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Stochastic diffusion equation with fractional Laplacian on the first quadrant

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Veröffentlicht/Copyright: 30. Juli 2019
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 22 Heft 3

Abstract

In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

MSC 2010: Primary 26A33; Secondary 31A10; 58J32
Key Words and Phrases: fractional Laplacian; Fokas method; Brownian motion; Monte Carlo integration

References

[1] J.L. Bona, L. Luo, Generalized Korteweg-de Vries equation in a quarter plane. Contemporary Mathematics221 (1999), 59–125.10.1090/conm/221/03118Suche in Google Scholar

[2] G. Casella, C.P. Robert, Monte Carlo Statistical Methods. Springer (2004).10.1007/978-1-4757-4145-2Suche in Google Scholar

[3] M.H. Chen, Q.M. Shao, J.G. Ibrahim, Monte Carlo Methods in Bayesian Computation. Springer (2000).10.1007/978-1-4612-1276-8Suche in Google Scholar

[4] L. Debnath, Recent applications of fractional calculus to science and engineering. Intern. J. of Math. and Math. Sci. 54 (2003), 3413–3442.10.1155/S0161171203301486Suche in Google Scholar

[5] M. Foondun, J.B. Mijena, E. Nane, Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1527–1553; 10.1515/fca-2016-0079; https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml.Suche in Google Scholar

[6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier (2006).10.3182/20060719-3-PT-4902.00008Suche in Google Scholar

[7] J. Liu, C.A. Tudor, Stochastic heat equation with fractional Laplacian and fractional noise: Existence of the solution and analysis of its density. Acta Math. Scientia37 (2017), 1545–1566.10.1016/S0252-9602(17)30091-7Suche in Google Scholar

[8] M. Mahdi, J. Saberi-Nadjafi, Gegenbauer spectral method for time-fractional convection-diffusion equations with variable coefficients. Math. Methods in Appl. Sci. 38 (2015), 3183–3194.10.1002/mma.3289Suche in Google Scholar

[9] D. Mantzavinos, A.S. Fokas, The unified transform for the heat equation: II. Non-separable boundary conditions in two dimensions. European J. of Appl. Math. 26 (2015), 887–916.10.1017/S0956792515000224Suche in Google Scholar

[10] C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu-Batlle, Fractional-order Systems and Controls. Springer-Verlag, London (2010).10.1007/978-1-84996-335-0Suche in Google Scholar

[11] S. Saha, S. Sahoo, Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection-dispersion equation involving nonlocal space fractional derivatives. Math. Methods in Appl. Sci. 38, (2015), 2840–2849.10.1002/mma.3267Suche in Google Scholar

[12] C. Pozrikidis, The Fractional Laplacian. CRC Press, Taylor & Francis (2016).10.1201/b19666Suche in Google Scholar

[13] H. Prado, M. Rivero, J.J. Trujillo, M.P. Velasco, New results from old investigation: A note on fractional m-dimensional differential operator. The fractional Laplacian. Fract. Calc. Appl. Anal. 18, No 2 (2015), 290–306; 10.1515/fca-2015-0020; https://www.degruyter.com/view/j/fca.2015.18.issue-2/issue-files/fca.2015.18.issue-2.xml.Suche in Google Scholar

[14] J.B. Walsh, An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 (1986), 265–439.10.1007/BFb0074920Suche in Google Scholar

[15] M.A. Tanner, Tools for Statistical Inference. Methods for the Exploration of Posterior Distributions and Likelihood Functions. Third Ed., Springer (1996).10.1007/978-1-4612-4024-2Suche in Google Scholar

[16] R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria; http://www.R-project.org/.Suche in Google Scholar

Received: 2018-03-09
Revised: 2019-05-17
Published Online: 2019-07-30
Published in Print: 2019-06-26

© 2019 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–3–2019)
  4. Survey Paper
  5. The probabilistic point of view on the generalized fractional partial differential equations
  6. Research Paper
  7. A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions
  8. Fractal convolution: A new operation between functions
  9. Unique continuation principle for the one-dimensional time-fractional diffusion equation
  10. Asymptotics of eigenvalues for differential operators of fractional order
  11. The asymptotic behaviour of fractional lattice systems with variable delay
  12. On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces
  13. Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics
  14. Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem
  15. Finite element approximations for fractional evolution problems
  16. Stochastic diffusion equation with fractional Laplacian on the first quadrant
  17. Stability of fractional variable order difference systems
  18. Chaotic dynamics of fractional Vallis system for El-Niño
https://doi.org/10.1515/fca-2019-0043
Schlagwörter für diesen Artikel
fractional Laplacian; Fokas method; Brownian motion; Monte Carlo integration

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–3–2019)
  4. Survey Paper
  5. The probabilistic point of view on the generalized fractional partial differential equations
  6. Research Paper
  7. A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions
  8. Fractal convolution: A new operation between functions
  9. Unique continuation principle for the one-dimensional time-fractional diffusion equation
  10. Asymptotics of eigenvalues for differential operators of fractional order
  11. The asymptotic behaviour of fractional lattice systems with variable delay
  12. On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces
  13. Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics
  14. Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem
  15. Finite element approximations for fractional evolution problems
  16. Stochastic diffusion equation with fractional Laplacian on the first quadrant
  17. Stability of fractional variable order difference systems
  18. Chaotic dynamics of fractional Vallis system for El-Niño
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