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Unformatted text preview: Lecture 33 ODE Boundary Value Problems and Finite Differences Steady State Heat and Diffusion If we consider the movement of heat in a long thin object (like a metal bar), it is known that the temperature, u ( x, t ), at a location x and time t satisfies the partial differential equation u t- u xx = g ( x, t ) , (33.1) where g ( x, t ) is the effect of any external heat source. The same equation also describes the diffusion of a chemical in a one-dimensional environment. For example the environment might be a canal, and then g ( x, t ) would represent how a chemical is introduced. Sometimes we are interested only in the steady state of the system, supposing g ( x, t ) = g ( x ) and u ( x, t ) = u ( x ). In this case u xx =- g ( x ) . This is a linear second-order ordinary differential equation. We could find its solution exactly if g ( x ) is not too complicated. If the environment or object we consider has length L , then typically one would have conditions on each end of the object, such as u (0) = 0, u ( L ) = 0. Thus instead of an initial value problem, we have a boundary value problem or BVP . Beam With Tension Consider a simply supported beam with modulus of elasticity E , moment af inertia I , a uniform load w , and end tension T (see Figure 33.1). If y ( x ) denotes the deflection at each point x in the beam, then y ( x ) satisfies the differential equation: y (1 + ( y ) 2 ) 3 / 2...
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.
- Fall '08