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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
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U(r, t) = J2 [knun(r, 0 + cnvn(r, 0] n=1
?
= Y2 [knJ0(knr) sinknat + cnJ0(knr) cosknat\. (17)
n=1
The initial conditions require that
?
u(r, 0) = Y2 CnJ0(knr) = f (r) (18)
n=1
and
?
Ut(r, 0) = ^2knaknJ0(knr) = 0. (19)
n=1
From Eq. (26) of Section 11.4 we obtain
f rf(r) J0(knr) dr kn = 0, Cn = ^-----------------; n = 1, 2,.... (20)
/ r[J0(knr)]2 dr J0
Thus, the solution of the partial differential equation (3) satisfying the boundary condition (4) and the initial conditions (5) and (6) is given by
?
u(r, t) = ^2, cnJ0(knr) cosknat (21)
n=1
with the coefficients cn defined by Eq. (20).
1. Consider Laplace’s equation uxx + uyy = 0 in the parallelogram whose vertices are (0, 0), (2, 0), (3, 2), and (1, 2). Suppose that on the side y = 2 the boundary condition is u(x, 2) = f(x) for 1 < x < 3, and that on the other three sides u = 0 (see Figure 11.5.1).
D '
(0, 2)
C
(2, 2)
B'(2, 0) (b)
FIGURE 11.5.1 The region in Problem 1.
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 667
(a) Show that there are no nontrivial solutions of the partial differential equation of the form u(x, y) = X(x)Y(y) that also satisfy the homogeneous boundary conditions.
(b) Let f = x — 1 y, n = /. Show that the given parallelogram in the xy-plane transforms into the square 0 < f < 2,0 < n < 2 in the fn-plane. Show that the differential equation transforms into
5 — uh u =0.
4 ff fn nn
How are the boundary conditions transformed?
(c) Show that in the fn-plane the differential equation possesses no solution of the form
u(f,n) = u (f)V (n).
Thus in the xy-plane the shape of the boundary precludes a solution by the method of the separation of variables, while in the fn-plane the region is acceptable but the variables in the differential equation can no longer be separated.
2. Find the displacement u (r, t) in a vibrating circular elastic membrane of radius 1 that satisfies the boundary condition
u(1, t) = 0, t > 0,
and the initial conditions
u(r, 0) = 0, ut(r, 0) = g(r), 0 < r < 1,
where g(1) = 0.
Hint: The differential equation to be satisfied is Eq. (3) of the text.
3. Find the displacement u(r, t) in a vibrating circular elastic membrane of radius 1 that satisfies the boundary condition
u(1, t) = 0, t > 0,
and the initial conditions
u(r, 0) = f(r), ut(r, 0) = g(r), 0 < r < 1,
where f (1) = g(1) = 0.
4. The wave equation in polar coordinates is
urr + (1/r)ur + (1/r2)uee = a—2utr
Show that if u(r,6, t) = R(r)®(9) T(t), then R, ©, and T satisfy the ordinary differential equations
r2 R" + rR + (k2r2 — n2) R = 0,
©" + n2® = 0,
T" + k2a2 T = 0.
5. In the circular cylindrical coordinates r,9, z defined by
x = r cos 9, y = r sin 9, z = z,
Laplace’s equation is
urr + (1/r)ur + (1/r2)u99 + uzz = °.
(a) Show that if u(r, 9, z) = R(r)®(9)Z(z), then R, ®, and Z satisfy the ordinary differential equations
r2 R" + rR + (k2r2 — n2) R = 0,
®" + n2® = 0,
Z"- k2 Z= 0.
668
Chapter 11. Boundary Value Problems and SturmLiouville Theory
(b) Show that if u(r, Q, z) is independent of Q, then the first equation in part (a) becomes
r2 R" + rR + k2r2 R = 0,
the second is omitted altogether, and the third is unchanged.
6. Find the steady-state temperature in a semi-infinite rod 0 < z < to, 0 < r < 1, if the temperature is independent of Q and approaches zero as z ^ to. Assume that the temperature u(r, z) satisfies the boundary conditions
u(1, z) = 0, z > 0,
u(r, 0) = f (r), 0 < r < 1.
Hint: Refer to Problem 5.
7. The equation
v + v + k2 v = 0
xx 1 yy
is a generalization of Laplace’s equation, and is sometimes called the Helmholtz (18211894) equation.
(a) In polar coordinates the Helmholtz equation is
vrr + (1/r)vr + (1/r2)vee + k2v = 0.
If v(r,Q) = R(r)©(Q), show that R and © satisfy the ordinary differential equations r2R'+ rR + (k2r2 - k2)R = 0, ©" + k2© = 0.
(b) Consider the Helmholtz equation in the disk r < c. Find the solution that remains bounded at all points in the disk, that is periodic in Q with period 2n, and that satisfies the boundary condition v(c, Q) = f (Q), where f is a given function on 0 < Q < 2n.
Hint: The equation for R is a Bessel equation. See Problem 3 of Section 11.4.
8. Consider the flow of heat in an infinitely long cylinder of radius 1:0 < r < 1,0 < Q < 2n, —to < z < to. Let the surface of the cylinder be held at temperature zero, and let the initial temperature distribution be a function of the radial variable r only. Then the temperature u is a function of r and t only, and satisfies the heat conduction equation
u2[urr + (1/r)ur] = ut, 0 < r < 1, t > 0,
and the following initial and boundary conditions:
u(r, 0) = f(r), 0 < r < 1,
u(1, t) = 0, t > 0.
Show that
TO
v > 21 2 *
u(r, t) = J2 cnJ0(knr)e-a knt,
n=1
where J0(kn) = 0. Find a formula for cn.
9. In the spherical coordinates p, Q, 0 (p > 0, 0 < Q < 2n, 0 < 0 < n) defined by the equations
x = p cos Q sin 0, y = p sin Q sin0, z = p cos 0,
Laplace’s equation is
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