Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

• Autorzy: Georgieva Atanaska

Identyfikatory

DOI

10.1515/dema-2021-0005

• Języki publikacji: EN

Abstrakty

EN

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Słowa kluczowe

EN

homotopy analysis method; two-dimensional nonlinear fuzzy Volterra integral equation; convergence; error estimation

PL

homotopijna metoda analizy; równanie całkowe Volterry dwuwymiarowe nieliniowe; nieliniowe równania całkowe; zbieżność; oszacowanie błędu

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2021
• Tom: Vol. 54, nr 1
• Strony: 11--24
• Opis fizyczny: Bibliogr. 31 poz., tab.

Twórcy

autor

Georgieva Atanaska

• University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4003 Plovdiv, Bulgaria

Bibliografia

• [1] C. Wu and M. Ma, On the integrals, series and integral equations of fuzzy set-valued functions, J. Harbin Inst. Tech. 21(1990), 11-19.
• [2] A. Alidema and A. Georgieva, Adomian decomposition method for solving two-dimensional nonlinear Volterra fuzzy integral equations, AIP Conference Proceedings 2048 (2018), 050009, DOI: https://doi.org/10.1063/1.5082108.
• [3] Sh. Sadigh Behzadi, Solving fuzzy nonlinear Volterra-Fredholm Integral equations by using Homotopy analysis and Adomian decomposition methods, J. Fuzzy Set Valued Anal. 35(2011), 1-13, DOI: https://doi.org/10.5899/2011/jfsva-00067.
• [4] A. M. Bica and S. Ziari, Open fuzzy cubature rule with application to nonlinear fuzzy Volterra integral equations in two dimensions, Fuzzy Sets Syst. 358(2019), 108-131, DOI: https://doi.org/10.1016/j.fss.2018.04.010.
• [5] A. Georgieva, Solving two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations by using Adomian decomposition method, Dynamic Systems and Applications 27(2018), no. 4, 819-837.
• [6] A. Georgieva and A. Alidema, Convergence of Homotopy perturbation method for solving of two-dimensional fuzzy Volterra functional integral equations, Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence, vol. 793, Springer, Nature Switzerland AG, 2019, pp. 129-145, DOI: https://doi.org/10.1007/978-3-319-97277-0_11.
• [7] A. Georgieva, A. Pavlova, and S. Enkov, Iterative method for numerical solution of two-dimensional nonlinear Urisohn fuzzy integral equations, Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence, vol. 793, Springer, Nature Switzerland AG, 2019, pp. 147–161, DOI: https://doi.org/10.1007/978-3-319-97277-0_12.
• [8] A. Georgieva, A. Pavlova, and I. Naydenova, Error estimate in the iterative numerical method for two-dimensional nonlinear Hammerstein-Fredholm fuzzy functional integral equations, Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence, vol. 728, Springer, Nature Switzerland AG, 2018, pp. 41–45, DOI: https://doi.org/10.1007/978-3-319-65530-7_5.
• [9] S. J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
• [10] M. Ghanbari, Numerical solution of fuzzy linear Volterra integral equations of the second kind by Homotopy analysis method, Int. J. Ind. Math. 2(2010), no. 2, 73–87.
• [11] S. J. Liao, Notes on the Homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14(2009), no. 4, 983–997, DOI: https://doi.org/10.1016/j.cnsns.2008.04.013.
• [12] M. A. Vali, M. J. Agheli, and S. G. Nezhad, Homotopy analysis method to solve two-dimensional fuzzy Fredholm integral equation, Gen. Math. Notes 22(2014), no. 1, 31–43.
• [13] S. Abbasbandy, E. Magyari, and E. Shivanian, The homotopy analysis method for multiple solutions of nonlinear boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 14(2009), no. 9–10, 3530–3536, DOI: https://doi.org/10.1016/j.cnsns.2009.02.008.
• [14] A. S. Bataineh, M. S. M. Noorani, and I. Hashim, Approximate analytical solutions of systems of PDEs by homotopy analysis method, Comput. Math. Appl. 55(2008), no. 12, 2913–2923, DOI: https://doi.org/10.1016/j.camwa.2007.11.022.
• [15] S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer, Higher Education Press, Berlin Beijing, 2012.
• [16] S. J. Liao, On the homotopy analysis method for nonlinear problem, Appl. Math. Comput. 147(2004), no. 2, 499–513,DOI: https://doi.org/10.1016/S0096-3003(02)00790-7.
• [17] A. Georgieva and S. Hristova, Homotopy analysis method to solve two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations, Fractal Fract. 4(2020), 9, DOI: https://doi.org/10.3390/fractalfract4010009.
• [18] A. Georgieva and I. Naydenova, Application of homotopy analysis method for solving of two-dimensional linear Volterra fuzzy integral equations, AIP Conf. Proc. 2159(2019), 030012, DOI: https://doi.org/10.1063/1.5127477.
• [19] G. J. Klir, U. S. Clair, and B. Yuan, Fuzzy Set Theory: Foundations and Applications, Prentice-Hall Inc, United States, 1997.
• [20] D. Dubois and H. Prade, Towards fuzzy differential calculus, Part 2: integration of fuzzy intervals, Fuzzy Sets Syst. 8(1982), no. 2, 105–116, DOI: https://doi.org/10.1016/0165-0114(82)90001-X.
• [21] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions, Fuzzy Sets Syst. 120(2001), no. 3, 523–532, DOI: https://doi.org/10.1016/S0165-0114(99)00057-3.
• [22] C. Wu and C. Wu, The supremum and infimum of these to fuzzy-numbers and its applications, J. Math. Anal. Appl. 210(1997), no. 2, 499–511, DOI: https://doi.org/10.1006/jmaa.1997.5406.
• [23] S. M. Sadatrasoul and R. Ezzati, Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations, Abstr. Appl. Anal. 2014(2014), 413570, DOI: https://doi.org/10.1155/2014/413570.
• [24] R. A. van Gorder and K. Vajravelu, On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simul. 14(2009), no. 12, 4078–4089, DOI: https://doi.org/10.1016/j.cnsns.2009.03.008.
• [25] M. Russo and R. A. van Gorder, Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems, Appl. Math. Comput. 219(2013), no. 12, 6494–6509, DOI: https://doi.org/10.1016/j.amc.2012.12.049.
• [26] Z. M. Odibat, A study on the convergence of homotopy analysis method, Appl. Math. Comput. 217(2010), no. 2, 782–789, DOI: https://doi.org/10.1016/j.amc.2010.06.017.
• [27] S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall-CRC Press, Boca Raton, 2003.
• [28] S. M. Sadatrasoul and R. Ezzati, Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula, J. Comput. Appl. Math. 292(2016), 430–446, DOI: https://doi.org/10.1016/j.cam.2015.07.023.
• [29] A. Bica and S. Ziari, Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions, Soft Comput. 21(2017), 1097–1108, DOI: https://doi.org/10.1007/s00500-016-2085-2.
• [30] Y. Cherruault, Convergence of Adomianas method, Kybernetes 18(1989), no. 2, 31–38, DOI: https://doi.org/10.1108/eb005812.
• [31] I. L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Appl. Math. Lett. 21(2008), no. 4, 372–376, DOI: https://doi.org/10.1016/j.aml.2007.05.008.

Uwagi

PL

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