Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Let U = C[0, 1 ]. Find the adjoint I * of the identity operator I: cfw_J U under the weighted
inner products
SOLUTION
I*[v ] = c(x)/(x) v(x) is a multiplication operator.
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Consider the linear operator
that maps u(x) C1 to the vectorvalued function whose components consist of the function
and its first derivative.
(a) Compute the adjoint L* with respect to the L2 inner products on both the domain and
target spaces, subject
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Does the inequality (11.94) hold when u(x) 0 is subject to the Neumann boundary
conditions u'(0) = u'() = 0?
SOLUTION
No. The boundary terms still vanish and so the integration by parts identity continues to
hold, but now u(x) = a constant make
The integr
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Prove that the complex differential operator
is selfadjoint with respect to the L2 Hermitian inner product
on the vector space of continuously differentiable, complexvalued, 2 n periodic functions:
n(x + 2) = u(x).
SOLUTION
Use integration by parts:
whe
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
For each of the following functionals and associated boundary conditions, (i) write down a
boundary value problem satisfied by the minimizing function, and (ii) find the minimizing
function u,(x):
a.
b.
c.
d.
SOLUTION
(a)
(b)
(i)
u = 1, u(0) = 2, u(1) = 3
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
(a) Determine the adjoint of the differential operator u = L[u] = u' + 2xu with respect to the
L2 inner products on [0, I] when subject to the fixed boundary conditions u(0) = u(1) = 0.
(b) Is the selfadjoint operator K = L* L is positive definite? Expla
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Compute the adjoint of the derivative operator v = D[u] = u' under the weighted inner
products (11.96) on, respectively, the displacement and strain spaces. Verify that all four
types of boundary' conditions are allowed. Choose one set of boundary conditi
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
(a) Show that a differential equation of the form a(x) u"+b(x) u' = /(x) is in selfadjoint form
(11.12) if and only if b(x) = a'(x).
(b) If b(x) a'(x) and a(x) 0 everywhere, show that you can multiply the differential
equation by a suitable integrating f
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
For each the following boundary value problems,
(i) write down a minimization principle, carefully specifying the space of functions, and
(ii) find the solution:
a. u" = cosx, u(0) = 1, u() = 2
b.
u(1) = 1
c. ex(u" + u) = 1, u(0) = 1, u(1) =0
d. xu" + 2u'
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Find a function u(x) such that
How do you reconcile this with the claimed positivity in (11.94)?
SOLUTION
Hence for the final analysis we can say that u(x) = x 2 satisfiess 10 u(x) u(x) dx = 2/3 .
Positivity of 10 u(x) u(x) dx holds only for functions tha
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Find the function u(x) that minimizes the integral
subject to the boundary conditions u(1) = 1, u'(2) = 0. Hint: Use Exercise 11.3.26.
SOLUTION
Solving the boundary value problem d/dx (x du/dx)
= 1/2 x2 with the given boundary conditions given u(x)
= 1/18
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
A bar 1 meter long has stiffness c(x) = 1 +x at position 0 < x < 1. It is subject to an external
force f(x) = 1 x. The left end of the bar is fixed, while the right end is extended 1 cm.
(a) Write out and solve the boundary value problem governing the dis
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Suppose
Prove that ail solutions to the inhomogeneous Neumann boundary value problem
are minimizers of the modified energy functional
SOLUTION
The extra boundary terms serve to cancel those arising during the integration by parts
computation:
Again, the m
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Explain how to solve the inhomogeneous boundary value problem u" f(x), u(0) = a, u(1)
= , by using the Greens function (11.59).
SOLUTION
The function ~u (x) = u(x) [ (1 x) + x] satisfies the homogeneous boundary conditions
Conditions
and so is given by t
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Prove that the functional
subject to the mixed boundary conditions u(0) = 0, u'(l) = I has no minimizer! Thus,
omitting the extra boundary term in (11.106) is a fatal mistake.
SOLUTION
For any > 0, the function u(x) =
boundary conditions, but J[u] = 1/3 ,
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
In Exercise 7.5.6, you determined the adjoint of the derivative operator D when acting on
the space of quadratic polynomials with respect to the L2 inner product
SOLUTION
Hence for the final analysis we can say that quadratic polynomials do not, in genera
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Find the function u.(x) that minimizes the integral
subject to the boundary conditions n(l) = 0, m(2) = 1.
SOLUTION
Solving the corresponding boundary value problem
u(1) = 0, u(2) = 1, gives u(x)
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Let c(x) C[a, b] be a continuous function. Prove that the linear multiplication operator
K[u) = c(x)u(x) is selfadjoint with respect to the L 2 inner product. What sort of boundary
conditions need to be imposed?
SOLUTION
K[u] , v =
No boundary conditions
Islamia College of Commerce, 5F, Satellite Town, Jhang
DBMS
IT 69

Fall 2015
Prove that the solution to the mixed boundary value problem
is the unique C2 function that minimizes the modified energy functional
when subject to the inhomogeneous boundary conditions. Hint: Mimic the derivation of
Theorem 11.10.
Remark: Physically, the