A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

Obaid Algahtani; M. A. Abdelkawy; António M. Lopes; Obaid Algahtani; M. A. Abdelkawy; António M. Lopes

doi:10.3934/math.2022846

AIMS Mathematics

2022, Volume 7, Issue 8: 15453-15470. doi: 10.3934/math.2022846

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A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

1.
Department of Mathematics, College of Sciences, King Saud University, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt
4.
LAETA/INEGI, Faculty of Engineering, University of Porto, Portugal

Received: 14 April 2022 Revised: 10 June 2022 Accepted: 13 June 2022 Published: 20 June 2022
MSC : 65Mxx, 44Axx, 45Dxx

A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.
- fractional Volterra integro-differential equation,
- Caputo fractional derivative,
- spectral collocation method
Citation: Obaid Algahtani, M. A. Abdelkawy, António M. Lopes. A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations[J]. AIMS Mathematics, 2022, 7(8): 15453-15470. doi: 10.3934/math.2022846

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Abstract

A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.

References

[1]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods: Fundamentals in single domains, New York: Springer-Verlag, 2006.
[2]	X. J. Yang, F. Gao, J. A. T. Machado, D. Baleanu, A new fractional derivative involving the normalized sinc function without singular kernel, Eur. Phys. J. Spec. Top., 226 (2017), 3567–3575. https://doi.org/10.1140/epjst/e2018-00020-2 doi: 10.1140/epjst/e2018-00020-2
[3]	R. Koskodan, E. Allen, Construction of consistent discrete and continuous stochastic models for multiple assets with application to option valuation, Math. Comput, Model., 48 (2008), 1775–1786. https://doi.org/10.1016/j.mcm.2007.06.032 doi: 10.1016/j.mcm.2007.06.032
[4]	J. A. T. Machado, A. M. Lopes, Rare and extreme events: The case of COVID-19 pandemic, Nonlinear Dyn., 100 (2020), 2953–2972. https://doi.org/10.1007/s11071-020-05680-w doi: 10.1007/s11071-020-05680-w
[5]	A. Ashyralyev, On modified Crank-Nicholson difference schemes for stochastic parabolic equation, Numer. Func. Anal. Optim., 29 (2008), 268–282. https://doi.org/10.1080/01630560801998138 doi: 10.1080/01630560801998138
[6]	M. Kamrani, S. M. Hosseini, The role of coefficients of a general SPDE on the stability and convergence of a finite difference method, J. Comput. Appl. Math., 234 (2010), 1426–1434. https://doi.org/10.1016/j.cam.2010.02.018 doi: 10.1016/j.cam.2010.02.018
[7]	C. Roth, A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations, Stoch. Anal. Appl., 24 (2006), 221–240. https://doi.org/10.1080/07362990500397764 doi: 10.1080/07362990500397764
[8]	E. Hausenblas, Finite element approximation of stochastic partial differential equations driven by Poisson random measures of jump type, SIAM J. Numer. Anal., 46 (2007), 437–471. https://doi.org/10.1137/050654141 doi: 10.1137/050654141
[9]	D. Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise, Commun. Math. Sci., 1 (2003), 361–375.
[10]	G. J. Lord, T. Shardlow, Post processing for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 45 (2007), 870–889. https://doi.org/10.1137/050640138 doi: 10.1137/050640138
[11]	J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), 1–43. https://doi.org/10.1007/s11118-004-2950-y doi: 10.1007/s11118-004-2950-y
[12]	Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), 1363–1384. https://doi.org/10.1137/040605278 doi: 10.1137/040605278
[13]	Z. Taheri, S. Javadi, E. Babolian, Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method, J. Comput. Appl. Math., 237 (2017), 336–347. https://doi.org/10.1016/j.cam.2017.02.027 doi: 10.1016/j.cam.2017.02.027
[14]	P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Berlin: Springer-Verlag, 1992. https://doi.org/10.1007/978-3-662-12616-5
[15]	G. N. Milstein, Numerical integration of stochastic differential equations, Dordrecht: Kluwer Academic Publishers, 1995. https://doi.org/10.1007/978-94-015-8455-5
[16]	N. Samadyar, F. Mirzaee, Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô‐Volterra integral equations of Abel type, Int. J. Numer. Model.: Electron. Networks, Devices Fields, 33 (2020), e2688.
[17]	F. Mirzaee, N. Samadyar, Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itô-Volterra integral equations, Multidiscip. Model. Mater. Struct., 15 (2019), 575–598. https://doi.org/10.1108/MMMS-04-2018-0075 doi: 10.1108/MMMS-04-2018-0075
[18]	F. Mirzaee, S. Alipour, Bicubic B-spline functions to solve linear two-dimensional weakly singular stochastic integral equation, Iran. J. Sci. Technol., Trans. A: Sci., 45 (2021) 965–972. https://doi.org/10.1007/s40995-021-01109-0
[19]	F. Mirzaee, S. Alipour, Quintic B-spline collocation method to solve n-dimensional stochastic Itô-Volterra integral equations, J. Comput. Appl. Math., 384 (2021), 113153. https://doi.org/10.1016/j.cam.2020.113153 doi: 10.1016/j.cam.2020.113153
[20]	S. Alipour, F. Mirzaee, An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation, Appl. Math. Comput., 371 (2020), 124947. https://doi.org/10.1016/j.amc.2019.124947 doi: 10.1016/j.amc.2019.124947
[21]	F. Mirzaee, S. Alipour, An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations, Math. Methods Appl. Sci., 43 (2020), 384–397. https://doi.org/10.1002/mma.5890 doi: 10.1002/mma.5890
[22]	N. Samadyar, F. Mirzaee, Numerical solution of two-dimensional weakly singular stochastic integral equations on non-rectangular domains via radial basis functions, Eng. Anal. Boundary Elem., 101 (2019), 27–36. https://doi.org/10.1016/j.enganabound.2018.12.008 doi: 10.1016/j.enganabound.2018.12.008
[23]	F. Mirzaee, N. Samadyar, On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions, Eng. Anal. Boundary Elem., 100 (2019), 246–255. https://doi.org/10.1016/j.enganabound.2018.05.006 doi: 10.1016/j.enganabound.2018.05.006
[24]	F. Mirzaee, S. Alipour, Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order, J. Comput. Appl. Math., 366 (2020), 112440. https://doi.org/10.1016/j.cam.2019.112440 doi: 10.1016/j.cam.2019.112440
[25]	F. Mirzaee, E. Solhi, S. Naserifar, Approximate solution of stochastic Volterra integro-differential equations by using moving least squares scheme and spectral collocation method, Appl. Math. Comput., 410 (2021), 126447. https://doi.org/10.1016/j.amc.2021.126447 doi: 10.1016/j.amc.2021.126447
[26]	E. H. Doha, M. A. Abdelkawy, A. Z. M. Amin, A. M. Lopes, Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations, Eng. Comput., 38 (2022), 1363–1373. https://doi.org/10.1007/s00366-020-01263-w doi: 10.1007/s00366-020-01263-w
[27]	X. Dai, W. Bu, A. Xiao, Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math., 356 (2019), 377–390. https://doi.org/10.1016/j.cam.2019.02.002 doi: 10.1016/j.cam.2019.02.002
[28]	X. Dai, A. Xiao, W. Bu, Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler-Maruyama approximation, Discrete Cont. Dyn. Syst.-B, 2021, 1–23. https://doi.org/10.3934/dcdsb.2021225
[29]	Y. H. Youssri, W. M. Abd‐Elhameed, M. Abdelhakem, A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials, Math. Methods Appl. Sci., 44 (2021), 9224–9236. https://doi.org/10.1002/mma.7347 doi: 10.1002/mma.7347
[30]	M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67 (2015), 773–791.
[31]	E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A: Math. Gen., 37 (2004), 657–675.
[32]	A. H. Bhrawy, E. A. Ahmed, D. Baleanu, An efficient collocation technique for solving generalized Fokker-Planck type equations with variable coefficients, Proc. Rom. Acad. Ser. A, 15 (2014), 322–330.
[33]	A. H. Bhrawy, A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222 (2013), 255–264. https://doi.org/10.1016/j.amc.2013.07.056 doi: 10.1016/j.amc.2013.07.056
[34]	M. Abbaszadeh, M. Dehghan, M. A. Zaky, A. S. Hendy, Interpolating stabilized element free Galerkin method for neutral delay fractional damped diffusion-wave equation, J. Funct. Spaces, 2021 (2021), 1–11. https://doi.org/10.1155/2021/6665420 doi: 10.1155/2021/6665420
[35]	A. S. Hendy, M. A. Zaky, Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations, Eng. Comput., 38 (2020), 1351–1363. https://doi.org/10.1007/s00366-020-01095-8 doi: 10.1007/s00366-020-01095-8
[36]	M. A. Zaky, A. S. Hendy, J. E. Macías-Díaz, Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions, J. Sci. Comput., 82 (2020), 1–27. https://doi.org/10.1007/s10915-019-01117-8 doi: 10.1007/s10915-019-01117-8
[37]	A. S. Hendy, M. A. Zaky, Global consistency analysis of $L$1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations, Appl. Numer. Math., 156 (2020), 276–302. https://doi.org/10.1016/j.apnum.2020.05.002 doi: 10.1016/j.apnum.2020.05.002
[38]	M. A. Zaky, A. S. Hendy, Convergence analysis of an $L$1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations, Int. J. Comput. Math., 98 (2020), 1420–1437. https://doi.org/10.1080/00207160.2020.1822994 doi: 10.1080/00207160.2020.1822994
[39]	A. H. Bhrawy, M. A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80 (2015), 101–116. https://doi.org/10.1007/s11071-014-1854-7 doi: 10.1007/s11071-014-1854-7
[40]	M. A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357 (2019), 103–122. https://doi.org/10.1016/j.cam.2019.01.046 doi: 10.1016/j.cam.2019.01.046
[41]	M. A. Zaky, An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math., 154 (2020), 205–222. https://doi.org/10.1016/j.apnum.2020.04.002 doi: 10.1016/j.apnum.2020.04.002
[42]	E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, R. A. V. Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014), 244–255. https://doi.org/10.1016/j.jcp.2014.01.003 doi: 10.1016/j.jcp.2014.01.003
[43]	A. H. Bhrawy, M. A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys., 294 (2015), 462–483. https://doi.org/10.1016/j.jcp.2015.03.063 doi: 10.1016/j.jcp.2015.03.063
[44]	A. H. Bhrawy, A. A. Al-Zahrani, Y. A. Alhamed, D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional Pantograph equations, Rom. J. Phys., 59 (2014), 646–657.
[45]	M. Abdelhakem, M. Biomy, S. A. Kandil, D. Baleanu, A numerical method based on Legendre differentiation matrices for higher order ODEs, Inf. Sci. Lett., 9 (2020), 1–7.
[46]	M. Abdelhakem, Y. H. Youssri, Two spectral Legendre's derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems, Appl. Numer. Math., 169 (2021), 243–255. https://doi.org/10.1016/j.apnum.2021.07.006 doi: 10.1016/j.apnum.2021.07.006
[47]	A. H. Bhrawy, M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations, Appl. Math. Model., 40 (2016), 832–845. https://doi.org/10.1016/j.apm.2015.06.012 doi: 10.1016/j.apm.2015.06.012
[48]	M. A. Zaky, E. H. Doha, J. A. Tenreiro Machado, A spectral framework for fractional variational problems based on fractional Jacobi functions, Appl. Numer. Math., 132 (2018), 51–72. https://doi.org/10.1016/j.apnum.2018.05.009 doi: 10.1016/j.apnum.2018.05.009

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