Laplace Equation ✬ 3 ✭ =0

8.1. Laplace Equation ✬ 3 ✭ =0

The three-dimensional Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes stationary temperature distribution in the absence of heat sources and sinks in the domain under study.

A regular solution of the Laplace equation is called a harmonic function. The first boundary value problem for the Laplace equation is often referred to as the Dirichlet problem, and the second boundary value problem, as the Neumann problem.

Extremum principle : Given a domain ✮ , a harmonic function ✪ in ✮ that is not identically constant in ✮ cannot attain its maximum or minimum value at any interior point of ✮ .

8.1.1. Problems in Cartesian Coordinates

The three-dimensional Laplace equation in the rectangular Cartesian system of coordinates is writ- ten as

8.1.1-1. Particular solutions and some relations.

1 ✑ . Particular solutions:

( ✜ , ✧ , ✯ ) = cos( ✴ 1 ✜ + ✴ 2 ✧ ) exp( ✵ ✴ ✯ ),

( ✜ , ✧ , ✯ ) = sin( ✴ 1 ✜ + ✴ 2 ✧ ) exp( ✵ ✴ ✯ ),

( ✜ , ✧ , ✯ ) = exp( ✴ 1 ✜ + ✴ 2 ✧ ) cos( ✴

( ✜ , ✧ , ) = exp( ✵ ✴ ✜

) cos( 1 + ) cos( 2 ✯ + ),

( ✜ , ✧ , ) = cosh( ✴ 1 ✜

) cosh( 2 ) cos( ✯ + ),

( ✜ , ✧ , ✯ ) = cosh( ✴ 1 ✜ ) sinh( ✴ 2 ✧ ) cos( ✴

( ✜ , ✧ , ) = cosh( ✴ ✜ ) cos( ✴

1 ✧ + ✰ ) cos( ✴ 2 ✯ + ),

( ✜ , ✧ , ) = sinh( ✴ ✯ 1 ✜ ) sinh( ✴ 2 ✧ ) sin( ✴ ✯ + ✱ ),

( ✜ , ✧ , ) = sinh( ✴ ✜ ✯ ) sin( ✴ 1 ✧ + ✰ ) sin( ✴ 2 ✯ + ✱ ),

where ✰ , ✱ , ✲ , ✮ , ✳ , ✴ 1 , and ✴ 2 are arbitrary constants, and ✴ = ✶ ✴ 2 1 + ✴ 2 2 .

2 ✷ . Fundamental solution:

3 ✷ . Suppose ✪ = ✪ ( ✜ , ✧ , ✯ ) is a solution of the Laplace equation. Then the functions

= 1 − 2( ❀ ✜ + ❁❃✧ + ❂ )+( ❀ 2 + ❁ 2 + ❂ ✽ 2 ✯ ) 2 , where ✰ , ✲ ❄ , ❀ , ❁ , ❂ , and ✺ are arbitrary constants, are also solutions of this equation. The signs

at ✺ ❅❇❆ in the expression of ✪ 1 can be taken independently of one another.

References : W. Miller, Jr. (1977), R. Courant and D. Hilbert (1989).

8.1.1-2. Domain: − ❈ < ❉ < ❈ ,− ❈ < ✧ < ❈ ,0≤ ✯ < ❈ . First boundary value problem.

A half-space is considered. A boundary condition is prescribed:

Reference ✯ : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).

8.1.1-3. Domain: − ❈ < ❉ < ❈ ,− ❈ < ✧ < ❈ ,0≤ ✯ < ❈ . Second boundary value problem.

A half-space is considered. A boundary condition is prescribed:

is an arbitrary constant.

Reference : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

8.1.1-4. Domain: 0 ≤ ❉ ≤ ❀ ,0≤ ✧ ≤ ❁ ,0≤ ✯ ≤ ❂ . First boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

)+ ❋ ❊ ❄ 1 ❄ P sinh[ ✺ ❄ 1 P ( ❀ − ❉ )]

sinh( ✺ ❄ 2 P ❙ )+ ❊ ❄ 3 P sinh[ ✺ ❄ 2 P ( ❁ − ❙ )]

sin ◗

sin ◗

sinh( ✺ 2 ❄ P ❁ )

=1 P =1

sinh( ✺ 3 ❄ P )+ ❊ ❄ 5 P

sinh[ ✺ 3 ( ❂ − )]

sin ❄ ◗ P ❀ ❚ sin ◗

sinh(

Example. The planes ♠ =0 and ♠ = ♥ have constant temperatures ❲ 1 and ❲ 2 , respectively. The other planes are

maintained at zero temperature ( ♦ 3 = ♦ 4 = ♦ 5 = ♦ 6 =0 ).

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984).

For the solution of other boundary value problems for the three-dimensional Laplace equation in the Cartesian coordinate system, see Subsection 8.2.2 for ❶ ≡ 0.

8.1.2. Problems in Cylindrical Coordinates

The three-dimensional Laplace equation in the cylindrical coordinate system is written as

8.1.2-1. Particular solutions:

= 0, 1, 2, ➊✞➊✞➊ ; , , ❿ ➁ , ➂ , ➃ , ➄ ❿ , and ➆ are arbitrary constants; the ➅ ( ➋ ) and ➇ ( ➋ ➀ ) are the Bessel functions; and the ➈ ( ➋ ) and ➉

( ➋ ) are the modified Bessel functions.

,− ➌ < ❞ < ➌ . First boundary value problem. An infinite circular cylinder is considered. A boundary condition is prescribed: ❹

8.1.2-2. Domain: 0 ≤ ❼ ≤ ,0≤ ≤2

= ( , ❞ ) at

Solution: ❜

( → ) cos ➣ ↔ + ❄ ( → ) sin ➣ ↔ ↕ exp ➙ − ➛ ❄ ➜ | → − ➝ | ➞ ➟ →

( ➢ ➛ ) = 0. The functions ❄ ( ➝ ) and ❄ ➀ ( ➝ ) are the coefficients of the Fourier series expansion of ➤ ( ↔ , ➝ ),

where 0 =1 ➧ 2 and ❄ = 1 for ➥ ➣ ➥ = 1, 2, ➨✞➨✞➨

If the surface temperature is independent of ↔ , i.e., ➤ ( ➝ , ↔ )= ➤ ( ➝ ), then the solution takes the form

where the ➛ ➜ are positive roots of the transcendental equation ➠ 0 ( ➢ ➯❇➲ ➛ ) = 0.

Reference : H. S. Carslaw and J. C. Jaeger (1984).

8.1.2-3. Domain: 0 ≤ ➡ ≤ ➢ ,0≤ ↔ ≤2 ✹ ,0≤ ➝ ≤ ➳ . First boundary value problem.

A circular cylinder of finite length is considered. Boundary conditions are prescribed: ➩ ➩ ➩

= ➵ 2 ( ➡ , ↔ ) at ➝ = ➳ . Solution:

where the ➠ ❄ ( ➡ ) are the Bessel functions, the ❄ ( ➡ ) are the modified Bessel functions, and ➔ ➜ ❄

is the th root of the equation ( ➾ ) = 0. The coefficients

, and are defined by

( ↔ , ➝ ) cos( ➣ ↔ ) sin ➺ ➻ ➼

( , ) sin( ) sin

References : V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

For the solution of other boundary value problems for the three-dimensional Laplace equation in the cylindrical coordinate system, see Subsection 8.2.3 for ✃ ≡ 0.

8.1.3. Problems in Spherical Coordinates

The three-dimensional Laplace equation in the spherical coordinate system is written as ➩ ➩ ➩

8.1.3-1. Particular solutions: ➩

(cos Ï )( ➚ cos

+ sin ),

are arbitrary constants; the Õ Ô ( ) are the Legendre polynomials; and the Õ Ô ( → ) are the associated Legendre functions that are expressed as

8.1.3-2. Domain: 0 ≤ ➡ ≤ × or × ≤ ➡ < Ø . First boundary value problem.

A boundary condition at the sphere surface is prescribed: ➩

= ➤ ( Ï , ↔ ) at ➡ = × .

1 Ù . Solution of the inner problem (for ➡ ≤ × ):

cos Ú = cos Ï cos Ï 0 + sin Ï sin Ï 0 cos( ↔ − ↔ 0 ).

This formula is conventionally called the Poisson integral for a sphere. Series solution:

( Ô ➜ cos ↔ + Ô ➜ sin ❮ ↔ × ❰ ➼ Ó ➼ ) Õ (cos Ô Ï ),

(cos ) cos

(cos Ï ) sin ↔ sin Ï ➟ Ï ➟ ↔ .

2 Ù . Solution of the outer problem (for ➡ ≥ × ):

2 −2 × ➡ cos Ú + × 2 3 Ö 2 sin 0 0 0 ,

where cos Ú is expressed in the same way as in the inner problem.

Series solution:

, ➜ ↔ )= ( ➜ cos ↔ + ➜ sin ↔ ) (cos Ï ),

where the coefficients ➔

and Ó Ô ➜ are defined by the same relations as in the inner problem.

References : G. N. Polozhii (1964), V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), A. N. Tikhonov and A. A. Samarskii (1990).

8.1.3-3. Domain: 0 ≤ ➡ ≤ × or × ≤ ➡ < Ø . Second boundary value problem.

A boundary condition at the sphere surface is prescribed: ➩

= ➤ ( Ï , ➡ ↔ ) at ➡ = × .

The function ❒

( Ï , ↔ ) must satisfy the solvability condition

( Ï , ↔ ) sin Ï ➟ Ï ➟ ➑ ↔ = 0.

1 Ù . Solution of the inner problem (for ➡ ≤ × ):

1 = Ð ➡ 2 −2 × ➡ cos Ú + × 2 , cos Ú = cos Ï cos Ï 0 + sin Ï sin Ï 0 cos( ↔ − ↔ 0 ). Series solution:

where the coefficients ➔

are expressed in the same way as in the inner first boundary value problem (see Paragraph 8.1.3-2), and ➚ is an arbitrary constant.

and Ó Ô

2 Ù . Solution of the outer problem (for ➡ ≥ × ):

1 = Ð ➡ 2 −2 × ➡ cos Ú + × 2 , cos Ú = cos Ï cos Ï 0 + sin Ï sin Ï 0 cos( ↔ − ↔ 0 ). Series solution:

sin ↔ ) (cos Ï )+ ➚ ,

where the coefficients ➔

and Ó Ô ➜ are expressed in the same way as in the inner first boundary value problem, and ➚ is an arbitrary constant.

are sought are also encountered in applications.

3 Ù . Outer boundary value problems where unbounded solutions as ➡ ß

is governed by the Laplace equation with the boundary conditions:

Example.

A potential translational flow of an ideal incompressible fluid about a sphere of radius à

, where è is the potential, ä the unperturbed flow velocity at infinity; the fluid velocity is expressed in terms of the potential

as v = ∇ é . Solution:

2 ã 3 ë cos å .

This solution is a special case of the second formula from Paragraph for 8.1.3-1 for ì =1 .

References : G. N. Polozhii (1964), V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), L. G. Loitsyanskii (1996).

For the solution of other boundary value problems for the three-dimensional Laplace equation in the spherical coordinate system, see Subsection 8.2.4 for ✃ ≡ 0.

8.1.4. Other Orthogonal Curvilinear Systems of Coordinates

The three-dimensional Laplace equation admits separation of variables in the eleven orthogonal coordinate systems that are listed in Table 26 .

For the general ellipsoidal and conical coordinate systems, the functions ➤ , ➵ , and í are deter- mined by Lam´e equations that involve the Jacobian elliptic function sn ➝ = sn( ➝ , î ). The solutions of these equations under some conditions can be represented in the form of finite series called Lam ´e polynomials. For details about the Lam´e equation and its solutions, see Whittaker and Watson (1963), Bateman and Erd´elyi (1955), Arscott (1964), and Miller, Jr. (1977).

There are also coordinate systems that allow the so-called ï -separation of variables of the three-dimensional Laplace equation. Such solutions in the new coordinate system, , ð , ñ , can be represented in the form ò = ó ï

( ➾ , , ) ( ) ( ) ( ). Coordinates that allow the -separation of variables are listed in Table 27 .

Only the bicylindrical and toroidal coordinate systems are fairly widely used in applications. In three subsequent coordinate systems, the functions õ = õ ( ô ) and ö = ö ( ñ ) are determined by identical equations. With the change of variables ô = sn 2 ( ÷ , î ), ñ = sn 2 ( ø , î ), where î = ù −1 Ö 2 , these equations are reduced to Lam´e equations ( ÷ and ø are the new independent variables).

References for Subsection 8.1.4: M. Bˆocher (1894), F. M. Morse and H. Feshbach (1953, Vols. 1–2), N. N. Lebedev, I. P. Skal’skaya, and Ya. S. Uflyand (1955), P. Moon and D. Spencer (1961), A. Makarov, J. Smorodinsky, K. Valiev, and P. Winternitz (1967), W. Miller, Jr. (1977).