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Nonlinear Dynamics

, Volume 85, Issue 2, pp 1185–1202 | Cite as

Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations

  • M. H. Heydari
  • M. R. Hooshmandasl
  • A. Shakiba
  • C. Cattani
Original Paper

Abstract

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) together with the Galerkin method is proposed for solving a class of nonlinear stochastic Itô–Volterra integral equations. For this purpose, a new stochastic operational matrix (SOM) for LWs is derived. A collocation method based on hat functions (HFs) is employed to derive a general procedure for forming this matrix. The LWs and their operational matrices of integration and stochastic Itô-integration and also some useful properties of these basis functions are used to transform such problems into corresponding nonlinear systems of algebraic equations, which can be simply solved to achieve the solution of such problems. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient. Furthermore as some useful applications, the proposed method is applied to obtain approximate solutions for some stochastic problems in the mathematics finance, biology, physics and chemistry.

Keywords

Legendre wavelets (LWs) Stochastic operational matrix (SOM) Nonlinear stochastic Itô–Volterra integral equations  Brownian motion process Stochastic volatility models Stochastic Lotka–Volterra model Duffing–Van der Pol Oscillator  Stochastic Brusselator problem 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • M. H. Heydari
    • 1
    • 2
  • M. R. Hooshmandasl
    • 1
    • 2
  • A. Shakiba
    • 1
    • 2
  • C. Cattani
    • 3
  1. 1.Faculty of MathematicsYazd UniversityYazdIran
  2. 2.Laboratory of Quantum Information ProcessingYazd UniversityYazdIran
  3. 3.Engineering SchoolTuscia UniversityViterboItaly

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