The Cauchy Problem for the Heat Equation with a Random Right Part from the Space *Sub*_{φ} (Ω)

ABSTRACT

The influence of random
factors should often be taken into account in solving problems of mathematical
physics. The heat equation with random factors is a classical problem of the parabolic
type of mathematical physics. In this paper, the heat equation with random right
side is examined. In particular, we give conditions of existence with probability,
one classical solutions in the case when the right side is a random field, sample
continuous with probability one from the space *Sub*_{φ} (Ω). Estimation for the distribution of the supremum
of solutions of such equations is founded.

Cite this paper

Kozachenko, Y. and Slyvka-Tylyshchak, A. (2014) The Cauchy Problem for the Heat Equation with a Random Right Part from the Space*Sub*_{φ} (Ω). *Applied Mathematics*, **5**, 2318-2333. doi: 10.4236/am.2014.515226.

Kozachenko, Y. and Slyvka-Tylyshchak, A. (2014) The Cauchy Problem for the Heat Equation with a Random Right Part from the Space

References

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[2] Beisenbaev, E. and Kozachenko, Yu.V. (1979) Uniform Convergence in Probability of Random Series, and Solutions of Boundary Value Problems with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 21, 9-23.

[3] Buldygin, V.V. and Kozachenko, Yu.V. (1979) On a Question of the Applicability of the Fourier Method for Solving Problems with Random Boundary Conditions. Random Processes in Problems Mathematical Physics, Academy of Sciences of Ukrain.SSR, Institute of Mathematics, Kuiv, 4-35.

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http://dx.doi.org/10.1515/rose.1995.3.3.201

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[6] Kozachenko, Yu.V. and Endzhyrgly (1994) Justification of Applicability of the Fourier Method to the Boundary-Value Problems with Random Initial Conditions II. Theory of Probability and Mathematical Statistics, 53, 58-68.

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[9] Kozachenko, Yu.V. and Slyvka, G.I. (2004) Justification of the Fourier Method for Hyperbolic Equations with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 69, 67-83.

http://dx.doi.org/10.1090/S0094-9000-05-00615-0

[10] Slyvka, A.I. (2002) A Boundary-Value Problem of the Mathematical Physics with Random Initials Conditions. Bulletin of University of Kyiv. Series: Physics & Mathematics, 5, 172-178.

[11] Slyvka-Tylyshchak, A.I. (2012) Justification of the Fourier Method for Equations of Homogeneous String Vibration with Random Initial Conditions. Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 38, 211-232.

[12] Kozachenko, Y.V. and Slyvka, G.I. (2007) Modelling a Solution of a Hyperbolic Equation with Random Initial Conditions. Theory Probability and Mathematical Statistics, 74, 59-75.

[13] Tylyshchak, A.I.S. (2012) Simulation of Vibrations of a Rectangular Membrane with Random Initial Conditions. Annales Mathematicae and Informaticae, 39, 325-338.

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[15] Kozachenko, Y.V. and Veresh, K.J. (2010) The Heat Equation with Random Initial Conditions from Orlicz Space. Theory of Probability and Mathematical Statistics, 80, 71-84.

http://dx.doi.org/10.1090/S0094-9000-2010-00795-2

[16] Kozachenko, Y.V. and Veresh, K.J. (2010) Boundary-Value Problems for a Nonhomogeneous Parabolic Equation with Orlicz Right Side. Random Operators and Stochastic Equations, 18, 97-119.

http://dx.doi.org/10.1515/rose.2010.005

[17] Angulo, J.M., Ruiz-Medina, M.D., Anh, V.V. and Grecksch, W. (2000) Fractional Diffusion and Fractional Heat Equation. Advances in Applied Probability, 32, 1077-1099.

http://dx.doi.org/10.1239/aap/1013540349

[18] Kozachenko, Y.V. and Leonenko, G.M. (2006) Extremal Behavior of the Heat Random Field. Extremes, 8, 191-205.

http://dx.doi.org/10.1007/s10687-006-7967-8

[19] Beghin, L., Kozachenko, Y., Orsingher, E. and Sakhno, L. (2007) On the Solution of Linear Odd-Order Heat-Type Equations with Random Initial. Journal of Statistical Physics, 127, 721-739.

http://dx.doi.org/10.1007/s10955-007-9309-x

[20] Ratanov N.E., Shuhov, A.G. and Suhov, Y.M. (1991) Stabilization of the Statistical Solution of the Parabolic Equation. Acta Applicandae Mathematicae, 22, 103-115.

[21] Buldygin, V.V. and Kozachenko, Y.V. (2000) Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Rhode.

[22] Antonini, R.G., Kozachenko, Y. and Nikitina, T. (2003) Spaces of φ-Subgaussian Random Variables. Memorie di Matematica e Applicazioni, Accademia Nazionale delle Scinze detta dei XL, Vol. 27, 95-124.

[23] Krasnoselsky, M.A. and Rutitcky, Y.B. (1961) Convex Functions and Orlicz Spaces. Noordhof, Gröningen.

[24] Kozachenko, Y.V. and Ostrovskij, E.V. (1986) Banach Spaces of Random Variables of Sub-Gaussian Type. Theory of Probability and Mathematical Statistics, 532, 42-53.

[25] Markovich, B.M. (2010) Equations of Mathematical Physics. Lviv Polytechnic Publishing House, Lviv, 384 p. (Ukrainian).

[26] Budylin, A.M. (2002) Fourier Series and Integrals. Saint Petersburg, 137 p.

[1] de Feriet, K. (1962) Statistical Mechanics of Continuous Media. Proceedings of Symposia in Applied Mathematics, American Mathematical Society, Providence, 165-198.

[2] Beisenbaev, E. and Kozachenko, Yu.V. (1979) Uniform Convergence in Probability of Random Series, and Solutions of Boundary Value Problems with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 21, 9-23.

[3] Buldygin, V.V. and Kozachenko, Yu.V. (1979) On a Question of the Applicability of the Fourier Method for Solving Problems with Random Boundary Conditions. Random Processes in Problems Mathematical Physics, Academy of Sciences of Ukrain.SSR, Institute of Mathematics, Kuiv, 4-35.

[4] de La Krus, E.B. and Kozachenko, Yu.V. (1995) Boundary-Value Problems for Equations of Mathematical Physics with Strictly Orlics Random Initial Conditios. Random Operators and Stochastic Equations, 3, 201-220.

http://dx.doi.org/10.1515/rose.1995.3.3.201

[5] Kozachenko, Yu.V. and Endzhyrgly (1994) Justification of Applicability of the Fourier Method to the Boundary-Value Problems with Random Initial Conditions I. Theory of Probability and Mathematical Statistics, 51, 78-89.

[6] Kozachenko, Yu.V. and Endzhyrgly (1994) Justification of Applicability of the Fourier Method to the Boundary-Value Problems with Random Initial Conditions II. Theory of Probability and Mathematical Statistics, 53, 58-68.

[7] Kozachenko, Yu.V. and Kovalchuk, Ya.A. (1998) Boundary Value Problems with Random Initial Conditions and Series of Functions of Subφ(Ω). Ukrainian Mathematical Journal, 50, 504-515.

[8] Dovgay, B.V., Kozachenko, Yu.V. and Slyvka-Tylyshchak, G.I. (2008) The Boundary-Value Problems of Mathematical Physics with Random Factors. Kyiv University, Kyiv, 173 p. (Ukrainian)

[9] Kozachenko, Yu.V. and Slyvka, G.I. (2004) Justification of the Fourier Method for Hyperbolic Equations with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 69, 67-83.

http://dx.doi.org/10.1090/S0094-9000-05-00615-0

[10] Slyvka, A.I. (2002) A Boundary-Value Problem of the Mathematical Physics with Random Initials Conditions. Bulletin of University of Kyiv. Series: Physics & Mathematics, 5, 172-178.

[11] Slyvka-Tylyshchak, A.I. (2012) Justification of the Fourier Method for Equations of Homogeneous String Vibration with Random Initial Conditions. Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 38, 211-232.

[12] Kozachenko, Y.V. and Slyvka, G.I. (2007) Modelling a Solution of a Hyperbolic Equation with Random Initial Conditions. Theory Probability and Mathematical Statistics, 74, 59-75.

[13] Tylyshchak, A.I.S. (2012) Simulation of Vibrations of a Rectangular Membrane with Random Initial Conditions. Annales Mathematicae and Informaticae, 39, 325-338.

[14] Dovgay, B.V. and Kozachenko, Y.V. (2005) The Condition for Application of Fourie Method to the Solution of Nongomogeneous String Oscillation Equation with φ-Subgaussianright Hand Side. Random Operators and Stochastic Equations, 13, 281-296.

[15] Kozachenko, Y.V. and Veresh, K.J. (2010) The Heat Equation with Random Initial Conditions from Orlicz Space. Theory of Probability and Mathematical Statistics, 80, 71-84.

http://dx.doi.org/10.1090/S0094-9000-2010-00795-2

[16] Kozachenko, Y.V. and Veresh, K.J. (2010) Boundary-Value Problems for a Nonhomogeneous Parabolic Equation with Orlicz Right Side. Random Operators and Stochastic Equations, 18, 97-119.

http://dx.doi.org/10.1515/rose.2010.005

[17] Angulo, J.M., Ruiz-Medina, M.D., Anh, V.V. and Grecksch, W. (2000) Fractional Diffusion and Fractional Heat Equation. Advances in Applied Probability, 32, 1077-1099.

http://dx.doi.org/10.1239/aap/1013540349

[18] Kozachenko, Y.V. and Leonenko, G.M. (2006) Extremal Behavior of the Heat Random Field. Extremes, 8, 191-205.

http://dx.doi.org/10.1007/s10687-006-7967-8

[19] Beghin, L., Kozachenko, Y., Orsingher, E. and Sakhno, L. (2007) On the Solution of Linear Odd-Order Heat-Type Equations with Random Initial. Journal of Statistical Physics, 127, 721-739.

http://dx.doi.org/10.1007/s10955-007-9309-x

[20] Ratanov N.E., Shuhov, A.G. and Suhov, Y.M. (1991) Stabilization of the Statistical Solution of the Parabolic Equation. Acta Applicandae Mathematicae, 22, 103-115.

[21] Buldygin, V.V. and Kozachenko, Y.V. (2000) Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Rhode.

[22] Antonini, R.G., Kozachenko, Y. and Nikitina, T. (2003) Spaces of φ-Subgaussian Random Variables. Memorie di Matematica e Applicazioni, Accademia Nazionale delle Scinze detta dei XL, Vol. 27, 95-124.

[23] Krasnoselsky, M.A. and Rutitcky, Y.B. (1961) Convex Functions and Orlicz Spaces. Noordhof, Gröningen.

[24] Kozachenko, Y.V. and Ostrovskij, E.V. (1986) Banach Spaces of Random Variables of Sub-Gaussian Type. Theory of Probability and Mathematical Statistics, 532, 42-53.

[25] Markovich, B.M. (2010) Equations of Mathematical Physics. Lviv Polytechnic Publishing House, Lviv, 384 p. (Ukrainian).

[26] Budylin, A.M. (2002) Fourier Series and Integrals. Saint Petersburg, 137 p.