Abstract
Uncertainties play a major role in stochastic mechanics problems. To study the trajectory involved in stochastic mechanics problems generally, probability distributions are considered. Accordingly, the stochastic mechanics problems govern by stochastic differential equations followed by Markov process. However, the observation still lacks some sort of uncertainties, which are important but ignored. These imprecise uncertainties involved in the various factors affecting the constants, coefficients, initial, and boundary conditions. Hence, there may be a possibility to model a more reliable strategy that will quantify the uncertainty with better confidence. In this context, a computational method for solving fuzzy stochastic Volterra-Fredholm integral equation, which is based on the Block Pulse Functions (BPFs) using fuzzy stochastic operational matrix, is presented. The developed model is used to investigate a test problem of fuzzy stochastic Volterra integral equation and the results are compared in special cases.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arqub OA (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural Comput Appl 28:1591–1610
Arqub OA, Al-Smadi M, Momani S, Hayat T (2016) Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput 20:3283–3302
Arqub OA, Al-Smadi M, Momani S, Hayat T (2017) Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput 21:7191–7206
Babolian E, Maleknejad K, Mordad M, Rahimi B (2011) A numerical method to solve Fredholm-Volterra integral equations in two dimensional spaces using block pulse functions and operational matrix. J Comput Appl Math 235(14):3965–3971
Berger MA, Mizel VJ (1980) Volterra equations with Ito integrals I. J Integr Eqn 2:187–245
Chakraverty S, Nayak S (2013) Fuzzy finite element method in diffusion problems. In: Mathematics of Uncertainty Modelling in the Analysis of Engineering and Science Problems. IGI global, pp 309–328
Cortes JC, Jodar L, Villafuerte L (2007) Numerical solution of random differential equations: a mean square approach. Math Comput Model 45:757–765
Etheridge A (2002) A course in financial calculus. Cambridge University Press
Jankovic S, Ilic D (2010) One linear analytic approximation for stochastic integro-differential eauations. Acta Mathematica Scientia 30(4):1073–1085
Jiang ZH, Schaufelberger W (1992) Block pulse functions and their applications in control systems. Springer
Khodabin M, Maleknejad K, Rostami M, Nouri M (2011) Numerical solution of stochastic differential equations by second order Runge-Kutta methods. Math Comput Model 53:1910–1920
Kloeden PE, Platen E (1999) Numerical solution of stochastic differential equations. In: Applications of mathematics. Springer, Berlin
Knight FH (2006) Risk, uncertainty and profit. Cosimo Classics, New York
Maleknejad K, Mahmoudi Y (2004) Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block pulse functions. Appl Math Comput 149:799–806
Maleknejad K, Shahrezaee M, Khatami H (2005) Numerical solution of integral equations system of the second kind by block pulse functions. Appl Math Comput 166:15–24
Malinowski MT, Michta M (2011) Stochastic fuzzy differential equations with an application. Kybernetika 47(1):123–143
Murge MG, Pachpatte BG (1990) Succesive approximations for solutions of second order stochastic integrodifferential equations of Ito type. Indian J Pure Appl Math 21(3):260–274
Nayak S, Chakraverty S (2016) Numerical solution of stochastic point kinetic neutron diffusion equation with fuzzy parameters. Nucl Technol 193(3):444–456
Nayak S, Chakraverty S (2016) Numerical solution of fuzzy stochastic differential equation. J Intell Fuzzy Syst 31:555–563
Nayak S, Marwala T, Chakraverty S (2018) Stochastic differential equations with imprecisely defined parameters in market analysis. Soft Comput. https://doi.org/10.1007/s00500-018-3396-2
Oksendal B (2003) Stochastic differential equations: an introduction with applications. Springer, Heidelberg
Prasada Rao G (1983) Piecewise constant orthogonal functions and their application to systems and control. Springer, Berlin
Tudor C, Tudor M (1995) Approximation schemes for Ito-Volterra stochastic equations. Boletin Sociedad Matemática Mexicana 3(1):73–85
Yong J (2006) Backward stochastic Volterra integral equations and some related problems. Stoch Process Appl 116:779–795
Zhang X (2008) Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J Diff Equat 244:2226–2250
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Nayak, S. (2020). Numerical Solution of Fuzzy Stochastic Volterra-Fredholm Integral Equation with Imprecisely Defined Parameters. In: Chakraverty, S., Biswas, P. (eds) Recent Trends in Wave Mechanics and Vibrations. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-0287-3_9
Download citation
DOI: https://doi.org/10.1007/978-981-15-0287-3_9
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-0286-6
Online ISBN: 978-981-15-0287-3
eBook Packages: EngineeringEngineering (R0)