Study on interval Volterra integral equations via parametric approach of intervals
DOI:
https://doi.org/10.24193/subbmath.2025.2.09Keywords:
Interval IVP, interval integral equation, parametric approach, successive approximation method, resolvent kernelAbstract
This work investigates the interval Volterra integral equation (IVIE) and its solution techniques through the parametric representation of intervals. First, the general form of the second-kind IVIE is expressed in both lower-upper bound format and its equivalent parametric form. Next, the methods of successive approximations and resolvent kernel are developed to solve the IVIE, utilizing parametric approaches and interval arithmetic. The solutions are presented in both parametric and lower-upper bound representations. Lastly, a series of numerical examples are provided to illustrate the application of these methods.
Mathematics Subject Classification (2010): 45G10, 45D05,45N05.
Received 15 September 2024; Accepted 15 October 2024.
References
1. Abbasbandy, S., Babolian, E., Alavi, M., Numerical method for solving linear Fredholm fuzzy integral equations of the second kind, Chaos Solitons Fractals, 31(2007), no. 1, 138-146.
2. Agarwal, R.P., O’Regan, D., Lakshmikantham, V., Fuzzy Volterra integral equations: A stacking theorem approach, Appl. Anal., 83(2004), no. 5, 521-532.
3. Agheli, B., Firozja, M.A., A fuzzy transform method for numerical solution of fractional Volterra integral equations, Int. J. Appl. Comput. Math., 6(2020), no. 1, 1-13.
4. Ahmady, N., A numerical method for solving fuzzy differential equations with fractional order, Int. J. Ind. Math., 11(2019), no. 2, 71-77.
5. An, T.V., Phu, N.D., Van Hoa, N., A note on solutions of interval-valued Volterra integral equations, J. Integral Equ. Appl., (2014), 1-14.
6. Attari, H., Yazdani, A., A computational method for fuzzy Volterra-Fredholm integral equations, Fuzzy Inf. Eng., 3(2011), no. 2, 147.
7. Babolian, E., Goghary, H.S., Abbasbandy, S., Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput., 161(2005), no. 3, 733-744.
8. Bica, A.M., Popescu, C., Approximating the solution of nonlinear Hammerstein fuzzy integral equations, Fuzzy Sets Syst., 245(2014), 1-17.
9. Buckley, J.J., Feuring, T., Fuzzy differential equations, Fuzzy Sets Syst., 110(2000), no. 1, 43-54.
10. da Costa, T.M., Chalco-Cano, Y., Lodwick, W.A., Silva, G.N., A new approach to linear interval differential equations as a first step toward solving fuzzy differential, Fuzzy Sets Syst., 347(2018), 129-141.
11. Kaleva, O., Fuzzy differential equations, Fuzzy Sets Syst., 24(1987), no. 3, 301-317.
12. Lupulescu, V., Van Hoa, N., Interval Abel integral equation, Soft Comput., 21(2017), no. 10, 2777-2784.
13. Malinowski, M.T., Interval differential equations with a second type Hukuhara derivative, Appl. Math. Lett., 24(2011), no. 12, 2118-2123.
14. Mao, X., Existence and uniqueness of the solutions of delay stochastic integral equations, Stochastic Anal. Appl., 7(1989), no. 1, 59-74.
15. Mao, X., Yuan, C., Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.
16. Mirzaee, F., Hamzeh, A., A computational method for solving nonlinear stochastic Volterra integral equations, J. Comput. Appl. Math., 306(2016), 166-178.
17. Mohammadi, F., Second kind Chebyshev wavelet Galerkin method for stochastic Ito-Volterra integral equations, Mediterr. J. Math., 13(2016), no. 5, 2613-2631.
18. Mordeson, J., Newman, W., Fuzzy integral equations, Inf. Sci., 87(1995), no. 4, 215-229.
19. Noeiaghdam, S., Araghi, M.A.F., Abbasbandy, S., Valid implementation of Sinc-collocation method to solve the fuzzy Fredholm integral equation, J. Comput. Appl. Math., 370(2020), 112632.
20. Ogawa, S., Stochastic integral equations for the random fields, Sémin. Probab. Strasbourg, 25(1991), 324-329.
21. Ogawa, S., Stochastic integral equations of Fredholm type, in "Harmonic, Wavelet and P-Adic Analysis", (2007), 331-342.
22. Otadi, M., Mosleh, M., Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network, Neurocomputing, 205(2016), 519-528.
23. Ramezanzadeh, M., Heidari, M., Fard, O., Borzabadi, A., On the interval differential equation: Novel solution methodology, Adv. Differ. Equ., (2015), 10.1186/s13662-015- 0671-8.
24. Samadyar, N., Mirzaee, F., Numerical solution of two-dimensional weakly singular stochastic integral equations on non-rectangular domains via radial basis functions, Eng. Anal. Bound. Elem., 101(2019), 27-36.
25. Samadyar, N., Mirzaee, F., Orthonormal Bernoulli polynomials collocation approach for solving stochastic Ito-Volterra integral equations of Abel type, Int. J. Numer. Model., 33(2020), no. 1, e2688.
26. Stefanini, L., Bede, B., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. Theory Methods Appl., 71(2009), no. 3-4, 1311-1328.
27. Subrahmanyam, P.V., Sudarsanam, S.K., A note on fuzzy Volterra integral equations, Fuzzy Sets Syst., 81(1996), no. 2, 237-240.
28. Vorobiev, D., Seikkala, S., Towards the theory of fuzzy differential equations, Fuzzy Sets Syst., 125(2002), no. 2, 231-237.
29. Wang, B., Zhu, Q., Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems, Syst. Control Lett., 105(2017), 55-61.
30. Yong, J.M., Backward stochastic Volterra integral equations – A brief survey, Appl. Math. J. Chin. Univ., 28 (2013), no. 4, 383-394.
31. Zakeri, K.A., Ziari, S., Araghi, M.A.F., Perfilieva, I., Efficient numerical solution to a bivariate nonlinear fuzzy Fredholm integral equation, IEEE Trans. Fuzzy Syst., (2019).
32. Ziari, S., Perfilieva, I., Abbasbandy, S., Block-pulse functions in the method of successive approximations for nonlinear fuzzy Fredholm integral equations, Differ. Equ. Dyn. Syst., (2019), 1-15.
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