fractional integral equations, J. Inequal. Appl. 2014(4) (2014), 1–12.10.1186/1029-242X-2014-4Search in Google Scholar

Received: 2021-04-12
Accepted: 2021-11-05
Published Online: 2022-12-04
Published in Print: 2022-12-16

© 2022 Mathematical Institute Slovak Academy of Sciences

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