99
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Euler–Maruyama methods for Caputo tempered fractional stochastic differential equations

, &
Received 29 Sep 2023, Accepted 28 Dec 2023, Published online: 10 Jan 2024

References

  • P. Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, Springer, Switzerland, 2017.
  • Z. Brzezniak, L. Debbi, and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Glob. Stoch. Anal. 1 (2014), pp. 149–174.
  • X.J. Dai, W.P. Bu, and A.G. Xiao, Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math. 356 (2019), pp. 377–390.
  • J.W. Deng, L.J. Zhao, and Y.J. Wu, Fast predictor-corrector approach for the tempered fractional differential equations, Numer. Algorithms. 74 (2017), pp. 717–754.
  • T.S. Doan, P.T. Huong, P.E. Kloeden, and H.T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl. 36 (2018), pp. 654–664.
  • T.S. Doan, P.T. Huong, P.E. Kloeden, and A.M. Vu, Euler–Maruyama scheme for Caputo stochastic fractional differential equations, J. Comput. Appl. Math. 380 (2020), pp. 112989.
  • A. Fernandez and H.M. Fahad, Weighted fractional calculus: a general class of operators, Fractal Fract.6 (2022), pp. 208.
  • X. Guo, Y.T. Li, and H. Wang, A fast finite difference method for tempered fractional diffusion equations, Commun. Comput. Phys. 24 (2018), pp. 531–556.
  • M.S. Heris and M. Javidi, A predictor–corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes, J. Supercomput. 75 (2019), pp. 8168–8206.
  • J.F. Huang, D.D. Yang, and L.O. Jay, Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations, Numer. Algorithms. 85 (2020), pp. 375–397.
  • J.F. Huang, J.N. Zhang, S. Arshad, and Y.F. Tang, A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations, Appl. Numer. Math. 159 (2021), pp. 159–173.
  • S.D. Jiang, J.W. Zhang, Q. Zhang, and Z.M. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys. 21 (2017), pp. 650–678.
  • C. Li, W.H. Deng, and L.J. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Continuous Dyn. Syst. Ser. B. 24 (2019), pp. 1989–2015.
  • Z.Y. Lin and Z.D. Bai, Probability Inequalities, Science Press, Beijing, 2010.
  • M. Medved and E. Brestovanska, Differential equations with tempered psi-fractional derivative, Math. Model. Anal. 26 (2021), pp. 631–650.
  • B.P. Moghaddam, J.A.T. Machado, and A. Babaei, A computationally efficient method for tempered fractional differential equations with application, Comput. Appl. Math. 37 (2018), pp. 3657–3671.
  • M.L. Morgado and M. Rebelo, Well-posedness and numerical approximation of tempered fractional terminal value problems, Fract. Calc. Appl. Anal. 20 (2017), pp. 1239–1262.
  • N.A. Obeidat and D.E. Bentil, New theories and applications of tempered fractional differential equations, Nonlinear. Dyn. 105 (2021), pp. 1689–1702.
  • B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 2003.
  • M.D. Ortigueira, G. Bengochea, and J.T. Machado, Substantial, tempered, and shifted fractional derivatives: three faces of a tetrahedron, Math. Methods. Appl. Sci. 44 (2021), pp. 9191–9209.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • Z.S. Qiu and X.N. Cao, Second-order numerical methods for the tempered fractional diffusion equations, Adv. Differ. Equ. 2019 (2019), pp. 485.
  • N.H. Sweilam, S.M. Al-Mekhlafi, and D. Baleanu, A hybrid stochastic fractional order Coronavirus (2019-nCov) mathematical model, Chaos, Solit. Fractals 145 (2021), pp. 110762.
  • S. Yadav, R.K. Pandey, and P.K. Pandey, Numerical approximation of tempered fractional Sturm-Liouville problem with application in fractional diffusion equation, Int. J. Numer. Methods. Fluids. 93 (2021), pp. 610–627.
  • Z.W. Yang, X.C. Zheng, Z.Q. Zhang, and H. Wang, Strong convergence of a Euler–Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise, Chaos Solit. Fractals 142 (2021), pp. 110392.
  • H.P. Ye, J.M. Gao, and Y.S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), pp. 1075–1081.
  • L.G. Yuan, S. Zheng, and Z.C. Wei, Comparison theorems of tempered fractional differential equations, Eur. Phys. J. Spec. Top. 231 (2022), pp. 2477–2485.
  • J.N. Zhang, Y.F. Tang, and J.F. Huang, A fast Euler–Maruyama method for fractional stochastic differential equations, J. Appl. Math. Comput. 69 (2022), pp. 273–291.
  • L. Zhao, C. Li, and F.Q. Zhao, Efficient difference schemes for the Caputo-tempered fractional diffusion equations based on polynomial interpolation, Comm. Appl. Math. Comput. 3 (2021), pp. 1–40.
  • Y. Zheng, S.Y. Qian, S. Arshad, and J.F. Huang, A modified Euler–Maruyama method for Riemann–Liouville stochastic fractional integro-differential equations, J. Stat. Comput. Simul. 93 (2023), pp. 249–265.
  • B.B. Zhou and L.L. Zhang, Local existence-uniqueness and monotone iterative approximation of positive solutions for p-Laplacian differential equations involving tempered fractional derivatives, J. Inequal. Appl. 2021 (2021), pp. 159.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.