موجک‌های چبیشف برای حل عددی معادلات انتگرال تصادفی ولترا با‌ روش کمترین مربعات

Document Type: Research Paper

Authors

گروه ریاضی، دانشکده علوم پایه، دانشگاه قم، قم، ایران

10.22072/wala.2019.102484.1216

Abstract

این مقاله با استفاده  از  موجک چبیشف و روش کمترین مربعات، یک روش تقریبی برای  حل معادله انتگرال ایتو-ولترا ارائه می‌دهد. معادله انتگرال ایتو-ولترا با روش کمترین مربعات به‌وسیله موجک چبیشف به یک دستگاه معادلات خطی تبدیل می‌شود که آنالیز خطای روش پیشنهادی، ارائه شده و  سرعت همگرایی  نیز اثبات شده است. همچنین مثال‌های عددی میزان دقت و کارآمدی این روش را  نسبت به روش ماتریس عملیاتی تصادفی نشان می‌دهند.

Keywords


[1] S.A. Broughton and K. Bryan, Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing, John Wiley $ & $ Sons, 2008.

[2] G.H. Choe, Stochastic Analysis for Finance with Simulations, Springer, 2016.

[3] P.A. Cioica and S. Dahlke,  Spatial besov regularity for semilinear stochastic partial
     differential equations on bounded Lipschitz domains, Int. J. Comput. Math., 89(18) (2012),
     2443-2459.
    
[4] J.C. Cortes, L. Jodar and L. Villafuerte,  Mean square numerical solution of random
     differential equations: facts and possibilities, Comput. Math. Appl., 53(7) (2007),
     1098--1106.
        
[5] J.C. Cortes, L. Jodar and L. Villafuerte, Numerical solution of random differential equations:
     a mean square approach, Math. Comput. Modelling, 45(7-8) (2007), 757--765.

[6] M. Ehler,  Shrinkage rules for variational minimization problems and applications to
     analytical ultracentrifugation, J. Inverse Ill-Posed Probl., 19(4–5) (2011), 593-614.
    
[7] K.D. Elworthy , A. Truman, H.Z. Zhao and J.G. Gaines,  Approximate traveling waves for
     generalized KPP equations and classical mechanics, Proc. R. Soc. Lond., Ser. A, 446(1928)
     (1994), 529-554.

[8] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek and C. Cattani,  A computational method
     for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix 
     for generalized hat basis functions, J. Comput. Phys., 270 (2014), 402-415.
    
[9] M.H. Heydari, M.R. Hooshmandasl, C. Cattani and F.M. Maalek Ghaini,  An efficient
     computational method for solving nonlinear stochastic It^{o} integral equations: Application
     for stochastic problems in physics, J. Comput. Phys., 283 (2015), 148-168.
    
[10] M.H. Heydari, M.R. Hooshmandasl and F.M. Mohammadi, Two-dimensional Legendre
       wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6(2) (2014),
       247-260.
    
[11] M.H. Heydari, F.M. Maalek Ghaini and M.R. Hooshmandasl,  Legendre wavelets method for
       numerical solution of time-fractional heat equation, Wavelets and Linear Algebra, 1(1)
       (2014) 19-31.

[12] D.J. Higham,  An algorithmic introduction to numerical simulation of stochastic differential
       equations, SIAM Rev., 43(3) (2001), 525-546.

[13] S. Jankovic and D. Ilic,  One linear analytic approximation for stochastic integro-ifferential
       equations, Acta Math. Sci., 30(4) (2010), 1073-1085.

[14] M. Khodabin, K. Maleknejad, M. Rostami and N. Nouri,  Interpolation solution in
       generalized stochastic exponential population growth model, Appl. Math. Modelling, 36(3)
       (2012), 1023-1033.

[15] M. Khodabin, K. Maleknejad, M. Rostami and N. Nouri, Numerical approach for solving
       stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput.
       Math. Appl., 64(6) (2012), 1903-1913.

[16] M. Khodabin, K. Maleknejad, M. Rostami and N. Nouri, Numerical solution of stochastic 
       differential equations by secind order Rune-Kutta mathods, Math. Comput. Modelling,
       53(9-10) (2011), 1910-1920.
    
[17] F.C. Klebaner, Introduction to Stochastic Calculus with Applications, World Scientific, 2012.

[18] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer,
       1999.

[19] J.J. Levin and J.A. Nobel , On a  system of integro-differential equations accurring in
       reactor dynamics, J. Math. Mech., 9(3) (1960), 347-368.

[20] K. Maleknejad, M. Khodabin and M. Rostami, Numerical solution of stochastic Volterra
       integral equations by a stochastic operational matrix based on block pulse functions, Math.
       Comput. Modelling, 55(3-4) (2012), 791-800.

[21] K. Maleknejad, M. Khodabin and M. Rostami,  A numerical method for solving
       m-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix,
       Comput. Math. Appl., 63(1) (2012), 133-143.

[22] J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, CRC press, 2002.
    
[23] R. K. Miller,  On a system of integro-differential equations occuring in reactor dynamic,
       SIAM J. Appl. Math., 14 (1966) 446-452.
    
[24] F. Mirzaee, N. Samadyar and S.F. Hoseini, Euler polynomial solutions of nonlinear
       stochastic It^{o}-Volterra integral equations, J. Comput. Appl. Math., 330 (2018), 574-585.
    
[25] F. Mirzaee and N. Samadyar, On the numerical solution of stochastic quadratic integral
       equations via operational matrix method, Math. Methods Appl. Sci., 41(2) (2018),
       4465-4479.
        
[26] F. Mohammadi, A computational wavelet method for numerical solution of stochastic
       Volterra-Fredholm integral equations, Wavelets and Linear Algebra, 3(1) (2016), 13-25.

[27] F. Mohammadi,  A efficient computational method for solving stochastic It^{o}-Volterra
       integral equations, TWMS J. App. Eng. Math., 5(2) (2015), 286-297.
    
[28] F. Mohammadi ,  A wavelet-based computational method for solving stochastic It^{o}-
       Volterra integral equations, J. Comput. Phys., 298 (2015), 254-265.
    
[29] F. Mohammadi , Haar wavelets approach for solving multidimensional  stochastic It^{o}-
       Volterra integral equations, Appl. Math. E-Notes, 15 (2015), 80-96.
    
[30] F. Mohammadi and A. Ciancio, Wavelet-based numerical method for solving fractional
       integro-differential equation with a weakly singular kernel, Wavelets and Linear Algebra, 4(1)
       (2017), 53-73.

[31] B.Kh. Mousavi, A. Askari Hemmat and M.H. Heydari, Wilson wavelets for solving nonlinear
       stochastic integral equations, Wavelets and Linear Algebra, 4(2) (2017), 33-48.

[32] M.G. Murge and B.G. Pachpatte,  Successive approximations for solutions of second order
       stochastic integro-differential equations of Ito type, Indian J. Pure Appl. Math., 21(3)
       (1990), 260-274.
    
[33] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, fifth ed.,
       Springer-Verlag, New York, 1998.
    
[34] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with
       Jumps in Finance}, Springer-Verlag, 2010.
    
[35] M. Saffarzadeh, G.B. Loghmani and  M. Heydari, An iterative technique for the numerical
       solution of nonlinear stochastic It^{o}-Volterra integral equations, J. Comput. Appl. Math.,
       333 (2018), 74-86.

[36] L.N. Trefethen , Is Gauss quadrature better than Clenshaw–Curtis, SIAM Rev., 50(1)
       (2008), 67-87.