MATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 5

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1 MATH 4 Fourier Series PDE- Spring SOLUTIONS to HOMEWORK 5 Problem (a: Solve the following Sturm-Liouville problem { (xu + λ x u = < x < e u( = u (e = (b: Show directly that the eigenfunctions are orthogonal with respect to the suitable inner product What is the length of an eigenfunction with respect to the norm related to this inner product? Solution: (a Multiplying by x both sides of the above equation we get ( x u + xu + λu = The indicial equation of this equation is s +λ = If λ <, the s +λ = has two distinct real solutions s = λ s = λ In this case the general solution of ( is given by u(x = Ax s + Bx s Then = u( = A +B so that A = B, ce u (x = s Ax s + s Bx s, = u (e = A e s e s s e s ] If A, then s e s = s e s Since s > s < the exponential function is positive, the equality s e s = s e s is impossible So, A = Then B = we get a trivial solution If λ =, then s = has a double solutions s = s = In this case the general equation of ( is equal to u(x = A+B lnx Then = u( = A ce u (x = B/x, = u (e = B Again the problem has a trivial solution If λ >, then the equation s = λ, has two complex solutions s = i λ s = i λ In this case the general solution of ( is u(x = A( λlnx + B ( λ lnx Then = u( = A ce u (x = B λ ( λ ln x x, it follows that = u (e = B λ ( λ e If B =, we get a trivial solution for B, we get λ = From this we find that the problem has a nontrivial solution if λ = (n+ for n Hence The eigenvalues the corresponding eigenfunctions are ( ( (n + (n + lnx λ n =, u n (x =, for all n (b The eigenfunctions are orthogonal with respect to the inner product f, g r = e f(xg(x x dx (To get (r(x = /x look at the original equation Ug substitution y = lnx, we find for n m, e ( ( (n + lnx (m + lnx = = = / / (n m ((n + y ((m + y dy ((n my (n my / ] / x dx ((n + my ] / (n + my (n + m =

2 To find the length of u n calculate e ( u n (n + lnx r = x dx = So, u n r = / / = ((n + y dy = = ] / ((n + y = (n + for all n ((n + y dy / Problem (a: Consider the following Sturm-Liouville problem { (x v + λv = < x < b v( = u(b = ((n + y dy where b > Find the eigenvalues eigenfunctions of the problem (b: Show directly that the eigenfunctions are orthogonal with respect to the suitable inner product What is the length of the eigenfunctions with respect to the norm related to this inner product? Solution: (a The equation is equivalent to ( x v + xv + λv = The indicial equation has the form s(s + s + λ = s + s + λ = Let s, s be zeros of this equation If s, s are distinct real solutions, then the general solution of ( is of the form u(x = Ax s + Bx s From the boundary conditions we get = v( = A+B = Ab s +Bb s = Ab s b s ] Since s s, A = so that also B = If s = s, then v(x = Ax s + Bx s lnx Hence = v( = A = v(b = Bb s so that B = ce b > If s, s are complex zeros, then 4λ < s = α + iβ = 4λ i s = α iβ = 4λ + The general solution in this case is equal to v(x = Ax α (β lnx+bx α (β lnx Hence = v( = A = v(b = Be αb (β which implies that (β = w So, β n = n/ for n Since β n = 4λ n /, the eigenvalues the corresponding eigenfunctions are λ n = n + 4, (b Substituting y = ln x v n, v m = b for n = m, v n = v n, v n = v n(x = x / ln b, we find for n m, ( ( n lnx m lnx x b = n lnx dx = dx = ( ny dy = n lnx, n (ny(my dy =, (ny dy x ] ny = ny

3 3 Hence the length of the eigenfunction v n is equal to v n =, n Problem 3 Use the method of separation of variables to solve the telegraphic equation with the initial boundary conditions u tt + u t u xx =, < x <, t > u(, t = u(, t =, t u(x, =, x u t (x, = x, x Solution: Write u(x, t = X(xT(t Then T (t T(t + T (t T(t = X (x X(x = λ for some constant λ This gives to equations (3 T + T + λt = (4 X + λx = X( = X( = The problem (4 has the following eigenvalues corresponding eigenfunctions ( n λ n =, Xn (x =, n For given λ n, we have to solve (3 It characteristic equation is s + s + λ s = Since 4λ n = n < for all n, the characteristic equation has two complex solutions s = i n s = + i n where n = n So the solutions T n of (3 with λ = λ n is given by T n (t = A n e t/ n t + B n e t/ n t with the product solution u(n(x, t = X n (xt n (x, the proposed solution for the problem is u(x, t = u n (x, t = e ] t/ n t n t A n + B n n n At t =, = u(x, = e t/ n A n so that A n = for all n Differentiating u with respect to t at t =, we get x = u t (x, = n B n n form which we obtain n B n = x, n

4 4 The above integral is equal to x dx = Hence = 4(n n + 4 n u(x, t = 8et/ x nx n n ] ] + n = 4(n+ n nx B n = 8(n+ n n n n n t Problem 4 Use the method of separation of variables to solve u t = u xx 4u < x <, t > u x (, t = u(, t = t u(x, = x x Is the solution found above a classical solution? Solution: Write u(x, t = X(xT(t Then This gives two equations T (t T(t + 4 = X (x X(x = λ (5 T + (4 + λt = (6 X + λx = X ( = X( = For λ <, the general solution is X(x = Ah λx + B h λx Since X (x = λah λx + B h λx], it follows that = X ( = B = X( = Ah λ so that also A = If λ =, then X(x = Ax + B the boundary conditions imply that A = B = If A >, then X(x = A λx+b λx Since X (x = λa λx+b λx], the boundary conditions imply that = X ( = λb so that B = = X( = A λ To get the nontrivial solution we need B λ = This is the case when λ = ( n+ for all n So, the eigenvalues corresponding eigenfunctions are ( n + λ n =, X n (x = For given λ n, the solution of (5 is T n (t = e 4t e λnt dx (n + x, for all n The product solution u n (x, t = X n (xt n (t = e 4t e λnt (n+x so that the formal solution of the problem is u(x, t = e 4t n A n e λnt (n + x

5 5 At t =, So, A n = The above integral is equal to (x x = u(x, = n (n + x The first integral is equal to x (n + x = = 4 (n + + n + = 4 (n + + n + = 4(n n + 6(n (n + 3 The second integral is equal to so that Hence = A n (x x x (n+x (n + 4 (n + ] (n + x (n + x (n + x x (n+x (n + 4 (n + 8 (n + (n + (n + x u(x, t = 6e4t = 4 n + ] A n = 6(n (n + 3 = 6(n+ (n + 3 n ( n+ (n + 3 e(n+/t (n + x (n + x x 8 (n + 6 (n + (n + 3 ] (n + x = 4(n n + (n + x (n + x