Parabolic Equations

Murray H. Protter and Hans F. Weinberger
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Murray H. Protter: University of California, Department of Mathematics
Hans F. Weinberger: University of Minnesota, Institute for Mathematics and its Applications

Chapter Chapter 3 in Maximum Principles in Differential Equations, 1984, pp 159-194 from Springer

Abstract: Abstract Suppose a long, thin rod of length l is situated on the interval (0, l) along the x-axis. We shall assume that the material of the rod is homogeneous. Heat may be put into or removed from the rod, and we assume that the temperature u at any point in the rod is a function only of x, the location of a particular cross section, and of t, the time. We write u = u(x, t). Under certain assumptions on the physical properties of the rod, the differential equation governing the flow of heat (in appropriate units) in the rod is given by $$ \frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}} = f(x,t). $$ . The function f is the rate of heat removal in the bar. The temperature function u(x, t) satisfies a maximum principle somewhat different from the one which was established for elliptic equations and inequalities.

Keywords: Maximum Principle; Parabolic Equation; Interior Point; Heat Equation; Uniqueness Theorem (search for similar items in EconPapers)
Date: 1984
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DOI: 10.1007/978-1-4612-5282-5_3

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