THERMAL SCIENCE

International Scientific Journal

Balachandar S. Raja

,

D. Uma

,

H. Jafari

,

S. G. Venkatesh

NUMERICAL SOLUTION FOR STOCHASTIC HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS

ABSTRACT
In this article, we propose a new technique based on 2-D shifted Legendre poly­nomials through the operational matrix integration method to find the numeri­cal solution of the stochastic heat equation with Neumann boundary conditions. For the proposed technique, the convergence criteria and the error estima­tion are also discussed in detail. This new technique is tested with two exam­ples, and it is observed that this method is very easy to handle such problems as the initial and boundary conditions are taken care of automatically. Also, the time complexity of the proposed approach is discussed and it is proved to be O[k(N + 1)4] where N denotes the degree of the approximate function and k is the number of simulations. This method is very convenient and efficient for solving other partial differential equations.
KEYWORDS
stochastic PDE, heat equation, error analysis, operational matrices, Shifted Legendre polynomials
PAPER SUBMITTED: 2022-04-01
PAPER REVISED: 2022-05-24
PAPER ACCEPTED: 2022-06-14
PUBLISHED ONLINE: 2023-04-08
DOI REFERENCE: https://doi.org/10.2298/TSCI23S1057R
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Special issue 1, PAGES [57 - 66]
REFERENCES
  1. Appleby, J. A., et al., Almost Sure Convergence of Solutions of Linear Stochastic Volterra Equations to Non-Equilibrium Limits, Journal Integral Equ. Appl., 19 (2007), 4, pp. 405-437
  2. Babuška, I., et al., A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Review, 52 (2010), 2, pp. 317-355
  3. Henderson, D., Plaschko, P., Stochastic Differential Equations in Science and Engineering (With Cd-rom), World Scientific, Singapore, 2006
  4. Heydari, M. H., et al., Wavelets Galerkin Method for Solving Stochastic Heat Equation, Int. J. Comput. Math., 93 (2016), 9, pp. 1579-1596
  5. Khodabin, M., et al., Interpolation Solution in Generalized Stochastic Exponential Population Growth Model, Appl. Math. Model., 36 (2012), 3, pp. 1023-1033
  6. Zhu, L., Fan, Q., Solving Fractional Non-Linear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet, Commun Non-Linear Sci. Numer Simul., 17 (2012), 6, pp. 2333-2341
  7. Fournier, N., Printems, J., Stability of the Stochastic Heat Equation in L1(
  8. Shang, S., Zhang, T., Global Well-Posedness to Stochastic Reaction-Diffusion Equations on the Real Line R with Superlinear Drifts Driven by Multiplicative Space-Time white Noise, On-line first, arXiv: 2106.02879, 2021
  9. Aidoo, A. Y., Wilson, M., A Review of Wavelets Solution Stochastic Heat Equation with Random Inputs, Appl. Math., 6 (2015), 14, pp. 2226-2239
  10. Chen, L., et al., A Boundedness Trichotomy for the Stochastic Heat Equation, Ann. Henri Poincare, 53 (2017), 4, pp. 1991-2004
  11. Benner, P., et al., A Linear Implicit Euler Method for the Finite Element Discretization of a Controlled Stochastic Heat Equation, arXiv preprint arXiv: 2006 (2020), 05370
  12. Mytnik, L., Perkins, E., Pathwise Uniqueness for Stochastic Heat Equations with Holder Continuous Coefficients: The white Noise Case, Probab. Theory Relat. Fields, 149 (2011),1, pp. 1-96
  13. Krishnaveni, K., et al., Approximate Solution of Space-Time Fractional Differential Equations Via Legendre Polynomials, Sci. Asia, 42 (2016), 5, pp. 356-361
  14. Mirzaee, F., Samadyar, N., Application of Operational Matrices for Solving System of Linear Stratonovich Volterra Integral Equation, Journal Comput. Appl. Math., 320 (2017), pp. 164-175
  15. Mao, X., Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing, Chichester, UK, 2007
  16. Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, Springer Science and Business Media, Berlin, Germany, 2013
PDF VERSION [DOWNLOAD]
Numerical solution for stochastic heat equation with Neumann boundary conditions

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence