# American Institute of Mathematical Sciences

April  2011, 4(2): 239-246. doi: 10.3934/dcdss.2011.4.239

## The position of the joint of shape memory alloy and bias springs

 1 Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193

Received  January 2009 Revised  June 2009 Published  November 2010

In our previous work we proposed the mathematical model for a device made of the standard spring and the shape memory alloy spring. The model was given by the system of partial differential equations with the dynamic boundary condition. Also, we have proved the existence and the uniquess theorems for the model. The purpose of this paper is to improve the existence theorem.
Citation: Toyohiko Aiki. The position of the joint of shape memory alloy and bias springs. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 239-246. doi: 10.3934/dcdss.2011.4.239
##### References:
 [1] T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Applicable Analysis, Applicable Analysis, 56 (1995), 71-94. doi: 10.1080/00036819508840311. [2] T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions, in "Nonlinear Analysis and Applications," Gakuto International Series, Mathematical Sciences and Applications, Vol. 7, (1996), 1-25. [3] T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Nonlinear Phenomena with Energy Dissipation, Mathematical Analysis, Modeling and Simulation," Gakuto International Series, Mathematical Sciences and Applications, Vol. 29, (2008), 1-18. [4] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795. doi: 10.1006/jmaa.1996.0053. [5] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996. [6] R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tôhoku Math. J. Ser. 1, 35 (1932), 260-275. [7] J. L. Lions, "Quelques Methods de Resolutions des Problems aux Limites Non Lineares," Dunod, Paris, 1969. [8] N. Sato and T. Aiki, Phase field equations with constraints under nonlinear dynamic boundary conditions, Commun. Appl. Anal., 5 (2001), 215-234.

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##### References:
 [1] T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Applicable Analysis, Applicable Analysis, 56 (1995), 71-94. doi: 10.1080/00036819508840311. [2] T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions, in "Nonlinear Analysis and Applications," Gakuto International Series, Mathematical Sciences and Applications, Vol. 7, (1996), 1-25. [3] T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Nonlinear Phenomena with Energy Dissipation, Mathematical Analysis, Modeling and Simulation," Gakuto International Series, Mathematical Sciences and Applications, Vol. 29, (2008), 1-18. [4] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795. doi: 10.1006/jmaa.1996.0053. [5] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996. [6] R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tôhoku Math. J. Ser. 1, 35 (1932), 260-275. [7] J. L. Lions, "Quelques Methods de Resolutions des Problems aux Limites Non Lineares," Dunod, Paris, 1969. [8] N. Sato and T. Aiki, Phase field equations with constraints under nonlinear dynamic boundary conditions, Commun. Appl. Anal., 5 (2001), 215-234.
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