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.
References
Mao, X. (1989). Existence and uniqueness of the solutions of delay stochastic integral equations. Stochastic analysis and applications, 7(1), 59-74.
Ogawa, S. (1991). Stochastic integral equations for the random fields. Séminaire de probabilités de Strasbourg, 25, 324-329.
Ogawa, S. (2007). Stochastic integral equations of Fredholm type. In Harmonic, Wavelet And P-Adic Analysis (pp. 331-342).
Mirzaee, F., & Hamzeh, A. (2016). A computational method for solving nonlinear stochastic Volterra integral equations. Journal of Computational and Applied Mathematics, 306, 166-178.
Yong, J. M. (2013). Backward stochastic Volterra integral equations—a brief survey. Applied Mathematics-A Journal of Chinese Universities, 28(4), 383-394.
Mohammadi, F. (2016). Second kind Chebyshev wavelet Galerkin method for stochastic Ito-Volterra integral equations. Mediterranean Journal of Mathematics, 13(5), 2613-2631.
Samadyar, N., Mirzaee, F. (2019). Numerical solution of two-dimensional weakly singular stochastic integral equations on non-rectangular domains via radial basis functions. Engineering Analysis with Boundary Elements, 101, 27-36.
Samadyar, N., Mirzaee, F. (2020). Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô‐Volterra integral equations of Abel type. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 33(1), e2688.
Subrahmanyam, P. V., & Sudarsanam, S. K. (1996). A note on fuzzy Volterra integral equations. Fuzzy Sets and Systems, 81(2), 237-240.
Agarwal, R. P., O'Regan, D., & Lakshmikantham, V. (2004). Fuzzy Volterra integral equations: a stacking theorem approach. Applicable Analysis, 83(5), 521-532.
Attari, H., Yazdani, A. (2011). A computational method for fuzzy Volterra-Fredholm integral equations. Fuzzy Information and Engineering, 3(2), 147.
Mordeson, J., & Newman, W. (1995). Fuzzy integral equations. Information Sciences, 87(4), 215-229.
Babolian, E., Goghary, H. S., Abbasbandy, S. (2005). Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Applied Mathematics and Computation, 161(3), 733-744.
Abbasbandy, S., Babolian, E., Alavi, M. (2007). Numerical method for solving linear Fredholm fuzzy integral equations of the second kind. Chaos, Solitons & Fractals, 31(1), 138-146.
Bica, A. M., Popescu, C. (2014). Approximating the solution of nonlinear Hammerstein fuzzy integral equations. Fuzzy Sets and Systems, 245, 1-17.
Zakeri, K. A., Ziari, S., Araghi, M. A. F., Perfilieva, I. (2019). Efficient Numerical Solution to a Bivariate Nonlinear Fuzzy Fredholm Integral Equation. IEEE Transactions on Fuzzy Systems.
Ziari, S., Perfilieva, I., Abbasbandy, S. (2019). Block-pulse functions in the method of successive approximations for nonlinear fuzzy Fredholm integral equations. Differential Equations and Dynamical Systems, 1-15.
Agheli, B., Firozja, M. A. (2020). A Fuzzy Transform Method for Numerical Solution of Fractional Volterra Integral Equations. International Journal of Applied and Computational Mathematics, 6(1), 1-13.
Noeiaghdam, S., Araghi, M. A. F., & Abbasbandy, S. (2020). Valid implementation of Sinc-collocation method to solve the fuzzy Fredholm integral equation. Journal of Computational and Applied Mathematics, 370, 112632.
Kaleva, O. (1987). Fuzzy differential equations. Fuzzy sets and systems, 24(3), 301-317.
Buckley, J. J., & Feuring, T. (2000). Fuzzy differential equations. Fuzzy sets and Systems, 110(1), 43-54.
Vorobiev, D., & Seikkala, S. (2002). Towards the theory of fuzzy differential equations. Fuzzy Sets and Systems, 125(2), 231-237.
Mao, X., & Yuan, C. (2006). Stochastic differential equations with Markovian switching. Imperial college press.
Stefanini, L., Bede, B. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71(3-4), 1311-1328.
Malinowski, M. T. (2011). Interval differential equations with a second type Hukuhara derivative. Applied Mathematics Letters, 24(12), 2118-2123.
Ramezanzadeh, M., Heidari, M., Fard, O., Borzabadi, A. (2015). On the interval differential equation: novel solution methodology. Advances in Difference Equations. 2015. 10.1186/s13662-015-0671-8.
Wang, B., Zhu, Q. (2017). Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Systems & Control Letters, 105, 55-61.
da Costa, T. M., Chalco-Cano, Y., Lodwick, W. A., & Silva, G. N. (2018). A new approach to linear interval differential equations as a first step toward solving fuzzy differential. Fuzzy Sets and Systems, 347, 129-141.
Ahmady, N. (2019). A Numerical Method for Solving Fuzzy Differential Equations with Fractional Order. International Journal of Industrial Mathematics, 11(2), 71-77.
An, T. V., Phu, N. D., & Van Hoa, N. (2014). A note on solutions of interval-valued Volterra integral equations. The Journal of Integral Equations and Applications, 1-14.
Otadi, M., & Mosleh, M. (2016). Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network. Neurocomputing, 205, 519-528.
Lupulescu, V., & Van Hoa, N. (2017). Interval Abel integral equation. Soft Computing, 21(10), 2777-2784.
