### The Thermostat Problem

#### Abstract

#### Full Text:

PDF (Português (Brasil))#### References

[1] H. Amann, “Linear and Quasilinear Parabolic Problems. Volume I: Abstract Linear Theory”, Birkh¨auser Verlag, 1995.

M. Brokate and A. Friedman, Optimal design for heat conduction problems with hysteresis, SIAM J. Control and Optimization, 27 (1989), 697-717.

J.R. Cannon, “The One-Dimensional Heat Equation”, Addison-Wesley Pub. Co., 1984.

A.K. Drangeid, The principle of linearized stability for quasilinear parabolic equations, Nonlin. Anal. TMA, 13 (1989), 1091-1113.

A. Friedman and L. Jiang, Periodic solutions for a thermostat control problem, Comm. in Partial Differential Equations, 13 (1988), 515-550.

J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields”, 3rd ed., Springer-Verlag, 1997.

P. Guidotti and S. Merino, Hopf bifurcation in a scalar reaction diffusion equation, J. Diff. Eqns., 140, (1997), 209-222.

P. Guidotti and S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation, Differential and Integral Equations, 13 (1997).

B. Jumarhon, “The One Dimensional Heat Equation and its Associated Volterra Integral Equations”, PhD Thesis, University of Strathclyde, 1994.

B. Jumarhon and S. McKee, On the heat equation with nonlinear and nonlocal boundary conditions, J. Math. Anal. Appl., 190 (1995), 806-820.

Y. Lin, “Parabolic Partial Differential Equations Subject to Non-local Boundary Conditions”, PhD Thesis, Washington State University, Pullman, WA, 1988.

Y. Lin, A nonlocal parabolic system in linear thermoelasticity, Dynamics of Continuous, Discrete and Impulsive System, 2 (1996), 267-283.

W.E. Schiesser, “The Numerical Method of Lines: Integration of Partial Differential Equations”, Academic Press, 1991.

J.L. Schiff, “The Laplace Transform: Theory and Applications”, Springer- Verlag, 1999.

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Diff. Integral Eqns., 8, (1995).

G. Simonett, Invariant manifolds and bifurcation for quasilinear reactiondiffusion systems, Nonlin. Anal. TMA, 23, (1997), 515-544.

G.D. Smith, “Numerical Solution of Partial Differential Equations: Finite Difference Methods”, Clarendon Press, Oxford, 1985.

A. Zafarullah, Application of the method of lines to parabolic partial differential equations with error estimates, J. ACM, 17, (1970), 294-302.

DOI: https://doi.org/10.5540/tema.2002.03.01.0015

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

**Trends in Computational and Applied Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in: