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34-pde

# 34-pde - HEAT AND WAVE EQUATION Math 21b Fall 2004 SOLVING...

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HEAT AND WAVE EQUATION Math 21b, Fall 2004 FUNCTIONS OF TWO VARIABLES. We consider functions f ( x, t ) which are for fixed t a piecewise smooth function in x . Analogously as we studied the motion of a vector ~v ( t ), we are now interested in the motion of a function f ( x, t ). While the governing equation for a vector was an ordinary differential equation (ODE), the describing equation will now be a partial differential equation (PDE). The function f ( x, t ) could denote the temperature of a stick at a position x at time t or the displacement of a string at the position x at time t . The motion of these dynamical systems can be understood using orthonormal Fourier basis 1 / 2 , sin( nx ) , cos( nx ) treated in an earlier lecture. The homework to this lecture is at the end of this 2 page handout. PARTIAL DERIVATIVES. We write f x ( x, t ) and f t ( x, t ) for the partial derivatives with respect to x or t . The notation f xx ( x, t ) means that we differentiate twice with respect to x . Example: for f ( x, t ) = cos( x + 4 t 2 ), we have f x ( x, t ) = - sin( x + 4 t 2 ) f t ( x, t ) = - 8 t sin( x + 4 t 2 ). f xx ( x, t ) = - cos( x + 4 t 2 ). One also uses the notation ∂f ( x,y ) ∂x for the partial derivative with respect to x . Tired of all the ”partial derivative signs”, we always write f x ( x, y ) or f t ( x, y ) in this handout. This is an official abbreviation in the scientific literature. PARTIAL DIFFERENTIAL EQUATIONS. A partial differential equation is an equation for an unknown function f ( x, t ) in which different partial derivatives occur. f t ( x, t ) + f x ( x, t ) = 0 with f ( x, 0) = sin( x ) has a solution f ( x, t ) = sin( x - t ). f tt ( x, t ) - f xx ( x, t ) = 0 has a solution f ( x, t ) = sin( x - t ) + sin( x + t ). THE HEAT EQUATION. The temperature distribution f ( x, t ) in a metal bar [0 , π ] satisfies the heat equation f t ( x, t ) = μf xx ( x, t ) This partial differential equation tells that the rate of change of the temperature at x is proportional to the second space derivative of f ( x, t ) at x . The function f ( x, t ) is assumed to be zero at both ends of the bar and f ( x ) = f ( x, t ) is a given initial temperature distribution. The constant μ depends on the heat conductivity properties of the material. Metals for example conduct heat well and would lead to a large

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34-pde - HEAT AND WAVE EQUATION Math 21b Fall 2004 SOLVING...

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