Some exact solutions to Stefan problems with fractional differential equations. (English) Zbl 1163.35043
Summary: Some exact solutions to the first, second and extended Stefan problems with fractional time derivative described in the Caputo sense are given by means of fractional Green’s function and Wright function in this paper. By the aid of simple calculations, many results of differential equations of integer order can be obtained as special cases of the results given by this paper.
MSC:
| 35R99 | Miscellaneous topics in partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 80A22 | Stefan problems, phase changes, etc. |
References:
| [1] | Bouchand, J. P.; Georges, A., Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195, 4-5, 127-293 (1990) |
| [2] | Srivastava, H. M.; Buschman, R. G., Theory and Applications of Convolution Integral Equations (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0755.45002 |
| [3] | Nonnenmacher, T. F.; Metzler, R., On the Riemann-Liouville fractional calculus and some recent applications, Fractals, 3, 3, 557-566 (1995) · Zbl 0868.26004 |
| [4] | Podlubny, I., Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some Their Applications (1999), Academic Press: Academic Press San Diego-New York-London · Zbl 0924.34008 |
| [5] | Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific Publishing Co. Pte. Ltd. · Zbl 0998.26002 |
| [6] | Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A, 278, 107-125 (2000) |
| [7] | Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461-580 (2002) · Zbl 0999.82053 |
| [8] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam · Zbl 1092.45003 |
| [9] | Mingyu, Xu; Wenchang, Tan, Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion, Sci. China Ser. A, 44, 11, 1387-1399 (2001) · Zbl 1138.76320 |
| [10] | Ibrahim, Rabha W.; Momani, Shaher, On the existence and uniqueness of solutions of a class of fractional differential equations, J. Math. Anal. Appl., 334, 1-10 (2007) · Zbl 1123.34302 |
| [11] | Zhang, Shuqin, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 136-148 (2003) · Zbl 1026.34008 |
| [12] | Lee, Chung-Yi; Srivastava, H. M.; Yueh, Wen-Chyuan, Explicit solutions of some linear ordinary and partial fractional differintegral equations, Appl. Math. Comput., 144, 11-15 (2003) · Zbl 1033.45006 |
| [13] | Junyi, Liu; Mingyu, Xu, An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices, Z. Angew. Math. Mech. (ZAMM), 84, 1, 22-28 (2004) · Zbl 1073.35214 |
| [14] | Li, X. C.; Xu, M. Y.; Wang, S. W., Analytical solutions to the moving boundary problems with time-space fractional derivatives in drug release devices, J. Phys. A, 40, 12131-12141 (2007) · Zbl 1134.35104 |
| [15] | Li, X. C.; Xu, M. Y.; Wang, S. W., Scale-invariant solutions to partial differential equations of fractional order with a moving boundary condition, J. Phys. A, 41, 155-202 (2008) · Zbl 1153.35088 |
| [16] | Mainardi, F., On the initial value problem for the fractional diffusion-wave equations, (Rionero, S.; Ruggeri, T., Waves and Stability in Continuous Media (1994), World Scientific: World Scientific Singapore), 246-251 |
| [17] | Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, 6, 23-28 (1996) · Zbl 0879.35036 |
| [18] | Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 9, 1461-1477 (1996) · Zbl 1080.26505 |
| [19] | Gorenflo, R.; Mainardi, F., Fractional calculus, integral and differential equations of fractional order, (Carpintevi, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics. Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, vol. 378 (1997), Springer-Verlag: Springer-Verlag Wien), 223-276 · Zbl 1438.26010 |
| [20] | Mingyu, Xu; Wenchang, Tan, Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions, Sci. China Ser. G, 46, 2, 145-157 (2004) |
| [21] | Gorenflo, R.; Mainardi, F.; Srivastava, H. M., Special functions in fractional relaxation and fractional diffusion-wave phenomena, (Bainov, D., Proc. VIII International Colloquium on Differential Equations. Proc. VIII International Colloquium on Differential Equations, Plovdiv, 1997 (1998), VSP: VSP Utrecht), 195-202 · Zbl 0921.33009 |
| [22] | Gorenflo, R.; Luchko, Yu. F.; Mainardi, F., Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math., 118, 1-2, 175-191 (2000) · Zbl 0973.35012 |
| [23] | Worster, M. G., Solidification of fluids, (Betchelor; etal., Perspectives in Fluid Dynamics, A Collective Introduction to Current Research (2000), Cambridge University Press), 393-446 · Zbl 0993.76085 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
