Mixed Boundary Value Problems Episode 10 potx

Transform Methods 261We can now solve for A n via Equation 4.3.353 and Equation 4.3.354 after we truncate the set of infinite equations to a finite number.. Step 1 : Using separation of va

In line with previous examples the solution that satisfies Equation 4.3.327through Equation 4.3.329 is

begins by showing that

permission of Oxford University Press; see also Lemczyk, T F., and M M Yovanovich, 1988: Thermal constriction resistance with convective boundary conditions–1 Half-space

contacts Int J Heat Mass Transfer , 31, 1861–1872.

Transform Methods 259

This follows from interchanging the order of integration and applying tion 1.4.9 Next, we view the quantity within the square brackets on the leftside of Equation 4.3.336 as the unknown in an integral equation of the Abeltype From Equation 1.2.13 and Equation 1.2.14, we have that

If we divide the left side of Equation 4.3.341 by t, we have the first term on

the left side of Equation 4.3.335 The second term can be evaluated fromintegral tables.63 Consequently Equation 4.3.335 becomes

sin(2θ) cos[(2n + 1)θ]



2[cos(2ϕ) − cos(2θ)] dθ (4.3.345)

=π8

from Equation 1.3.4 and Equation 1.3.5, where the prime denotes that

when-ever P −1(·) occurs, then it is replaced by P0(·) Similarly,

Transform Methods 261

We can now solve for A n via Equation 4.3.353 and Equation 4.3.354 after we

truncate the set of infinite equations to a finite number Then h(t) follows from Equation 4.3.343 Finally, A(k) is computed from Equation 4.3.334 while u(r, z) is obtained from Equation 4.3.331.

Step 1 : Show that

electrodes J Electroanal Chem., 222, 107–115; Gupta, S C., 1957: Slow broad side motion of a flat plate in a viscous liquid Z Angew Math Phys., 8, 257–261.

show that kA(k) = C sin(ka).

Step 4 : Using the relationship66

Wiley and Webster67 used this solution in an improved design for a circular

electrosurgical dispersive electrode

2 Solve Laplace’s equation68

SeeProblem 5 for a generalization of this problem

Step 1 : Using separation of variables or transform methods, show that the

general solution to the problem is

under circular dispersive electrode IEEE Trans Biomed Engng., BME-29, 381–385.

media J Appl Phys., 77, 110–117 This problem also appears while finding the

tempera-ture field in a paper by Florence, A L., and J N Goodier, 1963: The linear thermoelastic

problem of uniform heat flow disturbed by a penny-shaped insulated crack Int J Engng.

Sci., 1, 533–540.

Transform Methods 263

0 0.5 1 1.5 2

0 0.5 1 1.5 2

r z

satisfies both integral equations given in Step 2

Step 4 : Show that the solution to this problem is

3 Solve the potential problem70

Step 1: By using either separation of variables or transform methods, show

that the general solution to partial differential equation is

Step 3: Show that

r2− η2, integrating from 0 to r, and taking

the derivative with respect to r, show that

electrodynamics J Math Phys., 8, 518–522.

Transform Methods 265show that the integral equation in Step 4 simplifies to

2

π

d dr

Step 1 : Show that

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

disturbed by a penny-shaped crack in a constant axial magnetic field Eng Fract Mech.,

23, 977–982.

0 0.5 1 1.5 2

0 0.5 1 1.5 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

show that Equation (2) is automatically satisfied

Step 3 : By using Equation (1), show that h(t) = −2t/π and

kA(k) = − 2

πk2[sin(ka) − ka cos(ka)] Step 4 : Show that the solution to the problem is

The figure labeled Problem 4 illustrates this solution u(r, z).

5 Solve Laplace’s equation72

Astrophys Space Sci., 71, 195–201.

268 Mixed Boundary Value Problems

6 Solve the partial differential equation73

Step 1 : Show that

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

7 A generalization of a problem originally suggested by Popova74was given

by Kuz’min75 who solved

B Pryor, and R J Marhefka, 2003: Ohmic loss in frequency-selective surfaces J Appl.

Phys., 93, 5346–5358.

half-space J Engng Phys., 25, 934–935.

boundary conditions Sov Tech Phys., 11, 169–173.

0 0.5 1 1.5 2

0 0.5 1 1.5 2

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

r z

Problem 7

show that Equation (2) is automatically satisfied

Step 3 : By using Equation (1) and Equation 1.4.14 and noting that

boundary-value problem of thermoelasticity for a half-space Quart J Mech Appl Math., 20,

127–134.

Transform Methods 271subject to the boundary conditions

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

show that Equation (2) is automatically satisfied

Step 3 : By using Equation (1) and Equation 1.4.14 and noting that

0 0.5 1 1.5 2

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1

r z

Transform Methods 273

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

show that Equation (2) is automatically satisfied

Step 3 : By using Equation (1) and Equation 1.4.13, show that

Step 4 : Using the first integral equation in Step 2, show that

g(t) = 2π

general solution to the problem is

stress distributions in elastic solids containing cracks – I An external crack in an infinite

276 Mixed Boundary Value Problems

0 1 2 3 4

0 0.5 1 1.5 2

−0.5 0 0.5 1 1.5

rz

h(t) = −2

π

d dt

properties and colloid anisotropy: Towards a reliable pair potential for disc-like charged

particles Eur Phys J., Ser E, 15, 345–357.

subject to the boundary conditions

Step 1 : Show that

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

Step 2 : Setting x = r/a, ξ = ka and g(ξ) =

ξ2+ (κa)2A(ξ)/u0, show that

dual integral equations in Step 1 become

278 Mixed Boundary Value Problems

0 0.5 1 1.5

2 −1

−0.5 0 0.5 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z/ar/a

integrands that oscillate rapidly Finally, the potential is computed The

figure labeled Problem 12 illustrates an example when κa = 1.

13 A problem similar to Example 4.3.2 involves solving Laplace’s equation80

integrals with variable frequency J Comput Appl Math., 21, 87–99.

diffusion II Cylindrical punch J Appl Phys., 74, 4390–4397.

u z (r, 0) = 1, 0≤ r < 1, u(r, 0) = 0, 1 < r < ∞,

satisfies the partial differential equation and the boundary conditions provided

that A(k) satisfies the dual integral equations

matically satisfied Hint: Use Equation 1.4.13

Step 3 : Using Equation (1), show that f (t) is given by the integral equation

Theory for the determination of backside contact resistance of semiconductor wafers from

Transform Methods 281

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

rz

Problem 14

where f (t) is a real, even and continuous function, show that Equation (2) is

automatically satisfied Hint: SeeSection 2.2

Step 3 : Using Equation (1), show that f (t) is given by the integral equation

This integral equation must be solved numerically Because the integrals

involving cos[(t −η)k] and cos[(t+η)k] can oscillate rapidly, we use a numerical

scheme by Ehrenmark82 for their evaluation. A(k) and B(k) then follow

from f (t) Finally, the solution u(r, z) involves a numerical integration where the integrand includes both A(k) and B(k) The figure labeled Problem 14 illustrates this solution u(r, z) when γ = L = 1.

14 Solve Laplace’s equation in cylindrical coordinates:

u(r, 0 − ) = u(r, 0+) = 1, 0≤ r < 1, u(r, 0 − ) = u(r, 0+), u

satisfy the partial differential equation and the boundary conditions given by

Equation (1), Equation (2), Equation (4), and u(r, 0 − ) = u(r, 0+).

Step 2 : Show that the boundary condition given by Equation (3) yields the

dual integral equations

Step 3 : Verify that

0 0.2 0.4 0.6 0.8 1 1.2

rz

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/a z/a

The figure labeled Problem 16 illustrates this solution.83

4.4 TRIPLE AND HIGHER FOURIER-BESSEL INTEGRALS

In Section 4.3 we examined in detail mixed boundary value problemswhich yielded dual integral equations Here we extend our studies where weobtain triple integral equations

the solution of problem of motion of a circular disk in viscous liquid Philos Mag., Ser.

7 , 21, 546–564.

where a < 1.

Using transform methods or separation of variables, the general solution

to Equation 4.4.1, Equation 4.4.2, and Equation 4.4.3 is

To solve this set of integral equations, let us introduce the unknown

functions f (r) and g(r) such that

 ∞

0

kA(k)J0(kr) dk = f (r), 0≤ r < a, (4.4.9)

Geophysical Union Reproduced/modified by permission of American Geophysical Union.

Transform Methods 287and

dr, (4.4.23)

φ(r) = 2

π −2π

Belyaev85gave an alternative approach to this problem Again, we wish

to solve Laplace’s equation

(4.4.30)

apera-ture in a conducting plane Sov Tech Phys., 25, 12–16.