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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2022. Vol 41

Existence and Stability of Solutions for a Class of Stochastic Fractional Partial Differential Equation with a Noise

Author(s)
N. Bouteraa1,2

1University of Oran1 Ahmed Ben Bella, Algeria

2 Oran Graduate School of Economics, Algeria

Abstract
In this work, we will introduce a fractional Duhamel principle and use it to establish the well-boundedness and stability of a mild solution to an original fractional stochastic equation with initial data.
About the Authors
N. Bouteraa, University of Oran1 Ahmed Ben Bella, Oran, Algeria, bouteraa-27@hotmail.fr
For citation
Bouteraa N. Existence and Stability of Solutions for a Class of Stochastic Fractional Partial Differential Equation with a Noise. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 41, pp. 107–120. https://doi.org/10.26516/1997-7670.2022.41.107
Keywords
stochastic fractional partial differental equation, fractional derivative, mild solution, stability
UDC
518.517
MSC
34A08, 34B10, 34B15
DOI
https://doi.org/10.26516/1997-7670.2022.41.107
References
  1. Bensoussan A., Temam R. Equations stochastiques du type Navier-Stokes. J. Functional Analysis, 1973, vol. 13, pp. 195–222.
  2. Biagini F., Hu Y., Oksendal B., Zhang T. Stochastic calculus for fractional Brownian motion and applications. Springer, 2008.
  3. Bouteraa N., Inc M., Akgul A. Stability analysis of time-fractional differential equations with initial data. Math. Meth. Appl. Sci., 2021, pp. 1–9.
  4. Bouteraa N., Inc M., Akinlar M.A., Almohsen B. Mild solutions of fractional PDE with noise. Math. Meth. Appl. Sci., 2021, pp. 1–15.
  5. Caraballo T., Liu K. Exponential stability of mild solutions of stochastic partial differential equations with delays. Stoch. Anal. Appl., 1993, vol. 17, pp. 743–763.
  6. Djourdem H., Bouteraa N. Mild solution for a stochastic partial differential equation with noise. WSEAS Transactions on Systems, 2020, vol. 19, pp. 246–256.
  7. De Carvalho-Neto P.M., Gabriela P. Mild solutions to the time fractional NavierStokes equations in R𝑛. J. Differential Equations, 2015, vol. 259, pp. 2948–2980.
  8. Govindan T.E. Stability of mild solutions of stochastic evolutions with variable decay. Stoch. Anal. Appl., 2003, vol. 21, pp. 1059–1077.
  9. Hormander L. The Analysis of Linear Partial Differential Operators. Berlin, Springer, 2005.
  10. Inc M., Bouteraa N., Akinlar M.A., Chu Y.M., Weber G. W., Almohsen B. New positive solutions of nonlinear elliptic PDEs. Applied Sciences, 2020, vol. 10, 4863.https://doi.org/10.3390/app10144863
  11. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier, 2006.
  12. Liu K. Stability of Infnite Dimensional Stochastic Differential Equations with Applications. London, Chapman Hall, CRC, 2006.
  13. Luo J. Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. J. Math. Anal. Appl, 2008, vol. 342, no. 2, pp. 753–760.
  14. Mao X. Stochastic Differential Equations and Applications. Chichester, UK, Horwood Publishing Limited, 1997.
  15. Magin R.L. Fractional Calculus in Bioengineering. Begell House Publishers, 2006.
  16. Meerschaert M.M., Sikorskii A. Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics, Walter de Gruyter, Berlin/Boston, 2012, vol. 43.
  17. Metler R., Klafter J. The restaurant at the end of random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen., 2004, vol. 37, pp. 161–208.
  18. Rihan F.A., Alsakaji H.J., Rajivganthi C. Stochastic SIRC epidemic model with time delay for COVID-19. Adv. Differ. Equ., 2020, vol. 502.https://doi.org/10.1186/s13662-020-02964-8
  19. Podlubny I. Fractional Differential Equations. New York, Academic Press, 1993.
  20. Sakthivel R., Ren Y. Exponential stability of second-order stochastic evolution equations with Poisson jumps. Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, pp. 4517–4523.
  21. Seemab A., Rehman M. A note on fractional Duhamel’s principle and its application to a class of fractional partial differential equations. Applied Mathematics Letters,2017, vol. 64, pp. 8–14.
  22. Vashik M.J., Fursikov A.V. Mathematical problems of statistical hydromechanics. Kluver Dordreech, 1980.
  23. Umarov S., Saydamatov E. A Fractional Analog of the Duhamel Principle. Fract. Calc. Appl. Anal, 2006, vol. 9, pp. 57–70.
  24. Umarov S. On fractional Duhamel’s principle and its applications. J. Differential Equations, 2012, vol. 252, pp. 5217–5234.
  25. Wen Y., Zhou X.F., Wang J. Stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations. Advances in Difference Equations, 2017, vol. 230. https://doi.org/10.1186/s13662-017-1271-6
  26. Yang D. m-Dissipativity for Kolmogorov operator of fractional Burgers equation with space-time white noise. Potential anal, 2016, vol. 44, pp. 215–227.
  27. Zou G., Wang B. Stochastic Burgers equation with fractional derivative driven by multiplicative noise. Comput. Math. Appl, 2017, vol. 74, iss. 12, pp. 3195–3208. https://doi.org/10.1016/j.camwa.2017.08.023

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