Journal of Applied Mathematics and Stochastic Analysis

Volume 2006, Article ID 93502, 18 pages

http://dx.doi.org/10.1155/JAMSA/2006/93502

## Explicit solutions of some fractional partial differential equations via stable subordinators

^{1}Institut élie Cartan, Université Henri Poincaré – Nancy 1, B.P. 239, Vandoeuvre-lès-Nancy Cedex 54506, France^{2}Department of Mathematics, Faculty of Sciences, Ferhat Abbas University, El-Maabouda Sètif 19000, Algeria^{3}Department of Mathematics, Faculty of Sciences, University of M'sila, B.P. 166, Ichbilia, M'sila 28000, Algeria

Received 10 December 2004; Revised 5 May 2005; Accepted 10 May 2005

Copyright © 2006 Latifa Debbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- V. V. Anh and N. N. Leonenko, “Spectral analysis of fractional kinetic equations with random
data,”
*Journal of Statistical Physics*, vol. 104, no. 5-6, pp. 1349–1387, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ben Adda,
*Dérivation d'ordre réel, Etude, Application et Interprétation géométrique*, Thèse, Universitè Paris VI, Paris, 1997. - S. Benachour, B. Roynette, and P. Vallois, “Explicit solutions of some fourth order partial
differential equations via iterated Brownian motion,” in
*Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996)*, vol. 45 of*Progr. Probab.*, pp. 39–61, Birkhäuser, Basel, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Burdzy and A. Mądrecki, “An asymptotically $4$-stable process,”
*The Journal of Fourier Analysis and Applications*, pp. Special issue, 97–117, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Debbi, “On some properties of a higher fractional differential operator which
is not in general selfadjoint,”
*preprint,*, 2005. View at Google Scholar - L. Debbi and L. Abbaoui, “Explicit solution of some fractional heat equations via Lévy motion,” to appear in Maghreb Mathematical Review.
- L. Debbi and M. Dozzi, “On the solution of non linear stochastic fractional partial
differential equations,”
*Stochastic Processes and Their Applications*, vol. 115, no. 11, pp. 1764–1781, 2005. View at Publisher · View at Google Scholar - W. Feller, “On a generalization of Marcel Riesz' potentials and the semi-groups generated by them,”
*Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]*, vol. 1952, pp. Tome Supplementaire, 72–81, 1952. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Funaki, “Probabilistic construction of the solution of some higher
order parabolic differential equation,”
*Proceedings Japan Academy Series A Mathematical Sciences*, vol. 55, no. 5, pp. 176–179, 1979. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. W. Gardiner,
*Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences*, vol. 13 of*Springer Series in Synergetics*, Springer, Berlin, 1983. View at Zentralblatt MATH · View at MathSciNet - R. Gorenflo and F. Mainardi, “Random walk models for space-fractional diffusion processes,”
*Fractional Calculus & Applied Analysis*, vol. 1, no. 2, pp. 167–191, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. I. Henry and S. L. Wearne, “Fractional reaction-diffusion,”
*Phys. A*, vol. 276, no. 3-4, pp. 448–455, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - K. J. Hochberg, “A signed measure on path space related to Wiener measure,”
*The Annals of Probability*, vol. 6, no. 3, pp. 433–458, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Complex-valued Wiener measure: an approach via random walk
in the complex plane,”
*Statistics & Probability Letters*, vol. 42, no. 1, pp. 61–67, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Komatsu, “On the martingale problem for generators of stable processes
with perturbations,”
*Osaka Journal of Mathematics*, vol. 21, no. 1, pp. 113–132, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. J. Krylov, “Some properties of the distribution corresponding to the equation $\partial u/\partial t={(-1)}^{q+1}{\partial}^{2q}u/\partial {x}^{2q}$,”
*Soviet Mathematics Doklady*, vol. 132, pp. 1254–1257, 1960 (Russian), translated as Soviet Mathematics Doklady \textbf{1} (1960), 760–763. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Le Méhauté, J. A. T. Machado, J. C. Trigeassou, and J. Sabatier, “Fractional differentiation and its applications,” in
*Proceedings of the 1st IFAC Workshop on Fractional Differentiation and Its Applications (FDA '04)*, vol. 2004-1, pp. 353–358, ENSEIRB, Bordeaux, 2004. - X. Leoncini and G. M. Zaslavsky, “Jets, stickiness, and anomalous transport,”
*Physical Review E. Statistical, Nonlinear, and Soft Matter Physics.*, vol. 65, no. 4, pp. 046216-1–046216-16, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - E. Lukacs,
*Characteristic Functions*, Griffin's Statistical Monographs & Courses, no. 5, Hafner, New York, 1960, 2nd edition 1970. View at Zentralblatt MATH · View at MathSciNet - F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional
diffusion equation,”
*Fractional Calculus & Applied Analysis*, vol. 4, no. 2, pp. 153–192, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, 1993. View at Zentralblatt MATH · View at MathSciNet - T. Nakajima and S. Sato, “On the joint distribution of the first hitting time and the
first hitting place to the space-time wedge domain of a
biharmonic pseudo process,”
*Tokyo Journal of Mathematics.*, vol. 22, no. 2, pp. 399–413, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Nikitin and E. Orsingher, “On sojourn distributions of processes related to some
higher-order heat-type equations,”
*Journal of Theoretical Probability*, vol. 13, no. 4, pp. 997–1012, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Nishimoto,
*Fractional Calculus*, Descartes Press, Koriyama, 1994. - K. Nishioka, “Stochastic calculus for a class of evolution equations,”
*Japanese Journal of Mathematics.*, vol. 11, no. 1, pp. 59–102, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Nishioka, “A stochastic solution of a high order parabolic equation,”
*Journal of the Mathematical Society of Japan.*, vol. 39, no. 2, pp. 209–231, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Nishioka, “Boundary conditions for one-dimensional biharmonic pseudo
process,”
*Electronic Journal of Probability.*, vol. 6, no. 13, pp. 1–27, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives. Theory and Applications*, Gordon and Breach Science, Yverdon, 1993. View at Zentralblatt MATH · View at MathSciNet - G. Samorodnitsky and M. S. Taqqu,
*Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance*, Stochastic Modeling, Chapman & Hall, New York, 1994. View at Zentralblatt MATH · View at MathSciNet - K.-I. Sato,
*Lévy Processes and Infinitely Divisible Distributions*, vol. 68 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, 1999. View at Zentralblatt MATH · View at MathSciNet - L. Sirovich,
*Techniques of Asymptotic Analysis*, vol. 2 of*Applied Mathematical Sciences*, Springer, New York, 1971. View at Zentralblatt MATH · View at MathSciNet - V. V. Uchaikin and V. M. Zolotarev,
*Chance and Stability. Stable Distributions and Their Applications*, Modern Probability and Statistics, VSP, Utrecht, 1999. View at Zentralblatt MATH · View at MathSciNet