Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 10 - The Heat and Wave Equ

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 10 - The Heat and Wave Equ

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CHAPTER 10 The Heat and Wave Equations on an Unbounded Domain A t frst glance, the heat equation u t - κ u xx = 0 and the wave equation u tt - c 2 u = 0 appear very similar. Since κ and c 2 are always assumed to be positive constants, the only apparent distinction is between the u t in the heat equation and the u in the wave equation. As we shall see, this makes a profound diFFerence in the behavior oF the solutions oF these two equations. We begin this Chapter with a derivation oF these two pde s From basic physical principles. Then, we will solve both pde s on the unbounded one-dimensional spatial domain - < x < . 10.1. Derivation of the Heat and Wave Equations The heat equation arises in the context oF modeling diFFusive processes. ±or example, suppose heat is distributed within a long, thin wire that can be treated as one-dimensional. ±urther suppose that the wire is insulated so that heat is only transFerred within the wire (as opposed to radially outward). IF T ( x , t ) denotes the temperature at position x and time t , then T satisfes the heat equation. The Heat Equation. ±or the purposes oF deriving the heat equation, we actually have in mind another phenomenon that is modeled by the same pde . Namely, suppose that a dye (such as Food coloring) diFFuses in a motionless liquid that is confned to a long, thin “one-dimensional” pipe. Letting u ( x , t ) denote the concentration oF dye at position x and time t , our goal is to derive an equation that captures the dynamics oF u . Generally, dye diFFuses From regions oF higher concentration to regions oF lower concentration, and the relevant physical law that governs diFFusion is 250
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the heat and wave equations on an unbounded domain 251 Fick’s Law of Diffusion. The rate of motion of dye is proportional to the con- centration gradient. Fick’s Law will let us track how the total mass of dye in a given region of pipe, say [ x 0 , x 1 ] , changes over time. Mass is obtained by integrating concentration; i.e., the mass of dye in the region [ x 0 , x 1 ] at time t is given by M ( t )= # x 1 x 0 u ( x , t ) d x . Differentiating with respect to t yields d M d t = # x 1 x 0 u t ( x , t ) d x . Since d M d t measures the rate of change of mass, it is equal to the difference between the rate of ±ow of dye into the interval [ x 0 , x 1 ] and the rate of ±ow of dye out of the interval [ x 0 , x 1 ] . By Fick’s Law, the rate of ±ow of the dye is proportional to the (spatial) concentration gradient u x . Letting κ > 0 denote the proportionality constant, the rate of inward ±ow at the left endpoint x 0 is given by - κ u x ( x 0 , t ) . To explain why the negative sign is included, suppose u x ( x 0 , t ) > 0. Then the concentration is higher for x > x 0 than for x < x 0 , which implies that the ±ow at x 0 would be from right-to-left (see Figure 10 . 1 ). By contrast, if u x ( x 0 , t ) < 0, then the concentration is higher for x < x 0 than for x > x 0 , thereby resulting in a ±ow from left-to-right at x 0 . Similar reasoning shows that the rate of inward ±ow at the right endpoint x 1 is given by κ u x ( x 1 , t ) .
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Ordinary &amp; Partial Differential Equations - Reynolds (2000) - Chapter 10 - The Heat and Wave Equ

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