Cable Convergence. James K. Peterson. May 7, Department of Biological Sciences and Department of Mathematical Sciences Clemson University

Transcription

1 Cable Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 2018

2 Outline 1 Fourier Series Convergence Redux 2

3 Fourier Series Convergence Redux Let s review: (1): Assume f has continuous derivatives f and f on [0, L]. The Fourier Sine and Cosine series for f then converge uniformly on [0, L] to continuous functions S and C. We also know the Fourier Sine and Fourier Cosine series converge to f on [0, L]. Further, we know the Derived Series of the Fourier Sine and Fourier Cosine series converge uniformly on [0, L] Let s check the conditions of the derivative interchange theorem applied to the partial sums (T s n ) and (T c n ) for the Fourier Sine and Cosine Series. 1 T s n and T c n are differentiable on [0, L]: True. 2 (T s n ) and (T c n ) are Riemann Integrable on [0, L]: True as each is a polynomial of sines and cosines. 3 There is at least one point t 0 [0, L] such that the sequence (T s n (t 0 )) and (T c n (t 0 ) converges. True as these series converge on [0, L]. 4 (Tn s ) unif W s and (Tn c ) unif W c on [0, L] where W s and W c are continuous. True because of our coefficient theorem.

4 Fourier Series Convergence Redux The conditions of the Derivative Interchange Theorem are thus satisfied and we can say there are functions U s and U c on [0, L] so that Tn s unif U s on [0, L] with (U s ) = W s and Tn c unif U c on [0, L] with (U c ) = W c. Since limits are unique, we then have U s = f with f = W s. and U c = f with f = W c, This is the statement that we can take the derivative of the Fourier Sine and Fourier Cosine Series termwise. That is f (t) = f (t) = ( A n sin( nπ ) L x) = ( B n cos( nπ ) L x) = nπ L A n cos( nπ L x) nπ L B n sin( nπ L x) (2): Now assume f also has a continuous third derivative on [0, L]. The arguments we just presented can be used with some relatively obvious modifications to show similar results.

5 Fourier Series Convergence Redux f (t) = f (t) = ( nπ L A n cos( nπ ) L x) = ( nπ L A n sin( nπ ) L x) = n2 π 2 L 2 A n sin( nπ L x) n2 π 2 L 2 A n cos( nπ L x) (3): If we assume the continuity of higher order derivatives, we can argue in a similar fashion to show that term by term higher order differentiation is possible.

6 The formal series solution is Φ(x, t) = B 0 e 1 α t + ( ) nπ B n cos L x e L2 +n 2 π 2 β 2 αl 2 t. We can show how we establish that Φ is indeed the solution to the cable model by setting all the parameters here to the value of 1 to make the analysis more transparent. Hence, we set L = 1, α = 1 and π β = 1. The solution is then Φ(x, t) = B 0 e t + B n cos(nπx) e (1+n2 )t. Now, we can estimate the B n coefficients as B 0 1 L B n 2 L L 0 L 0 L f (x) dx 1 f (x) dx, L 0 ( ) f (x) nπ cos L x dx 2 L L 0 f (x) dx.

7 Letting C = 2 L L 0 f (x) dx, we see B n C for all n 0. Thus, for any fixed t > 0, we have the estimate B n cos(nπx) e (1+n2 )t e t C e n2t. and by the Wierstrass Theorem for Uniform Convergence of Series, we see the series on the left hand side converges uniformly to Φ(x, t) on the interval [0, 1] as long as t > 0. Since each of the individual functions in the series on the left hand side is continuous and convergence is uniform, we know the limit function for the left hand side series is continuous on [0, 1] for each t > 0. Hence, the function Φ(x, t) is continuous on the domain [0, 1] [t, ) for any positive t. At t = 0, the series for Φ(x, t) becomes Φ(x, 0) = B 0 + B n cos(nπx). which is the Fourier cosine series for f on [0, 1].

8 Now from our previous work with the Fourier cosine series, we know this series converges uniformly to f (x) as long as f is continuous on [0, 1] with a derivative. So if we assume f is continuous on [0, 1], we know Φ(x, 0) converges uniformly to a continuous function of x which matchs the data f. Indeed, we see ( lim Φ(x, t) = lim B t e t + t 0 + = f (x) = Φ(x, 0). ) B n cos(nπx) e (1+n2 )t Hence, we know Φ(x, t) is continuous on [0, 1] [0, ). Now let s look at the partial derivatives. If we take the partial derivative with respect to t of the series for Φ(x, t) term by term, we find the function D t (x, t) given by D t (x, t) = B 0 e t B n (1 + n 2 ) cos(nπx) e (1+n2 )t

9 The series portion for D t (x, t) satisfies the estimate B n (1 + n 2 ) cos(nπx) e (1+n2 )t C(1 + n 2 ) e n2t. For t > 0, applying the Weierstrass Uniform Convergence Test, the series for D t (x, t) converges uniformly. Since it is built from continuous functions, we know limit function is continuous for t > 0 because the convergence is uniform. Finally, applying the Derivative Interchange Theorem, we see the series for Φ t is the same as the series D t (x, t) we found by differentiating term by term. At t = 0, we have the series D t (x, 0) = B 0 B n (1 + n 2 ) cos(nπx)

10 From our earlier discussions, if we know f is continuous, the series n2 B n cos(nπx) converges uniformly and differentiation term by term is permitted. Also, the series B n cos(nπx) converges uniformly since we know f is continous. We conclude Φ t = t (B 0 e t + ) B n cos(nπx) e (1+n2 )t The formal partial derivatives of the cable series solution with respect to x are then D x (x, t) = D xx (x, t) = B n nπ sin(nπx) e (1+n2 )t B n n 2 π 2 cos(nπx) e (1+n2 )t Now we can estimate the B n coefficients as usual with the constant C as before. Hence, B n C for all n 0.

11 The series for the two partial derivatives with respect to x satisfies and B n nπ sin(nπx) e (1+n2 )t B n n 2 π 2 sin(nπx) e (1+n2 )t e t Cnπ e n2 t e t Cn 2 π 2 e n2 t Again, these estimates allow us to use the Weierstrass Theorem for Uniform Convergence to conclude the series for D x (x, t) and D xx (x, t) converge uniformly. Since these series are built from continuous functions, we then know the limit function is continuous for t > 0 since the convergence is uniform. Thus, the series for Φ x is the same as the series D x (x, t) we find by differentiating term by term. Further, the series for 2 Φ x is the same as the series D 2 xx (x, t) we also find by differentiating term by term.

12 At t = 0, we have D x (x, 0) = D xx (x, 0) = B n nπ sin(nπx) B n n 2 π 2 cos(nπx) which converge uniformly if f is continuous and the Derivative Interchange Theorem tells us that differentiation is justified term by term.

13 Hence, on [0, 1] (0, ) we can compute 2 Φ x 2 Φ Φ t = B n n 2 π 2 cos(nπx) e (1+n2 )t B 0 e t B n cos(nπx) e (1+n2 )t +B 0 e t + B n (1 + n 2 π 2 ) cos(nπx) e (1+n2 )t = 0. So that we see our series solution satisfies the partial differential equation. To check the boundary conditions, because of continuity, we can see ( B n nπ sin(nπx) e )t) (1+n2 = 0 Φ (0, t) = x Φ (1, t) = x ( B n nπ sin(nπx) e )t) (1+n2 x=0 x=1 = 0.

14 and finally, the data boundary condition gives Φ(x, 0) = f (x). Note, we can show that higher order partial derivatives of Φ are also continuous even though we have only limited smoothness in the data function f. Hence, the solutions of the cable equation smooth irregularities in the data. However, we cannot see this from our analysis because we must assume continuity in the third derivative of the data to insure this. But there are other techniques we can use to show these solutions are very smooth which we cover in more advanced classes. The analysis for the actual cable model solution is handled similarly, of course. The arguments we presented in this chapter are quite similar to the ones we would use to analyze the smoothness qualities of the solutions to other linear partial differential equations with boundary conditions.

15 We didn t want to go through all of them in this text as we don t think there is a lot of value in repeating these arguments over and over. Just remember that to get solutions with this kind of smoothness requires that the series we formally compute converge uniformly. If we did not have that information, our series solutions would not have this amount of smoothness. But that is another story! Also, if the data f was a C bump function on [0, L] centered at L/2, we would have continuity in all orders of the data f and hence all order partials of the solution Φ would be smooth.

16 Homework 37 Let π n = {t i } n i=0 be the uniform partition of the interval [0, L] with π n = L/n. Let φ i be the C bump function defined on [t i, t i+1 ] centered at the midpoint (t i + t i+1 )/2 of height 1. Let f be any continuous function on [0, L] Prove the functions φ i form an orthornormal set Prove < f, φ i > (L/n) f av,i, where f av,i is the average value of f on the subinterval [t i, t i+1 ] Show that there is some constant A > 0 so that φ i 2 = A for all i Find the best approximation y of f to the subspace of C([0, L]) spanned by {φ i } n i=1 and prove y bounded above by L f av /A 2 where f av is the average value of f on [0, L].