Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

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9.10 Exercises 351

For n = 0, equation (9.9.12) takes the form

′′

(y)=

(y) − f

(y)]

and hence,

′

(y)=



(τ) − f

(τ)] dτ + C,

where C is an integration constant. Employing the condition (9.9.14) for

n = 0, we ﬁnd

C =



(x) dx.

Thus, we have

′

(y)=





(τ) − f

(τ)] dτ +



(x) dx



Consequently,

′

(b)=



(τ) − f

(τ)] dτ +



(x) dx

Also from equation (9.9.14), we have

′

(b)=



(x) dx.

It follows from th ese two expressions for Y

′

(b)that



(y) − f

(y)] dy +



(x) − g

(x)] dx =0.

which is the necessary condition for the existence of a solution to the Neu-

mann problem for a rectangle.

9.10 Exercises

1. Reduce the Neumann problem to the Dirichlet problem in the two-

dimensional case.

2. Reduce the wave equation

= c

+ u

)

352 9 Boundary-Value Problems and Applications

to the Laplace equation

+ u

ττ

by letting τ = ict where i =

√

−1. Obtain the solution of the wave equa-

tion in cylindrical coordinates via the solution of the Laplace equation.

Assume that u (r, θ, z, τ ) is independent of z.

3. Prove that, if u (x, t) satisﬁes

= ku

for 0 ≤ x ≤ 1, 0 ≤ t ≤ t

, then the maximum value of u is attained

either at t = 0 or at the end points x =0orx =1for0≤ t ≤ t

.This

is called the maximum principle for the heat equation.

4. Prove that a function which is harmonic everywhere on a plane and is

bounded either above or below is a constant. This is called the Liouville

theorem.

5. Show that the compatibility condition for the Neumann problem

∇

u = f in D

∂u

∂n

= g on B



fdS+



gds=0,

where B is the boundary of domain D.

6. Show that the second degree polynomial

P = Ax

+ Bxy + Cy

+ Dyz + Fz

+ Fxz

is harmonic if

E = −(A + C)

and obtain

P = A



− z



+ Bxy + C



− z



+ Dyz + Fxz.

7. Prove that a solution of the Neumann problem

∇

u = f in D

u = g on B

diﬀers from another solution by at most a constant.

9.10 Exercises 353

8. Determine the solution of each of the following problems:

(a) ∇

u =0, 1 <r<2, 0 <θ<π,

u (1,θ)=sinθ, u (2,θ)=0, 0 ≤ θ ≤ π,

u (r, 0) = 0,u(r, π)=0, 1 ≤ r ≤ 2.

(b) ∇

u =0, 1 <r<2, 0 <θ<π,

u (1,θ)=0,u(2,θ)=θ (θ − π) , 0 ≤ θ ≤ π,

u (r, 0) = 0,u(r, π)=0, 1 ≤ r ≤ 2.

u =0, 1 <r<3, 0 <θ<π/2,

u (1,θ)=0,u(3,θ)=0, 0 ≤ θ ≤ π/2,

u (r, 0) = (r − 1) (r − 3) ,u





=0, 1 ≤ r ≤ 3.

(d) ∇

u =0, 1 <r<3, 0 <θ<π/2,

u (1,θ)=0,u(3,θ)=0, 0 ≤ θ ≤ π,

u (r, 0) = 0,u





= f (r) , 1 ≤ r ≤ 3.

9. Solve the boundary-value problem

∇

u =0, a<r<b, 0 <θ<α,

u (a , θ)=f (θ) ,u(b, θ)=0, 0 ≤ θ ≤ α,

u (r, α)=0,u(r, 0) = f (r) ,a≤ r ≤ b.

10. Verify directly that the Poisson integral is a solution of the Laplace

equation.

11. Solve

∇

u =0, 0 <r<a, 0 <θ<π,

u (r, 0) = 0,u(r, π)=0,

u (a , θ)=θ (π − θ) , 0 ≤ θ ≤ π,

u (0,θ) is bounded.

354 9 Boundary-Value Problems and Applications

12. Solve

∇

u + u =0, 0 <r<a, 0 <θ<α,

u (r, 0) = 0,u(r, α)=0,

u (a , θ)=f (θ) , 0 ≤ θ ≤ α,

u (0,θ) is bounded.

13. Find the solution of the Dirichlet problem

∇

u = −2,r<a,0 <θ<2π,

u (a , 0) = 0.

14. Solve the following problems:

(a) ∇

u =0, 1 <r<2, 0 <θ<2π,

(1,θ)=sinθ, u

(2,θ)=0, 0 ≤ θ ≤ 2π,

(b) ∇

u =0, 1 <r<2, 0 <θ<2π,

(1,θ)=0,u

(2,θ)=θ − π, 0 ≤ θ ≤ 2π.

15. Solve

∇

u =0, a<r<b, 0 <θ<2π,

(a, θ)=f (θ) ,u

(b, θ)=g (θ) , 0 ≤ θ ≤ 2π,

where



r=a

fds+



r=b

gds=0.

16. Solve the Robin problem for a semicircular disk

∇

u =0,r<R,0 <θ<π,

(R, θ)=sinθ, 0 ≤ θ ≤ π,

u (r, 0) = 0,u(r, π)=0.

9.10 Exercises 355

17. Solve

∇

u =0, a<r<b, 0 <θ<α,

(a, θ)=0,u

(b, θ)=f (θ) , 0 ≤ θ ≤ α,

u (r, 0) = 0,u(r, α)=0, a<r<b.

18. Determine the solution of th e mixed boundary-value problem

∇

u =0,r<R, 0 <θ<2π,

(R, θ)+hu (R, θ )=f (θ) ,h= constant.

19. Solve

∇

u =0, a<r<b, 0 <θ<2π,

(a, θ)+hu(a, θ)=f (θ) ,u

(b, θ)+hu(b, θ)=g (θ) .

20. Find a solution of the Neumann problem

∇

u = −r

sin 2θ, r

<r<r

, 0 <θ<2π,

,θ)=0,u

,θ)=0, 0 ≤ θ ≤ 2π.

21. Solve the Robin problem

∇

u = −r

sin 2θ,

u (r

,θ)=0,u

,θ)=0.

22. Solve the following Dirichlet problems:

(a) ∇

u =0, 0 <x<1, 0 <y<1,

u (x , 0) = x (x − 1) ,u(x, 1) = 0, 0 ≤ x ≤ 1,

u (0,y)=0,u(1,y)=0, 0 ≤ y ≤ 1.

(b) ∇

u =0, 0 <x<1, 0 <y<1,

u (x , 0) = 0,u(x, 1) = sin (πx) , 0 ≤ x ≤ 1,

u (0,y)=0,u(1,y)=0, 0 ≤ y ≤ 1.

356 9 Boundary-Value Problems and Applications

u =0, 0 <x<1, 0 <y<1,

u (x , 0) = 0,u(x, 1) = 0, 0 ≤ x ≤ 1,

u (0,y)=



cos

πy

− 1



cos



πy



, u (1,y)=0, 0 ≤ y ≤ 1.

(d) ∇

u =0, 0 <x<1, 0 <y<1,

u (x , 0) = 0,u(x, 1) = 0, 0 ≤ x ≤ 1,

u (0,y)=0,u(1,y)=sinπy cos πy, 0 ≤ y ≤ 1.

23. Solve the following Neumann problems:

(a) ∇

u =0, 0 <x<π, 0 <y<π,

(0,y)=



y −



(π, y)=0, 0 ≤ y ≤ π,

(x, 0) = x, u

(x, π)=x, 0 ≤ x ≤ π.

(b) ∇

u =0, 0 <x<π, 0 <y<π,

(0,y)=0,u

(π, y)=2cosy, 0 ≤ y ≤ π,

(x, 0) = 0,u

(x, π)=0, 0 ≤ x ≤ π.

u =0, 0 <x<π, 0 <y<π,

(0,y)=0,u

(π, y)=0, 0 ≤ y ≤ π,

(x, 0) = cos x, u

(x, π)=0, 0 ≤ x ≤ π.

(d) ∇

u =0, 0 <x<π, 0 <y<π,

(0,y)=y, u

(π, y)=y, 0 ≤ y ≤ π,

(x, 0) = x, u

(x, π)=x. 0 ≤ x ≤ π.

24. The steady-state temperature distribution in a rectangular plate of

length a and width b is described by

∇

u =0, 0 <x<a, 0 <y<b.

At x = 0, the temp eratu re is kept at zero degrees, while at x = a,the

plate is insulated. The temperature is prescribed at y =0.Aty = b,

9.10 Exercises 357

heat is allowed to radiate freely into the surrounding medium of zero

degree temperature. That is, the boundary conditions are

u (0,y)=0,u

(a, y)=0, 0 ≤ y ≤ b,

u (x , 0) = f (x) ,u

(x, b)+hu(x, b)=0, 0 ≤ x ≤ a.

Determine the temperature distribution.

25. Solve the Dirichlet problem

∇

u = −2y, 0 <x<1, 0 <y<1,

u (0,y)=0,u(1,y)=0, 0 ≤ y ≤ 1,

u (x , 0) = 0,u(x, 1) = 0, 0 ≤ x ≤ 1.

26. Find the harmonic function which vanishes on the hypotenuse and has

prescribed values on the other two sides of an isosceles right-angled

triangle formed by x =0,y =0,andy = a − x,wherea is a constant.

27. Find a solution of the Neumann problem

∇

u =



− y



, 0 <x<a, 0 <y<a,

(0,y)=0,u

(a, y)=0, 0 ≤ y ≤ a,

(x, 0) = 0,u

(x, a)=0, 0 ≤ x ≤ a.

28. Solve the third boundary-value problem

∇

u =0, 0 <x<1, 0 <y<1,

(0,y)+hu(0,y)=0, 0 ≤ y ≤ 1,h= constant

(1,y)+hu(1,y)=0, 0 ≤ y ≤ 1,

(x, 0) + hu(x, 0) = 0, 0 ≤ x ≤ 1,

(x, 1) + hu(x, 1) = f (x) , 0 ≤ x ≤ 1.

358 9 Boundary-Value Problems and Applications

29. Determine the solution of th e boundary-value problem

∇

u =1, 0 <x<π, 0 <y<π,

u (0,y)=0,u

(π, y)=0, 0 ≤ y ≤ π,

(x, 0) = 0,u

(x, π)+hu(x, π)=f (x) , 0 ≤ x ≤ π.

30. Obtain the integral representation of the Neumann problem

∇

u = f in D,

∂u

∂n

= g on boundary B of D.

31. Find the solution in terms of the Green’s function of the Poisson prob-

lem

∇

u = f in D,

∂u

∂n

+ hu = g on boundary B of D.

32. Find the steady-state t emperature distribution inside a circular annular

region governed by the boundary-value problem

θθ

=0, 1 <r<2, −π<0 <π,

u (1,θ)=sin

θ, u

(2,θ)=0, −π<θ<π.

33. Consider a radially symmetric steady-state problem in a solid homoge-

neous cylinder of radius unity and height h. The temperature distribu-

tion function u (r, z) satisﬁes the equation

∇

u = u

+ u

=0, 0 <r<1, 0 <z<h.

Solve the Dirichlet boundary-value problem with the following b ound-

ary conditions:

(a) u (1,z)=f (z) ,u(r, 0) = 0,u(r, h)=0,

(b) u (1,z)=0,u(r, 0) = g (r) ,u(r, h)=0,

(d) u (1,z)=a sin



3πz



,u(r, 0) = 0 = u (r, h) .

9.10 Exercises 359

34. Show that the solution of the p rob lem 33(c) is given by

u (r, z)=

∞



n=1

sinh k

where J

)=0,n =1, 2, 3,....

35. Solve the following boundary-valu e problem:

(a) ∇

u =0, 0 <r<1, 0 <z<1,

u (1,z)=0=u (r, 0) ,u(r, 1) = 1 − r

, 0 ≤ r ≤ 1.

(b) ∇

u =0, 0 <r<1, 0 <z<π,

u (1,z)=A = constant,u(r, 0) = 0,u

(r, π)=0.

u =0, 0 <r<a, 0 <z<h,

u (a , z)=f (z) ,u(r, 0) = 0 = u (r, h) , 0 ≤ r ≤ a.

(d) ∇

u =0, 0 <r<a, 0 <z<h,

u (a , z)=z (h − z) ,u(r, 0) = 0 = u (r, h) .

In each of the above problems (a)–(d),

∇

u = u

+ u