Modified Block-Pulse Functions Scheme for Solve of Two-Dimensional Stochastic Integral Equations

Year 2020, , 38 - 46, 24.04.2020

Mohsen Fallahpour Morteza Khodabin

https://doi.org/10.33187/jmsm.561172

Abstract

In this paper, two-dimensional modified block-pulse functions (2D-MBPFs) method is introduced for approximate solution of 2D-linear stochastic Volterra-Fredholm integral equations so the ordinary and stochastic operrational matrices of integration are utilized to reduce the computation of such equations into some algebraic equations. Convergence analysis of this method is discussed. Finally an illustrative example is given to show the accuracy of the proposed method so the results of it is compared with the block-pulse functions (BPFs) method.

Keywords

Modified block-pulse functions; Two-dimensional integral equation;, Stochastic integral equation; Volterra-Fredholm integral equation; Operational

References

• [1] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.
• [2] A. J. Jerri, Introduction to Integral Equations with Applications, John Wiley and Sons, INC, 1999.
• [3] M. Fallahpour, M. Khodabin, K. Maleknejad, Approximation solution of two-dimensional linear stochastic Volterra-Fredholm integral equation via two-dimensional Block-pulse functions, Int. J. Indus. Math., 8(4) (2016), IJIM-00774.
• [4] M. Fallahpour, M. Khodabin, K. Maleknejad, Approximation solution of two-dimensional linear stochastic fredholm integral equation by applying the Haar wavelet, Math. Model. Comp., 5 (2015), 361- 372.
• [5] M. Fallahpour, M. Khodabin, K. Maleknejad, Theoretical error analysis and validation in numerical solution of two-dimensional linear stochastic Volterra–Fredholm integral equation by applying the block-pulse functions, Cog. Math., 4 (2017), 1296750.
• [6] M. Fallahpour, M. Khodabin, R. Ezzati, A new computational method based on Bernstein operational matrices for solving two-dimensional Linear stochastic Volterra integral equations, Differ. Equat. Dynam. Syst., (2019), doi.org/10.1007/s12591-019-00474-y.
• [7] M. Fallahpour, M. Khodabin, K. Maleknejad, Theoretical error analysis of solution for two-dimensional stochastic Volterra integral equations by Haar wavelet, Int. J. Appl. Comput. Math, (2019), doi.org/10.1007/s40819-019-0739-3.
• [8] F. Mirzaee, E. Hadadiyan, Using modified two-dimensional block-pulse functions for the numerical solution of nonlinear two-dimensional Volterra integral equations, J. Hyperst., 3(1) (2014), 68-80.
• [9] K. Maleknejad, B. Rahimi, Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind, Com. Non. Sci. Num. Sim, 16 (2011), 2469-2477.
• [10] K. Maleknejad, S. Sohrabi, B. Baranji, Two-dimensional PCBFs: Application to nonlinear Volterra integral equations, Proce. Wor. Cong. Engin., (2009), Vol II WCE 2009, July 1 - 3.
• [11] Z. H. Jiang, W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag, 1992.
• [12] K. Maleknejad, M. Khodabin, F. Hosseini Shekarabi, Modified block pulse functions for numerical solution of stochastic Volterra integral equations, Appl. Math., (2014), doi. org/10.1155/2014/469308.
• [13] F. Mirzaee, E. Hadadiyan, Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations via modified block-pulse functions, J. Assoc. Arab. Uni. Bas. Appl. Sci., 12 (2012), 65–73.