A Review of Wavelets Solution to Stochastic Heat Equation with Random Inputs

Anthony Y. Aidoo¹, Matilda Wilson²

¹Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, USA.
²Department of Computer Science, University of Ghana, Legon, Ghana.

DOI: 10.4236/am.2015.614196                

Abstract

We consider a wavelet-based solution to the stochastic heat equation with random inputs. Computational methods based on the wavelet transform are analyzed for solving three types of stochastic heat equation. The methods are shown to be very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. The results reveal that the wavelet algorithms are very accurate and efficient.

Keywords

Wavelets, Stochastic, Heat Equation, Collocation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Gunzberger, M., Webster, C.G. and Zhang, G. (2002) An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Stochastic Partial Differential Equations, ORNL Report, 21-33.
[2]	Frauenfelder, P., Schwab, C. and Todor, R.A. (2005) Finite Elements for Elliptic Problems with Stochastic Coefficients. Computer Methods in Applied Mechanics and Engineering, 194, 205-228. http://dx.doi.org/10.1016/j.cma.2004.04.008
[3]	Li, J. and Xiu, D. (2009) A Generalized Polynomial Chaos Based Ensemble Kalman Filter with High Accuracy. Journal of Computational Physics, 228, 5454-5469. http://dx.doi.org/10.1016/j.jcp.2009.04.029
[4]	Rannacher, R. (2005) Adaptive Solution of PDE-Constrained Optimal Control Problems, In: Liu, W., Ed., The 2nd International Conference on Scientific Computing and Partial Differential Equations (SCPDE05).
[5]	Vasilyev, O.V., Paolucci, S. and Sen, M. (1995) A Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain. Journal of Computational Physics, 120, 33-47. http://dx.doi.org/10.1006/jcph.1995.1147
[6]	Allosa, E., Mazetb, O. and Nualartb, D. (2000) Stochastic Calculus with Respect to Fractional Brownian Motion with Hurst Parameter Lesser than 1/2. Stochastic Processes and Their Applications, 86,121-139. http://dx.doi.org/10.1016/S0304-4149(99)00089-7
[7]	Kythe, P.K. and Schäferkotter, M.R. (2004) Handbook of Computational Methods for Integration. CRC Press, Boca Raton, 401. http://dx.doi.org/10.1201/9780203490303
[8]	Vasilyev, O.V. and Bowman, C. (2000) Second Generation Wavelet Collocation Method for the Solution of Partial Differential Equations. Journal of Computational Physics, 165, 660-693. http://dx.doi.org/10.1006/jcph.2000.6638
[9]	Sweldens, W. (1998) The Lifting Scheme: A Construction of Second Generation Wavelets. SIAM Journal of Mathematical Analysis, 29, 511-546. http://dx.doi.org/10.1137/S0036141095289051
[10]	Alam, S.M., Kevlahan, N.K.-R. and Vasilyev, O.V. (2006) Simultaneous Space-Time Adaptive Wavelet Solution of Nonlinear Parabolic Differential Equations. Journal of Computational Physics, 214, 829-857. http://dx.doi.org/10.1016/j.jcp.2005.10.009
[11]	Cioica, P.A., Dahlke, S., Döhring, N., Kinzel, S., Lindner, F., Raasch, T., Ritter, K. and Schilling, R.L. (2011) Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations. BIT, Preprint. http://www.dfg-sppl324.de/publications.php
[12]	Fukutani, K., Toyoshima, S., Yutaka, H. and Yamamoto, A. (2003) Numerical Computation of Diameter Fluctuation in Optical Fiber Drawing from Silica Glass Preform by Perturbation Method. Transactions of the Japan Society of Mechanical Engineers, 69, 2403-2410. http://dx.doi.org/10.1299/kikaic.69.2403
[13]	Chiba, R. (2012) Chap. 9. Stochastic Analysis of Heat Conduction and Thermal Stresses in Solids: A Review. In: Kazi, S.N., Ed., Heat Transfer Phenomena and Applications, InTech. http://dx.doi.org/10.5772/50994
[14]	Prasad, R.C., Karmeshu and Bharadwaj, K.K. (2002) Stochastic Modeling of Heat Exchange Response to Data Uncertainties. Applied Mathematical Modelling, 26, 715-726. http://dx.doi.org/10.1016/S0307-904X(01)00082-8
[15]	Fouque, J.P., Papanicolaou, G. and Sircar, R. (2004) Stochastic Volatility and Correction to the Heat Equation, Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability, 58, 267-276.
[16]	Fouque, J.P., Papanicolaou, G. and Sircar, R. (2004) Stochastic Volatility and Correction to the Heat Equation. Progress in Probability, 58, 265-274. http://dx.doi.org/10.1007/978-3-0348-7943-9_16
[17]	Barcy, E., Mallat, S. and Papanicolaou, G. (1992) A Wavelet Based Space-Time Adaptive Numerical Method for Partial Differential Equations. RAIRO-Modélisation Mathématique et Analyse Numérique, 26, 793-834.
[18]	Cheng, M., Hou, T.Y., Yan, M. and Zhang, Z. (2013) A Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients. SIAM/ASA Journal on Uncertainty Quantification, 1, 452-493.
[19]	Assing, S. and Manthey, R. (2003) Invariant Measures for Stochastic Heat Equation with Unbounded Coefficients. Stochastic Processes and Their Applications, 103, 237-256. http://dx.doi.org/10.1016/S0304-4149(02)00211-9
[20]	Balan, R. and Kim, D. (2008) The Stochastic Heat Equation Driven by a Gaussian Noise: Germ Markov Property. Communications on Stochastic Analysis, 2, 229-249.
[21]	Ghanem, R.G. and Spanos, P.D. (1991) Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-3094-6
[22]	Le Maître, O.P., Knio, O.M., Debusschere, B.J., Najm, H.N. and Ghanem, R.G. (2003) A Multigrid Solver for Two- Dimensional Stochastic Diffusion Equation. Computer Methods in Applied Mechanics and Engineering, 192, 4723- 4744. http://dx.doi.org/10.1016/S0045-7825(03)00457-2
[23]	Kovács, M., Larsson, S. and Urban, K. (2013) On Wavelet-Galerkin Methods for Semilinear Parabolic Equations with Additive Noise. In: Dick, J., Kuo, F.Y., Peters, G.W. and Sloan, I.H., Eds., Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Mathematics and Statistics, Vol. 65, Springer, Berlin Heidelberg, 481-499. http://dx.doi.org/10.1007/978-3-642-41095-6_24
[24]	Xiu, D. and Karniadakis, G.E. (2003) A New Stochastic Approach to Transient Heat Conduction Modeling with Uncertainty. International Journal of Heat and Mass Transfer, 46, 4681-4693. http://dx.doi.org/10.1016/S0017-9310(03)00299-0
[25]	Lisei, H. and Soós, A. (2004) Wavelet Approximations of the Solution of Some Stochastic Differential Equations. Pure Mathematics and Applications, 15, 213-223.
[26]	Du, Q. and Zhang, T. (2002) Numerical Approximation of Some Linear Stochastic Partial Differential Equation Driven by Special Additive Noises. SIAM Journal on Numerical Analysis, 40, 1421-1445. http://dx.doi.org/10.1137/S0036142901387956
[27]	Swanson, J. (2007) Variations of the Solution to a Stochastic Heat Equation. The Annals of Probability, 35, 2122-2159. http://dx.doi.org/10.1214/009117907000000196
[28]	Wells Jr., R.O. and Zhou, X.D. (1995) Wavelet Solutions for the Dirichlet Problem. Numerische Mathematik, 70, 379-396. http://dx.doi.org/10.1007/s002110050125