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# spacetime2

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# spacetime2   { SPACETIME2.PDE

This example is a modification of SPACETIME1.PDE, showing the solution of

one-dimensional transient heatflow with differing material properties,

cast as a boundary-value problem.

The time variable is represented by Y, and the temperature by u(x,y).

We specify two regions of differing conductivity, KX.

The initial Temperature is given as a truncated parabola along y=0.

We specify reflective boundary conditions in X (natural(u)=0) along

the sides x=0 and x=1.

The value of u is thus assigned everywhere on the boundary except

along the segment y=1, 0<x<1.  Along that boundary, we use the

natural boundary condition,

natural(u) = 0,

since this corresponds to the application of no boundary sources.

}

title "1-D Transient Heatflow as a Boundary-Value Problem"

Variables

u              { define U as the system variable }

definitions

kx              { declare KX as a parameter, but leave the value for later }

Initial values

u = 0          { unimportant, since this problem is masquerading

as a linear boundary-value problem }

equations           { define the heatflow equation }

U: dy(u)  =  dx(kx*dx(u))

boundaries

region 1

kx = 0.1                { conductivity = 0.1 in region 1 }

start(0,0)

value(u)=2.025-10*x^2   { define the temperature at t=0, x<=0.45 }

line to (0.45,0)

value(u) = 0           { force zero temperature for t=0, x>0.45 }

line to (1,0) to (1,1)

natural(u) = 0          { no flux across x=1 boundary }

line to (1,1)

natural(u) =  0         { no sources on t=1 boundary }

line to (0,1)

natural(u) = 0        { no flux across x=0 boundary }

line to close

region 2

kx = 0.01                { low conductivity in region 2 }

start(0.45,0)           { lay region 2 over center strip of region 1 }

line to (0.55,0)

to (0.55,1)

to (0.45,1)

to close

monitors

contour(u)

plots

contour(u)

surface(u)

end