Analytic-numerical solution of random parabolic models: a mean square fourier transform approach
DOI: https://doi.org/10.3846/mma.2018.006Abstract
This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an infinite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example.
Keywords:
mean square random calculus, random parabolic models, analytic-numerical solution, random mean square quadrature formulae, random Fourier transformHow to Cite
Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Analytic-numerical solution of random parabolic models: a mean square fourier transform approach. Mathematical Modelling and Analysis, 23(1), 79-100. https://doi.org/10.3846/mma.2018.006
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Copyright (c) 2018 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.-C. Casaban, R. Company, J.-C. Cortes and L. Jodar. Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24):5922-5933, 2014. <a href= "https://doi.org/10.1016/j.apm.2014.04.063" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2014.04.063</a>
M.-C. Casaban, J.-C. Cortes, B. Garcia-Mora and L. Jodar. Analyticnumerical solution of random boundary value heat problems in a semi-infinite bar. Abstract and Applied Analysis, 2013(Article ID 676372):1-9, 2013. <a href= "https://doi.org/10.1155/2013/676372" target="_blank" rel="noopener"> https://doi.org/10.1155/2013/676372</a>
M.-C. Casaban, J.-C. Cortes and L. Jodar. A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259:654-667, 2015. <a href= "https://doi.org/10.1016/j.amc.2015.02.091" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2015.02.091</a>
M.-C. Casaban, J.-C. Cortes and L. Jodar. Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22):9362-9377, 2016. <a href= "https://doi.org/10.1016/j.apm.2016.06.017" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2016.06.017</a>
R. Chiba. Stochastic heat conduction analysis of a functionally grade annular disc with spatially random heat transfer coefficients. Applied Mathematical Modelling, 33(1):507-523, 2009. <a href= "https://doi.org/10.1016/j.apm.2007.11.014" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2007.11.014</a>
J.-C. Cortes, L. Jodar, M.-D. Rosello and L. Villafuerte. Solving initial and two-point boundary value linear random differential equations: A mean square approach. Applied Mathematics and Computation, 219(4):2204-2211, 2012. <a href= "https://doi.org/10.1016/j.amc.2012.08.066" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2012.08.066</a>
J.-C. Cortes, L. Jodar, L. Villafuerte and R. J. Villanueva. Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3):44-48, 2007. <a href= "https://doi.org/10.1016/j.matcom.2007.01.020" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2007.01.020</a>
M. C. C. Cunha and F. A. Dorini. Statistical moments of the solution of the random Burgers-Riemann problem. Mathematics and Computers in Simulation, 79(5):1440-1451, 2009. <a href= "https://doi.org/10.1016/j.matcom.2008.06.001" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2008.06.001</a>
P. J. Davis and P. Rabinowitz. Computer Science and Applied Mathematics, second edition. Academic Press, Inc., San Diego, 1984. [14] L. M. Delves and J. L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, New York, 1985.
D. C. Dibben and R. Metaxas. Time domain finite element analysis of multimode microwave applicators. IEEE Transactions on Magnetics, 32(3):942-945, 1996. <a href= "https://doi.org/10.1109/20.497397" target="_blank" rel="noopener"> https://doi.org/10.1109/20.497397</a>
F. A. Dorini and M. C. C. Cunha. On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4):679-690, 2011. <a href= "https://doi.org/10.1016/j.matcom.2011.10.008" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2011.10.008</a>
S. J. Farlow. Partial Differential Equations for Scientists and Engineers. Dover, New York, 1993.
I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series and products, fifth edition. Academic Press, Inc., San Diego, 1994.
G. R. Grimmett and D. R. Stirzaker. Probability and Random Processes. Clarendon Press, New York, 2000.
A. F. Harvey. Microwave Engineering. Academic Press, New York, 1963.
A. Hussein and M. M. Selim. Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique. Applied Mathematics and Computation, 213(1):250-261, 2009. <a href= "https://doi.org/10.1016/j.amc.2009.03.016" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2009.03.016</a>
A. Hussein and M. M. Selim. A general analytical solution for the stochastic Milne problem using Karhunen{Loeve (K-L) expansion. Quantitative Spectroscopy and Radiative Transfer, 125:84-92, 2013. <a href= "https://doi.org/10.1016/j.jqsrt.2013.03.018" target="_blank" rel="noopener"> https://doi.org/10.1016/j.jqsrt.2013.03.018</a>
L. Jodar, J. I. Castano, J. A. Sanchez and G. Rubio. Accurate numerical solution of coupled time dependent parabolic initial value problems. Applied Numerical Mathematics, 47(3-4):467-476, 2003. <a href= "https://doi.org/10.1016/S0168-9274(03)00086-2" target="_blank" rel="noopener"> https://doi.org/10.1016/S0168-9274(03)00086-2</a>
Y. Li and S. Long. A finite element model based on statistical two-scale analysis for equivalent heat transfer parameters of composite material with random grains. Applied Mathematical Modelling, 33(7):3157-3165, 2009. <a href= "https://doi.org/10.1016/j.apm.2008.10.018" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2008.10.018</a>
M. Loeve. Probability Theory I, series: Graduate Texts in Mathematics, Vol. 45. Springer-Verlag, New York, 1977.
B. McLaughlin, J. Peterson and M. Ye. Stabilized reduced order models for the advection-diffusion-reaction equation using operator splitting. Computers and Mathematics with Applications, 71(11):2407-2420, 2016. <a href= "https://doi.org/10.1016/j.camwa.2016.01.032" target="_blank" rel="noopener"> https://doi.org/10.1016/j.camwa.2016.01.032</a>
M. Necati Ozisik. Boundary Value Problems of Heat Conduction. Dover Publications, New York, 1968.
T. T. Soong. Random Differential Equations in Science and Engineering. Academic Press, New York, 1973.
S.R.S. Varadhan and N. Zygouras. Behavior of the solution of a random semilinear heat equation. Communications on Pure and Applied Mathematics, 61(9):1298-1329, 2008. <a href= "https://doi.org/10.1002/cpa.20256" target="_blank" rel="noopener"> https://doi.org/10.1002/cpa.20256</a>
L. Villafuerte, C.A. Braumann, J.-C. Cortes and L. Jodar. Random differential operational calculus: theory and applications. Computers and Mathematics with Applications, 59(1):115-125, 2010. <a href= "https://doi.org/10.1016/j.camwa.2009.08.061" target="_blank" rel="noopener"> https://doi.org/10.1016/j.camwa.2009.08.061</a>
N. Bellomo, L. M. De Socio and R. Monaco. Random heat equation: solutions by the stochastic adaptive interpolation method. Computers and Mathematics with Applications, 16(9):759-766, 1988. <a href= "https://doi.org/10.1016/0898-1221(88)90011-9" target="_blank" rel="noopener"> https://doi.org/10.1016/0898-1221(88)90011-9</a>
A. T. Bharucha-Reid. Probabilistic Methods in Applied Mathematics. Academic Press, London, 1973.
G. Calbo, J.-C. Cortes, L. J_odar and L. Villafuerte. Analytic stochastic process solutions of second-order random differential equations. Applied Mathematics Letters, 23(12):1421-1424, 2010. <a href= "https://doi.org/10.1016/j.aml.2010.07.011" target="_blank" rel="noopener"> https://doi.org/10.1016/j.aml.2010.07.011</a>
M.-C. Casaban, R. Company, J.-C. Cortes and L. Jodar. Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24):5922-5933, 2014. <a href= "https://doi.org/10.1016/j.apm.2014.04.063" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2014.04.063</a>
M.-C. Casaban, J.-C. Cortes, B. Garcia-Mora and L. Jodar. Analyticnumerical solution of random boundary value heat problems in a semi-infinite bar. Abstract and Applied Analysis, 2013(Article ID 676372):1-9, 2013. <a href= "https://doi.org/10.1155/2013/676372" target="_blank" rel="noopener"> https://doi.org/10.1155/2013/676372</a>
M.-C. Casaban, J.-C. Cortes and L. Jodar. A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259:654-667, 2015. <a href= "https://doi.org/10.1016/j.amc.2015.02.091" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2015.02.091</a>
M.-C. Casaban, J.-C. Cortes and L. Jodar. Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22):9362-9377, 2016. <a href= "https://doi.org/10.1016/j.apm.2016.06.017" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2016.06.017</a>
R. Chiba. Stochastic heat conduction analysis of a functionally grade annular disc with spatially random heat transfer coefficients. Applied Mathematical Modelling, 33(1):507-523, 2009. <a href= "https://doi.org/10.1016/j.apm.2007.11.014" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2007.11.014</a>
J.-C. Cortes, L. Jodar, M.-D. Rosello and L. Villafuerte. Solving initial and two-point boundary value linear random differential equations: A mean square approach. Applied Mathematics and Computation, 219(4):2204-2211, 2012. <a href= "https://doi.org/10.1016/j.amc.2012.08.066" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2012.08.066</a>
J.-C. Cortes, L. Jodar, L. Villafuerte and R. J. Villanueva. Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3):44-48, 2007. <a href= "https://doi.org/10.1016/j.matcom.2007.01.020" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2007.01.020</a>
M. C. C. Cunha and F. A. Dorini. Statistical moments of the solution of the random Burgers-Riemann problem. Mathematics and Computers in Simulation, 79(5):1440-1451, 2009. <a href= "https://doi.org/10.1016/j.matcom.2008.06.001" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2008.06.001</a>
P. J. Davis and P. Rabinowitz. Computer Science and Applied Mathematics, second edition. Academic Press, Inc., San Diego, 1984. [14] L. M. Delves and J. L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, New York, 1985.
D. C. Dibben and R. Metaxas. Time domain finite element analysis of multimode microwave applicators. IEEE Transactions on Magnetics, 32(3):942-945, 1996. <a href= "https://doi.org/10.1109/20.497397" target="_blank" rel="noopener"> https://doi.org/10.1109/20.497397</a>
F. A. Dorini and M. C. C. Cunha. On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4):679-690, 2011. <a href= "https://doi.org/10.1016/j.matcom.2011.10.008" target="_blank" rel="noopener"> https://doi.org/10.1016/j.matcom.2011.10.008</a>
S. J. Farlow. Partial Differential Equations for Scientists and Engineers. Dover, New York, 1993.
I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series and products, fifth edition. Academic Press, Inc., San Diego, 1994.
G. R. Grimmett and D. R. Stirzaker. Probability and Random Processes. Clarendon Press, New York, 2000.
A. F. Harvey. Microwave Engineering. Academic Press, New York, 1963.
A. Hussein and M. M. Selim. Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique. Applied Mathematics and Computation, 213(1):250-261, 2009. <a href= "https://doi.org/10.1016/j.amc.2009.03.016" target="_blank" rel="noopener"> https://doi.org/10.1016/j.amc.2009.03.016</a>
A. Hussein and M. M. Selim. A general analytical solution for the stochastic Milne problem using Karhunen{Loeve (K-L) expansion. Quantitative Spectroscopy and Radiative Transfer, 125:84-92, 2013. <a href= "https://doi.org/10.1016/j.jqsrt.2013.03.018" target="_blank" rel="noopener"> https://doi.org/10.1016/j.jqsrt.2013.03.018</a>
L. Jodar, J. I. Castano, J. A. Sanchez and G. Rubio. Accurate numerical solution of coupled time dependent parabolic initial value problems. Applied Numerical Mathematics, 47(3-4):467-476, 2003. <a href= "https://doi.org/10.1016/S0168-9274(03)00086-2" target="_blank" rel="noopener"> https://doi.org/10.1016/S0168-9274(03)00086-2</a>
Y. Li and S. Long. A finite element model based on statistical two-scale analysis for equivalent heat transfer parameters of composite material with random grains. Applied Mathematical Modelling, 33(7):3157-3165, 2009. <a href= "https://doi.org/10.1016/j.apm.2008.10.018" target="_blank" rel="noopener"> https://doi.org/10.1016/j.apm.2008.10.018</a>
M. Loeve. Probability Theory I, series: Graduate Texts in Mathematics, Vol. 45. Springer-Verlag, New York, 1977.
B. McLaughlin, J. Peterson and M. Ye. Stabilized reduced order models for the advection-diffusion-reaction equation using operator splitting. Computers and Mathematics with Applications, 71(11):2407-2420, 2016. <a href= "https://doi.org/10.1016/j.camwa.2016.01.032" target="_blank" rel="noopener"> https://doi.org/10.1016/j.camwa.2016.01.032</a>
M. Necati Ozisik. Boundary Value Problems of Heat Conduction. Dover Publications, New York, 1968.
T. T. Soong. Random Differential Equations in Science and Engineering. Academic Press, New York, 1973.
S.R.S. Varadhan and N. Zygouras. Behavior of the solution of a random semilinear heat equation. Communications on Pure and Applied Mathematics, 61(9):1298-1329, 2008. <a href= "https://doi.org/10.1002/cpa.20256" target="_blank" rel="noopener"> https://doi.org/10.1002/cpa.20256</a>
L. Villafuerte, C.A. Braumann, J.-C. Cortes and L. Jodar. Random differential operational calculus: theory and applications. Computers and Mathematics with Applications, 59(1):115-125, 2010. <a href= "https://doi.org/10.1016/j.camwa.2009.08.061" target="_blank" rel="noopener"> https://doi.org/10.1016/j.camwa.2009.08.061</a>
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2018-02-20
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Copyright (c) 2018 The Author(s). Published by Vilnius Gediminas Technical University.
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This work is licensed under a Creative Commons Attribution 4.0 International License.
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Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Analytic-numerical solution of random parabolic models: a mean square fourier transform approach. Mathematical Modelling and Analysis, 23(1), 79-100. https://doi.org/10.3846/mma.2018.006