Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 3 ppsx

We impose the terminal constraint xT = h, and we wish to find the particular thrust function U t which will minimize T assuming that the total thrust of the rocket engineover the entire

Thus the general solution for u is

Z 2π 0

f (θ) dθ

an= 1π

Z 2π 0

f (θ) cos(nθ) dθ

bn= 1π

Z 2π 0

g(θ) dθ = 0

The coefficients are

an = 1nπ

Z 2π 0

g(θ) cos(nθ) dθ

bn = 1nπ

Z 2π 0

g(θ) sin(nθ) dθ

Consider the problem

r2

∂2u

∂θ2 = 0, 0 ≤ r < a, −π < θ ≤ π,with the boundary condition

u(a, θ) = θ2

So far this problem only has one boundary condition By requiring that the solution be finite, we get the boundarycondition

|u(0, θ)| < ∞

By specifying that the solution be C1, (continuous and continuous first derivative) we obtain

u(r, −π) = u(r, π) and ∂u

Substituting into the partial differential equation,

Θ00(θ) + λΘ(θ) = 0, −π < θ ≤ π,subject to

Θ(−π) = Θ(π) and Θ0(−π) = Θ0(π)

We consider the following three cases for the eigenvalue, λ,

λ < 0 No linear combination of the solutions, Θ = exp(√

−λθ), exp(−√−λθ), can satisfy the boundary conditions.Thus there are no negative eigenvalues

λ = 0 The general solution solution is Θ = a + bθ By applying the boundary conditions, we get Θ = a Thus wehave the eigenvalue and eigenfunction,

An = cos(nθ) and Bn= sin(nθ)

The equation for R is

R = a log r + bRequiring that the solution be bounded at r = 0 yields (to within a constant multiple)

R0 = 1

For λn = n2, n ≥ 1, we have

r2R00+ rR0− n2R = 0Recognizing that this is an Euler equation and making the substitution R = rα,

α(α − 1) + α − n2 = 0

α = ±n

R = arn+ br−n.requiring that the solution be bounded at r = 0 we obtain (to within a constant multiple)

Rn = rnThe general solution to the partial differential equation is a linear combination of the eigenfunctions

We determine the coefficients of the expansion with the boundary condition

Part VI Calculus of Variations

Chapter 48

Calculus of Variations

48.1 Exercises

Exercise 48.1

Discuss the problem of minimizing Rα

0 ((y0)4 − 6(y0)2) dx, y(0) = 0, y(α) = β Consider both C1[0, α] and C1

p[0, α],and comment (with reasons) on whether your answers are weak or strong minima

Discuss finding a weak extremum for the following:

1 R01((y00)2− 2xy) dx, y(0) = y0(0) = 0, y(1) = 1201

2 R1

0

1

2(y0)2+ yy0+ y0+ y
dx

3 Rab(y2+ 2xyy0) dx, y(a) = A, y(b) = B

4 R01(xy + y2− 2y2y0) dx, y(0) = 1, y(1) = 2

in D, and σ is known on Γ

where the fluid motion is described by φ(x, y, t) and g is the acceleration of gravity Show that all these equations may

be obtained by varying the functions φ(x, y, t) and h(x, t) in the variational principle

δ

Z Z

R

Z h(x,t) 0

Consider the functional R √y ds where ds is the arc-length differential (ds = p(dx)2+ (dy)2) Find the curve orcurves from a given vertical line to a given fixed point B = (x1, y1) which minimize this functional Consider both theclasses C1 and C1

p.Exercise 48.8

A perfectly flexible uniform rope of length L hangs in equilibrium with one end fixed at (x1, y1) so that it passes over

a frictionless pin at (x2, y2) What is the position of the free end of the rope?

Exercise 48.9

The drag on a supersonic airfoil of chord c and shape y = y(x) is proportional to

D =

Z c 0

 dydx

2

dx

Find the shape for minimum drag if the moment of inertia of the contour with respect to the x-axis is specified; that

is, find the shape for minimum drag if

Z c 0

y2dx = A, y(0) = y(c) = 0, (c, A given)

of the natural frequencies

Exercise 48.11

A boatman wishes to steer his boat so as to minimize the transit time required to cross a river of width l The path ofthe boat is given parametrically by

x = X(t), y = Y (t),for 0 ≤ t ≤ T The river has no cross currents, so the current velocity is directed downstream in the y-direction v0 isthe constant boat speed relative to the surrounding water, and w = w(x, y, t) denotes the downstream river current atpoint (x, y) at time t Then,

˙X(t) = v0cos α(t), Y (t) = v˙ 0sin α(t) + w,

where α(t) is the steering angle of the boat at time t Find the steering control function α(t) and the final time T thatwill transfer the boat from the initial state (X(0), Y (0)) = (0, 0) to the final state at X(t) = l in such a way as tominimize T

and obtain the equations of motion

2 In the case where the displacement of the suspended mass from equilibrium is small, show that the suspendedmass performs small vertical oscillations and find the period of these oscillations

Exercise 48.13

A rocket is propelled vertically upward so as to reach a prescribed height h in minimum time while using a given fixedquantity of fuel The vertical distance x(t) above the surface satisfies,

m¨x = −mg + mU (t), x(0) = 0, ˙(x)(0) = 0,where U (t) is the acceleration provided by engine thrust We impose the terminal constraint x(T ) = h, and we wish

to find the particular thrust function U (t) which will minimize T assuming that the total thrust of the rocket engineover the entire thrust time is limited by the condition,

Z T 0

U2(t) dt = k2.Here k is a given positive constant which measures the total amount of fuel available

to dock the vehicle in minimum time; that is, we seek a thrust function U (t) which will minimize the final time Twhile bringing the vehicle to rest at the origin with x(T ) = 0, ˙x(T ) = 0 Find U (t), and in the (x, ˙x)-plane plot thecorresponding trajectory which transfers the state of the system from (a, b) to (0, 0) Account for all values of a and b.Exercise 48.15

Find a minimum for the functional I(y) = Rm

Exercise 48.17

Consider the integral R 1+y 2

(y 0 ) 2 dx between fixed limits Find the extremals, (hyperbolic sines), and discuss the Jacobi,Legendre, and Weierstrass conditions and their implications regarding weak and strong extrema Also consider the value

of the integral on any extremal compared with its value on the illustrated strong variation Comment!

PiQi are vertical segments, and the lines QiPi+1 are tangent to the extremal at Pi+1

Exercise 48.18

Consider I =Rx1

x 0 y0(1 + x2y0) dx, y(x0) = y0, y(x1) = y1 Can you find continuous curves which will minimize I if

(i) x0 = −1, y0 = 1, x1 = 2, y1 = 4,(ii) x0 = 1, y0 = 3, x1 = 2, y1 = 5,(iii) x0 = −1, y0 = 1, x1 = 2, y1 = 1

(φψx− ψφx) dx + 1

2Z

(y0)2− y2 − 2xy
dx, y(0) = 0 = y(1)

For this problem take an approximate solution of the form

y = x(1 − x) (a0+ a1x + · · · + anxn) ,

and carry out the solutions for n = 0 and n = 1.

•

Z 2 0

(y0)2+ y2 + 2xy
dx, y(0) = 0 = y(2)

•

Z 2 1

x(y0)2 −x

K(x − y)f (y) dy

1 Show that the spectrum of T consists of the range of the Fourier transform ˆK of K, (that is, the set of all valuesˆ

K(y) with −∞ < y < ∞), plus 0 if this is not already in the range (Note: From the assumption on K it followsthat ˆK is continuous and approaches zero at ±∞.)

2 For λ in the spectrum of T , show that λ is an eigenvalue if and only if ˆK takes on the value λ on at least someinterval of positive length and that every other λ in the spectrum belongs to the continuous spectrum

3 Find an explicit representation for (T − λI)−1f for λ not in the spectrum, and verify directly that this resultagrees with that givenby the Neumann series if λ is large enough

Exercise 48.22

Let U be the space of twice continuously differentiable functions f on [−1, 1] satisfying f (−1) = f (1) = 0, and

W = C[−1, 1] Let L : U 7→ W be the operator dxd22 Call λ in the spectrum of L if the following does not occur:There is a bounded linear transformation T : W 7→ U such that (L − λI)T f = f for all f ∈ W and T (L − λI)f = ffor all f ∈ U Determine the spectrum of L

Exercise 48.23

Solve the integral equations

1 φ(x) = x + λ

Z 1 0

x2y − y2
φ(y) dy

2 φ(x) = x + λ

Z x 0

K(x, y)φ(y) dywhere

K(x, y) =

(sin(xy) for x ≥ 1 and y ≤ 1,

2 Find the eigenvalues and eigenfunctions of the operator K ≡ −dtd +t42 in the space of functions u ∈ L2(−∞, ∞).(Hint: L1 = 2t +dtd, L2 = 2t − d

dt e−t2/4 is the eigenfunction corresponding to the eigenvalue 1/2.)Exercise 48.25

Prove that if the value of λ = λ1 is in the residual spectrum of T , then λ1 is in the discrete spectrum of T∗

sin(k(s − t))u(s) ds = f (t), u(0) = u0(0) = 0

u(x) = λ

Z π 0

K(x, s)u(s) dswhere

K(x, s) = 1

2log

sin x+s2
sin x−s2

... 3< /sup>

2

3? ?4− 6? ?2− +√3? ?(4 − λ2)3/ 2.E(w2) is negative for −1 < λ < √

3 and E(w3< /sub>)...

!+ 3 βα

4

− 6< sup> βα

2

3? ?4− 6? ?2− −√3? ?(4 − λ2)3/ 2,

E(w3< /sub>)... √

3 and E(w3< /sub>) is negative for −√

3 < λ < This implies that the weakminimum ˆy = βx/α is not a strong local minimum for |λ| < √

3| Since E(w1)