2000 character limit reached
Comparison of two efficient numerical techniques based on Chelyshkov polynomial for solving stochastic Itô-Volterra integral equation (2312.12445v2)
Published 7 Nov 2023 in math.NA and cs.NA
Abstract: In this study, two reliable approaches to solving the nonlinear stochastic It^o-Volterra integral equation are provided. These equations have been evaluated using the orthonormal Chelyshkov spectral collocation technique and the orthonormal Chelyshkov spectral Galerkin method. The techniques presented here transform this problem into a collection of nonlinear algebraic equations that have been numerically solved using the Newton method. Also, the convergence analysis has been studied for both approaches. Two illustrative examples have been provided to show the efficacy, plausibility, proficiency, and applicability of the current approaches.
- Gokme, E., Yuksel G. and Sezer M., 2017, “A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays”, Journal of Computational and Applied Mathematics, 311, pp. 354-363.
- Karimi S. and Jozi M., 2015, “A new iterative method for solving linear Fredholm integral equations using the least squares method”, Applied Mathematics and Computation, 250, pp. 744-758.
- Mirzaee F., Hadadiyan E., 2016, “Numerical solution of Volterra–Fredholm integral equations via modification of hat functions”, Applied Mathematics and Computation, 280, pp. 110-123.
- Viggiano B., Friedrich J., Volk R., Bourgoin M., Cal R.B. and Chevillard L., 2020, “ Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes”, Journal of Fluid Mechanics, 900, p.A27.
- Mao X., Marion G. and Renshaw E., 2002, “ Environmental Brownian noise suppresses explosions in population dynamics”, Stochastic Processes and their Applications, 97(1), pp.95-110.
- Emery A.F., 2004, “ Solving stochastic heat transfer problems”, Engineering Analysis with Boundary Elements, 28(3), pp.279-291.
- Bouchaud J.P. and Sornette D., 1994, “The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes”, Journal de Physique I, 4(6), pp.863-881.
- Shaw S. and Whiteman J.R., 2000, “Adaptive space–time finite element solution for Volterra equations arising in viscoelasticity problems”, Journal of Computational and Applied Mathematics, 125(1-2), pp.337-345.
- Babaei A., Jafari H., and Banihashemi S., 2020, “A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise”, Symmetry, 12(6), p.904.
- Behera S., Saha Ray S., 2020, “An operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro-differential equations”, Applied Mathematics and Computation, 367, 18 pages.
- Mirzaee F. and Hadadiyan E., 2014, “ A collocation technique for solving nonlinear stochastic Itô–Volterra integral equations”, Applied Mathematics and Computation, 247, pp.1011-1020.
- Wan Z., Chen Y. and Huang Y., 2009, “Legendre spectral Galerkin method for second-kind Volterra integral equations”, Frontiers of Mathematics in China, 4, pp.181-193.
- Yalçinbaş S., 2002, “ Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations”, Applied Mathematics and Computation, 127(2-3), pp.195-206.
- Maleknejad K., Basirat B. and Hashemizadeh E., 2011, “ Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra–Fredholm integro-differential equations”, Computers & Mathematics with applications, 61(9), pp.2821-2828.
- Khajehnasiri A.A., 2016, “ Numerical solution of nonlinear 2D Volterra–Fredholm integro-differential equations by two-dimensional triangular function”, International Journal of Applied and Computational Mathematics, 2, pp.575-591.
- Mohammadi F., 2016, “ Second kind Chebyshev wavelet Galerkin method for stochastic Ito-Volterra integral equations”, Mediterranean Journal of Mathematics, 13, pp.2613-2631.
- Heydari M.H., Hooshmandasl M.R., Ghaini F.M. and Cattani C., 2014, “ A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions”, Journal of Computational Physics, 270, pp.402-415.
- Sahu P.K. and Saha Ray S., 2015, “ A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method”, Mathematical Methods in the Applied Sciences, 38(2), pp.274-280.
- Saha Ray S., Singh P., 2021, “Numerical solution of stochastic Itô-Volterra integral equation by using Shifted Jacobi operational matrix method”, Applied Mathematics and Computation, 410, p.126440.
- Khan S.U., Ali M. and Ali I., 2019, “ A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis”, Advances in Difference Equations, 2019(1), pp.1-14.
- Øksendal B., 2003, “ Stochastic differential equations”, In stochastic differntial equations, Springer, Berlin, Heidelberg, pp. 65-84.
- Heydari M.H., Mahmoudi M.R., Shakiba A. and Avazzadeh Z., 2018, “Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion”, Communications in Nonlinear Science and Numerical Simulation, 64, pp.98-121.
- Canuto C., Hussaini M.Y., Quarteroni A. and Zang T.A., 2007, “ Spectral methods: fundamentals in single domains”, Springer Science & Business Media.