Math 337, Summer 2010 Assignment 5

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1 Math 337, Summer Assignment 5 Dr. T Hillen, University of Alberta Exercise.. Consider Laplace s equation r r r u + u r r θ = in a semi-circular disk of radius a centered at the origin with boundary conditions u(r, ) =, <r a, u(r, ) =, <r a, u(a, θ) =sinθ, θ, u(r, θ) < as r +. Solve this problem using separation of variables. Solution to Exercise.: Assuming a solution of the form u(r, θ) =R(r) ϕ(θ), and separating variables, we have the following problems for R and ϕ : r(rr ) + λr =, <r a, ϕ + λϕ =, θ, R() <, ϕ() =, ϕ() =. We solve the θ-problem first. The eigenvalues and corresponding eigenfunctions are λ n = n and ϕ n (x) =sinnθ for n. The corresponding r-equation is the Cauchy-Euler equation r R + rr n R =, with general solution R(r) =A n r n + B n r n

2 for n. The boundedness condition R() < requires that B n =, so that for n. Using the superposition principle, we write u(r, θ) = R n (r) =A n r n A n r n sin nθ for <r a, θ, and from the boundary condition we want n= sin θ = u(a, θ) = A n a n sin nθ. From the orthogonality of the eigenfunctions on the interval [, ], we have n= aa = and A n =,n, so the solution is u(r, θ) = r a sin θ for r a, θ. Exercise.. Assume that f(x) isabsolutelyintegrableanda is a given real constant. Show that F e iax f(x) (ω) = f(ω a). Solution to Exercise.: Since e iax =andf is absolutely integrable on (, ), then e iax f(x) isalsoabsolutelyintegrableon(, ) andwehave for all ω R. F e iax f(x) (ω) = e iax f(x)e iωx dx = f(x)e i(ω a)x dx = f(ω a)

3 3 Exercise.3. Assume that f (t) isabsolutelyintegrableand lim f(t) = and lim f (t) =. t t Show that F s (f )(ω) = ω F s (f)(ω)+ ωf(). Solution of.3: Assuming that lim f(t) =, and lim f (t) =, and integrating by parts t t we have F s (f )(ω) = f (t)sinωt dt = f (t)sinωt ω f (t)cosωt dt = ω f (t)cosωt dt = ωf(t)cosωt + ω f(t)sinωt dt = ωf() ω F s (f)(ω). We used the fact that f (t)sinωt f (t) as t, and f(t)cosωt f(t) as t.

4 4 Exercise.4. Let f(x) = cos x x <, x >. (a) Find the Fourier integral of f. (b) For which values of x does the integral converge to f(x)? (c) Evaluate the integral for <x<. λ sin λ cos λx λ dλ Solution to.4: (a) The Fourier integral representation of f is given by f(x) [A(λ)cosλx + B(λ)sinλx] dλ, <x< where A(λ) = f(t)cosλt dt and B(λ) = for λ. f(t)sinλt dt Since f(t) isanevenfunctionontheinterval <t<, then B(λ) =forall λ, and A(λ) = f(t)cosλt dt = for all λ. f(t)cosλt dt

5 5 Now, for λ =, we have that is, for λ, λ=. Now, for λ =, we have A() = Therefore A(λ) = = = = cos t cos λt dt [cos( + λ)t +cos( λ)t] dt sin( + λ)t ( + λ) sin( + λ) ( + λ) sin λ = ( + λ) = cos tdt= λ sin λ ( λ ), A(λ) = + + sin( λ)t ( λ) sin( λ) ( λ) + sin λ ( λ) λ sin λ ( λ ) [ + cos t] dt = t + sin t = =. λ sin λ for λ, λ= A(λ) = ( λ ) for λ =. (b) Since f(x) iscontinuousforallx = ±, then from Dirichlet s theorem, the Fourier integral representation converges to f(x) forallsuchx, that is, λ sin λ cos x for x < f(x) = A(λ)cosλx dλ = cos λx dλ = ( λ ) for x >. for all x = ±. When x = ±, from Dirichlet s theorem the Fourier integral representation converges to f( + )+f( ) = = and f( + )+f( ) = + =.

6 6 (c) From part (b) above, we have λ sin λ cos λx λ dλ = cos x for x < 4 for x > for x =. Exercise.5. Besides linear equations, some nonlinear equations can also result in traveling wave solutions of the form u(x, t) =φ(x ct). Fisher s equation, which models the spread of an advantageous gene in a population, where u(x, t) isthedensityofthegeneinthepopulationattimetand location x, is given by u t = u + u( u). x Show that Fisher s equation has a solution of this form if φ satisfies the nonlinear ordinary differential equation φ + cφ + φ( φ) =. Solution to Exercise.5: If u(x, t) =φ(x ct), then and Fisher s equation becomes u x = φ (x ct) u x = φ (x ct) u t = cφ (x ct), cφ (x ct) =φ (x ct)+φ(x ct)( φ(x ct)), for all x and t, so that if φ satisfies the nonlinear ordinary differential equation φ (s)+cφ (s)+φ(s)( φ(s)) =, <s<, then u(x, t) =φ(x ct) is a traveling wave solution to Fisher s equation.