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 

References

  1. 1.
    Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M.M., Fereidouni, F.: Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions. Eng. Anal. Boundary Elem. 37, 1331–1338 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Sohrabi, S.: Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation. Ain Shams Eng. J. 2, 249–254 (2011)CrossRefGoogle Scholar
  3. 3.
    Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)Google Scholar
  4. 4.
    Spencer, J.B.F., Bergman, L.A.: On the numerical solution on the Fokker–Planck equation for nonlinear stochastic systems. Nonlinear Dyn. 4, 357–372 (1993)CrossRefGoogle Scholar
  5. 5.
    Zeng, C., Yang, Q., Chen, Y.Q.: Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach. Nonlinear Dyn. 67, 2719–2726 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Mamontov, Y.V., Willander, M.: Long asymptotic correlation time for non-linear autonomous itô stochastic differential equation. Nonlinear Dyn. 12, 399–411 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    der Wouw, N.V., Nijmeijer, H., Campen, D.H.V.: A Volterra series approach to the approximation of stochastic nonlinear dynamics. Nonlinear Dyn. 4, 397–409 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mahmoudkhani, M., Haddadpour, H.: Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations. Nonlinear Dyn. 1, 165–188 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Levin, J.J., Nohel, J.A.: On a system of integro-differential equations occurring in reactor dynamics. J. Math. Mech. 9, 347–368 (1960)MathSciNetMATHGoogle Scholar
  10. 10.
    Miller, R.K.: On a system of integro-differential equations occurring in reactor dynamics. SIAM J. Appl. Math. 14, 446–452 (1966)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Interpolation solution in generalized stochastic exponential population growth model. Appl. Math. Model. 36, 1023–1033 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Oǧuztöreli, M.N.: Time-Lag Control Systems. Academic Press, New York (1966)Google Scholar
  13. 13.
    Maleknejad, K., Khodabin, M., Rostami, M.: Numerical solutions of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions. Math. Comput. Model. 55, 791–800 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical approach for solving stochastic Volterra–Fredholm integral equations by stochastic operational matrix. Comput. Math. Appl. 64, 1903–1913 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Maleknejad, K., Khodabin, M., Rostami, M.: A numerical method for solving \(m\)-dimensional stochastic Itô–Volterra integral equations by stochastic operational matrix. Comput. Math. Appl. 63, 133–143 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cao, Y., Gillespie, D., Petzold, L.: Adaptive explicit–implicit tau-leaping method with automatic tau selection. J. Chem. Phys. 126, 1–9 (2007)Google Scholar
  17. 17.
    Ru, P., Vill-Freixa, J., Burrage, K.: Simulation methods with extended stability for stiff biochemical kinetics. BMC Syst. Biol. 4(110), 1–13 (2010)Google Scholar
  18. 18.
    Elworthy, K., Truman, A., Zhao, H., Gaines, J.: Approximate traveling waves for generalized KPP equations and classical mechanics. Proc. R. Soc. Lond. Ser. A 446(1928), 529–554 (1994)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  20. 20.
    Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge–Kutta methods. Appl. Math. Model. 53, 1910–1920 (2011)MathSciNetMATHGoogle Scholar
  21. 21.
    Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)MATHGoogle Scholar
  22. 22.
    Cortes, J.C., Jodar, L., Villafuerte, L.: Numerical solution of random differential equations: a mean square approach. Math. Comput. Model. 45, 757–765 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Cortes, J.C., Jodar, L., Villafuerte, L.: Mean square numerical solution of random differential equations: facts and possibilities. Comput. Math. Appl. 53, 1098–1106 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Oksendal, B.: Stochastic Differential Equations: An introduction with applications. Springer, New York (1998)Google Scholar
  25. 25.
    Holden, H., Oksendal, B., Uboe, J., Zhang, T.: Stochastic Partial Differential Equations. Springer, Berlin (1996)Google Scholar
  26. 26.
    Abdulle, A., Blumenthal, A.: Stabilized multilevel Monte Carlo method for stiff stochastic differential equations. J. Comput. Phys. 251, 445–460 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Berger, M., Mizel, V.: Volterra equations with Ito integrals I. J. Integral Equ. 2, 187–245 (1980)MathSciNetMATHGoogle Scholar
  28. 28.
    Murge, M., Pachpatte, B.: On second order Ito type stochastic integro-differential equations. Analele Stiintifice ale Universitatii. I. Cuza din Iasi, Mathematica, 30(5), 25–34 (1984)Google Scholar
  29. 29.
    Murge, M., Pachpatte, B.: Successive approximations for solutions of second order stochastic integro-differential equations of Ito type. Indian J. Pure Appl. Math. 21(3), 260–274 (1990)MathSciNetMATHGoogle Scholar
  30. 30.
    Zhang, X.: Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J. Differ. Equ. 244, 2226–2250 (2008)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jankovic, S., Ilic, D.: One linear analytic approximation for stochastic integro-differential equations. Acta Math. Sci. 308(4), 1073–1085 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zhang, X.: Stochastic Volterra equations in Banach spaces and stochastic partial differential equation. Acta J. Funct. Anal. 258, 1361–1425 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Yong, J.: Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779–795 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Djordjevic, A., Jankovic, S.: On a class of backward stochastic Volterra integral equations. Appl. Math. Lett. (2013). doi:10.1016/j.aml.2013.07.006 MathSciNetMATHGoogle Scholar
  35. 35.
    Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F.: Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl. Math. Comput. 234, 267–276 (2014)MathSciNetMATHGoogle Scholar
  36. 36.
    Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F.: Two-dimensional Legendre wavelets for solving time-fractional telegraph equation. Adv. Appl. Math. Mech. 6(2), 247–260 (2014)MathSciNetMATHGoogle Scholar
  37. 37.
    Heydari, M.H., Hooshmandasl, M.R., Cattani, C., Li, M.: Legendre wavelets method for solving fractional population growth model in a closed system. Math. Probl. Eng. 2013, 1–8 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M., Mohammadi, F.: Wavelet collocation method for solving multiorder fractional differential equations. J. Appl. Math. 2012, 1–19 (2012)Google Scholar
  39. 39.
    Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M.M., Cattani, C.: Wavelets method for the time fractional diffusion-wave equation. Phys. Lett. A 379, 71–76 (2015)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Heydari, M. H., Hooshmandasl, M. R., Maalek Ghaini, F. M., Fatehi Marji, M., Dehghan, R., Memarian, M. H.: A new wavelet method for solving the Helmholtz equation with complex solution. Numer. Methods Partial Differ. Equ. 32(3), 741–756 (2016)Google Scholar
  41. 41.
    Heydari, M.H., Hooshmandasl, M.R., Barid Loghmania, Gh., Cattani, C.: Wavelets Galerkin method for solving stochastic heat equation. Int. J. Comput. Math. (2015). doi:10.1080/00207160.2015.1067311
  42. 42.
    Babolian, E., Mordad, M.: A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis function. Comput. Math. Appl. 62, 187–198 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Tripathi, M.P., Baranwal, V.K., Pandey, R.K., Singh, O.P.: A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions. Commun. Nonlinear Sci. Numer. Simul. 18, 1327–1340 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M.M., Cattani, C.: A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. J. Comput. Phys. 270, 402–415 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Heydari, M.H., Hooshmandasl, M.R., Cattani, C., Ghaini, F.M.M.: An efficient computational method for solving nonlinear stochastic Itô–Volterra integral equations: application for stochastic problems in physics. J. Comput. Phys. 283, 148–168 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1995)MATHGoogle Scholar
  47. 47.
    Schwartz, E.S.: The stochastic behavior of commodity prices: implications for valuation and hedging. J. Finance 52, 923–973 (1997)CrossRefGoogle Scholar
  48. 48.
    Jackwerth, J., Rubinstein, M.: Recovering probability distributions from contemporaneous security prices. J. Finance 51(5), 1611–1631 (1996)CrossRefGoogle Scholar
  49. 49.
    Rubinstein, M.: Nonparametric tests of alternative option pricing models. J. Finance 40(2), 455–480 (1985)CrossRefGoogle Scholar
  50. 50.
    Aarató, M.: A famous nonlinear stochastic equation (Lotka–Volterra model with diffusion). Math. Comput. Model. 38, 709–726 (2003)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Henderson, D., Plaschko, P.: Stochastic Differential Equations in Science and Engineering. World Scientific, Singapore (2006)CrossRefMATHGoogle Scholar

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

Personalised recommendations