Partial Differential Equations PowerTool

by Dr. Jim Herod

Section 4.5: Structure for Solutions of the Heat Equation

We review structure for solutions of the heat equation:

The Partial Differential Equation:            =  .

The Boundary Conditions:       u(t, 0) =   and u(t, a ) = .

The Initial Conditions: u (0, x ) = f ( x ), where f ( x ) is continuous on [0, a ] and f (0) =  , f ( a ) = .

Existence of solutions:   There is one and only one solution for the above problem and it exists for t  in [0,  ) and for x  in [0, a]. Moreover, the solution u  is infinitely differentiable for   t  in [0,  ) and for x  in [0, a ].

Note that this implies that the solutions have an intolerance for sharp corners. The initial conditions may have some points where the derivative fails to exist, but the solutions become infinitely differentiable instantaneously. We illustrate with an initial value having a sharp corner and examine the solution near t  = 0. We expect to see that the sharp corner is immediately smoothed.

>

restart:

Here is the definition of the function with a sharp corner.

>	f:=x->Heaviside(1/2-x)*x^2+Heaviside(x-1/2)*(x-1)^2;

>	plot(f(x),x=0..1);

We make the Fourier coefficients.

>	for n from 1 to 40 do    c[n]:=int(f(x)*sin(n*Pi*x),x=0..1)/int(sin(n*Pi*x)^2,x=0..1): od: n:='n';

With the coefficients, we make a solution for the heat equation.

>	u:=(t,x)->sum(c[n]*exp(-n^2*Pi^2*t)*sin(n*Pi*x),n=1..40);

Check to see that the graph is smooth for any t > 0.

>	plot3d(u(t,x),x=0..1,t=0..1/10,axes=NORMAL,orientation=[-60,75]);

It is interesting to view that the smoothing happens immediately.

>	plot([f(x),u(1/10000,x)],x=0..1,color=[red,blue,green]);

This example was created to have another property. The initial value is concave up at any x  except 1/2. It should be that if you pick any x  in the interval [0, 1] except 1/2, then there is a positive number t  for which u ( t , x ) > f ( x ). This is true because the second derivative with respect to x  is positive, so u  must be increasing at that x .

>

Asymptotic Properties for Solutions:  As t  increases, u ( t , x ) approaches a solution which is independent of t . For the above equation, this time independent solution would be

               (  ) ( x  - a ) / a  +  .

That is, no matter what initial value, f ( x ), we begin with, all solutions for the above equation would have limit this steady state solution. We compute the solution for the above equation with

                        f ( x ) = ( 1+sin(   x  / 2) ) ( x  + 1)

and with    = 1,  = 2.  We compute the solution and check the results.

>

restart:

>	f:=x->(sin(Pi*x)+1)*(x+1); g:=x->f(x)-x-1;

We make the Fourier coefficients.

>	for n from 1 to 20 do    c[n]:=int(g(x)*sin(n*Pi*x),x=0..1)/int(sin(n*Pi*x)^2,x=0..1): od: n:='n':

We make the solution.

>	u:=(t,x)->1+x+sum(c[n]*exp(-n^2*Pi^2*t)*sin(n*Pi*x),n=1..20);

We check that this really is a solution, checking that it satisfies the PDE and boundary condition.

>	diff(u(t,x),t)-diff(u(t,x),x,x); u(t,0); u(t,1);

We check to see that we are close to the initial condition.

>	plot([u(1/1000,x),f(x)],x=0..1,color=[red,black]);

Finally, we draw the graph of f . What do we expect? We expect that the solution begins looking like the initial value and ends looking like the graph of x  + 1.

>	plot3d(u(t,x),x=0..1,t=0..1/2,axes=NORMAL,orientation=[-125,65]);

>

The next subject is the maximal principle. We ask where can the maximum value of solutions lie. One might try the following experiment. Take an initial distribution of heat with two or more peaks and see if, as the heat diffuses, the heat from one peak might combine with the heat from another peak to produce a temperature greater than that for either of the two initial peaks.

Too bad. That will not work.

Maximal Principle for the Heat Equation:  The solution for the above heat equation for t  in [0, T] and x  in [0, a] attains it maximum and minimum on one of four sides of the boundary of the region in the plane described as all points given by

      t  = 0 and x  in [0, a ], or

      t  in [0, T] and x  = 0, or

      t  in [0, T] and x  = a , or

      t  = T and x  in [0, a ].

That is, the temperature in the interval [0, a ] can never exceed the larger of the maximum of the initial temperature or the maximum temperature at the ends of the interval [0, a ] as time increases. In a similar manner, the temperature in the interval [0, a ] is not smaller than the minimum of the initial temperature or the minimum temperature at the ends of the interval [0, a ] as time increases.

Here's an example to try out for t  in the interval [0, 10] and x  in the interval [0, /2]. We ask what is the maximum value of u ( t , x ) and where does it occur?

>	u:=(t,x)->exp(-t)*sin(x)+x+1;

Check that this u  satisfies the heat equation with some boundary conditions.

>	diff(u(t,x),t)-diff(u(t,x),x,x); u(t,0); u(t,Pi/2); u(0,x);

Now, ask what is the maximum on the four boundaries.

>	maximize(u(0,x),x=0..evalf(Pi/2)); maximize(u(t,0),t=0..10); maximize(u(t,Pi/2),t=0..10); maximize(u(t,x),t=0..10,x=0..evalf(Pi/2));

See if it is clear where the maximum occurs.

>	plot3d(u(t,x),x=0..Pi/2,t=0..10,axes=NORMAL,orientation=[-125,60]);

>	evalf(u(0,Pi/2));

>

A consequence of the Maximum Principle for the Heat Equation is that order is preserved. Consider this: Suppose that u satisfies the differential equation above and that v satisfies the equation

The Partial Differential Equation:            =  .

The Boundary Conditions:        v ( t , 0) =   and v ( t , a ) = .

The Initial Conditions: v (0, x ) = g ( x ), where g( x ) is continuous on [0, a ] and g (0) =  , g(a) =

Suppose also that . We argue that for all t  and x  in the appropriate intervals,  . The suggestion for doing this problem is to consider the function w ( t , x ) = u ( t , x ) - v ( t , x ) and ask what equation w  satisfies. You will see that w  satisfies the equation.

The Partial Differential Equation:            =  .

The Boundary Conditions:        w ( t , 0) =   and w ( t , 1) = .

The Initial Conditions: w (0, x ) = f ( x ) - g ( x ).

We can conclude that  because the maximum must occur on the boundary and that is zero.

In this section, we have discussed the theoretical structure for solutions of the heat equation. We will see that not everything that happens here is true for the wave equation. There, sharp edges can propagate in time. It will not longer be true that solutions must be differentiable. This is an important change which we will illustrate.

EMAIL: herod@math.gatech.edu   or   jherod@tds.net

URL: http://www.math.gatech.edu/~herod

Copyright ©  2003  by James V. Herod

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