Laplace Equation ❢ 2 ❣ =0

7.1. Laplace Equation ❢ 2 ❣ =0

The 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 steady-state 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 ❤ .

7.1.1. Problems in Cartesian Coordinate System

The Laplace equation with two space variables in the rectangular Cartesian system of coordinates is written as

7.1.1-1. Particular solutions and a method for their construction.

1 ▼ . Particular solutions: ❦ ❧

( ❦ , ❧ ) = exp( ♣ q ❦ )( ♠ cos q ❦ + ♥ sin q ❧ ),

( ❦ , ❧ )=( ♠ cos q ❦ + ♥ sin q ) exp( ❦ ♣ q ), ❧

( ❦ , ❧ )=( ♠ sinh q ❦ + ♥ cosh ❦ q )( ♦ cos ❧ q + ❤ sin q ❧ ),

, ❧ )=( cos ❦

+ sin ❧ )( ❧ sinh

, ♥ , ♦ , ❤ , 0 , 0 , and q are arbitrary constants.

2 ▼ . Fundamental solution:

tt

ln ,

. If ✐ ( , ) is a solution of the Laplace equation, then the functions

2 = ♠ ✐ ( cos ❦ ③ + sin ❧ ③ ,− sin ③ + cos ③ ),

are also solutions everywhere they are defined; ♠ , ♦ 1 , ♦ 2 , ③ , and ② are arbitrary constants. The signs at ② in ✐ 1 are taken independently of each other.

4 ① ❦ . A fairly general method for constructing particular solutions involves the following. Let ❧ ❦ ❧ ❦ ❧ ⑥ ( ⑦ )=

= + ⑨ ( and ⑩ are real functions of the real variables and ; ⑨ 2 = −1). Then the real and imaginary parts of ⑥ both satisfy the two-dimensional Laplace equation, ❷

, ⑧ ( , ) be any analytic function of the complex variable

Recall that the Cauchy–Riemann conditions

are necessary and sufficient conditions for the function ⑥ to be analytic. Thus, by specifying analytic functions ⑥ ( ⑦ ) and taking their real and imaginary parts, one obtains various solutions of the two-dimensional Laplace equation. ❸❭❬

References : M. A. Lavrent’ev and B. V. Shabat (1973), A. G. Sveshnikov and A. N. Tikhonov (1974), A. V. Bitsadze and D. F. Kalinichenko (1985).

7.1.1-2. Specific features of stating boundary value problems for the Laplace equation.

1 ① . For outer boundary value problems on the plane, it is (usually) required to set the additional condition that the solution of the Laplace equation must be bounded at infinity.

2 ① . The solution of the second boundary value problem is determined up to an arbitrary additive term.

3 ① . Let the second boundary value problem in a closed bounded domain ❤ with piecewise smooth boundary ❹

be characterized by the boundary condition*

= ⑥ (r) for r ❻ ❹ ,

where ❼ ❽ is the derivative along the (outward) normal to ❹ . The necessary and sufficient condition

of solvability of the problem has the form ❾ ❿

(r) ➁ ❹ = 0.

The same solvability condition occurs for the outer second boundary value problem if the domain is infinite but has a finite boundary. ❸❭❬

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

* More rigorously, ➊ must satisfy the Lyapunov condition [see Babich, Kapilevich, Mikhlin, et al. (1964) and Tikhonov and Samarskii (1990)].

7.1.1-3. Domain: − ➍ < ➎ < ➍ ,0≤ ➏ < ➍ . First boundary value problem.

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

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

7.1.1-4. Domain: − ➍ < ➎ < ➍ ,0≤ ➏ < ➍ . Second boundary value problem.

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

where ❸❭❬ ➙ is an arbitrary constant.

Reference : V. S. Vladimirov (1988).

7.1.1-5. Domain: 0 ≤ ➎ < ➍ ,0≤ ➏ < ➍ . First boundary value problem.

A quadrant of the plane is considered. Boundary conditions are prescribed:

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

7.1.1-6. Domain: − ➍ < ➎ < ➍ ,0≤ ➏ ≤ ➜ . First boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

= ⑥ 2 ( ➎ ) at ➏ = ➜ . Solution:

− ➑ cosh[ ( ➎ − ➓ ) ➝ ➜ ] − cos( ➏ ➝ ➜ )

+ ➓ sin

2 − cosh[ ( − ) ] + cos( )

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

7.1.1-7. Domain: − ➍ < ➎ < ➍ ,0≤ ➏ ≤ ➜ . Second boundary value problem. An infinite strip is considered. Boundary conditions are prescribed:

) ln ➑ ➞ cosh[ ( ➎ − ➓ ) ➝ ➜ ] − cos( ➏ ➝ ➜ ) ➟ ➁ ➓

) ln ➑ cosh[ ( − ) ] + cos( ➏ ➝ ➜ ) ➟ ➁ ➓ + ➙ ,

where ➙ is an arbitrary constant.

7.1.1-8. Domain: 0 ≤ ➎ < ➍ ,0≤ ➏ ≤ ➜ . First boundary value problem.

A semiinfinite strip is considered. Boundary conditions are prescribed:

) ➐ at ➎ = 0, ➐ = ⑥ 2 ( ➎ ) at ➏ = 0, = ⑥ 3 ( ➎ ) at ➏ = ➜ . Solution: ➒

cosh[ ( ➎ − ➓ ) ➝ ➜ ] − cos( ➏ ➝ ➜ ) cosh[ ( ➎ + ➓ ) ➝ ➜ ] − cos( ➏ ➝ ➜ ➦ ➧ 2 ( ) )

sin ➨

] + cos( ➑ ➏ ➝ ➜ ➦ ➧ 3 ( ) . )

cosh[ ( ➎ − ➓ ) ➝ ➜ ] + cos( ➏ ➝ ➜

) ➑ cosh[ ( ➎ + ➓ ) ➝ ➜

Example. Consider the first boundary value problem for the Laplace equation in a semiinfinite strip with ➫ 1 ( ➭ ) = 1 and

2 ( ➯ )= ➫ 3 ( ➯ ) = 0. Using the general formula and carrying out transformations, we obtain the solution

( ➯ , ➭ )= ➲ arctan ➳ sin(

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

7.1.1-9. Domain: 0 ≤ ➎ ≤ ➜ ,0≤ ➏ ≤ ➼ . First boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

) ➐ at ➎ = 0, = 2 ( ➏ ) at ➎ = ➜ ,

= ➧ 4 ( ➎ ) at ➏ = ➼ . Solution:

sinh ➾ ➢ ( ➜ − ➎ ) ➚ sin ➨ ➢ ➏

where the coefficients

, ➙ , and

are expressed as

References : M. M. Smirnov (1975), H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-10. Domain: 0 ≤ Ï

≤ ❒ ,0≤ Ð ≤ ➼ . Second boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

1 ( Ð ) at Ï = 0,

2 ( ) at

= ➧ 4 ( Ï ) at Ð = ➼ . Solution:

= 3 ( Ï ) at ➧ Ð = 0,

cosh Ø ➢ Ù ➬ ❒ ( − Ï ) Ú cos Û ➢ Ù ➬ Ð Ü + Ô

cosh Û ➢ ➱ Ù ➬ Ï Ü cos Û ➢ Ù ➬ Ð Ü

cos Û ➢ ❒ ➬ Ï Ü cosh Ø ➢ ❒ ➬ ( − Ð ) Ú + Ô

cos Û ➢ ❒ ➬ Ï Ü cosh Û ➢ ❒ ➬ Ð Ü ,

where ❐ is an arbitrary constant, and the coefficients , , , , , and are expressed as

The solvability condition for the problem in question has the form (see Paragraph 7.1.1-2, ➱

Item 3 ➤

7.1.1-11. Domain: 0 ≤ Ù

≤ ,0≤ Ð ≤ . Third boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

For the solution, see Paragraph 7.2.2-14 with Þ

7.1.1-12. Domain: 0 ≤ Ï

≤ . Mixed boundary value problems.

1 ß . A rectangle is considered. Boundary conditions are prescribed:

) at Ï = 0,

= â ( Ð ) at Ï

= ä ( Ï ) at Ð = . Solution:

cosh Ø ➬ Ù ➢ ( − Ï ) Ú sin Û ➬ ➢ Ù Ü +

cosh Û ➬ ➢ Ù Ü sin Û ➬ ➢ Ù Ü

cos ➤ Û ➱ ➬ ❒ ➢ Ü sinh Ø ➤ ➬ ❒ ➢ ( − Ð ) Ú + Ô

cos

sinh

Reference : M. M. Smirnov (1975).

2 ß . A rectangle is considered. Boundary conditions are prescribed:

) at Ï

= 0, ❒ = â ( Ð ) at Ï = ,

= ä ( Ï ) at Ð = , where (0) = ã (0).

sin Û sinh Û

Reference : M. M. Smirnov (1975).

7.1.2. Problems in Polar Coordinate System

The two-dimensional Laplace equation in the polar coordinate system is written as

7.1.2-1. Particular solutions:

= 1, 2, ö✄ö✄ö ; , , ô , and are arbitrary constants.

7.1.2-2. Domain: 0 ≤ ≤ ù or ù ≤ < ú . First boundary value problem. The condition

= ñ ( ) at

is set at the boundary of the circle; Þ

( ) is a given function.

. Solution of the inner problem ( ≤ ù ):

This formula is conventionally referred to as the Poisson integral. Solution of the outer problem in series form:

2 ß . Bounded solution of the outer problem ( ≥ ù ):

Bounded solution of the outer problem in series form:

, and ì are defined by the same relations as in the inner problem. In hydrodynamics and other applications, outer problems are sometimes encountered in which ð þ

where the coefficients Ù 0 ,

one has to consider unbounded solutions for

Example. The potential flow of an ideal (inviscid) incompressible fluid about a circular cylinder of radius ÿ with a constant incident velocity

at infinity is characterized by the following boundary conditions for the stream function:

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

7.1.2-3. Domain: 0 ≤ ≤ ù or ù ≤ < ú . Second boundary value problem. The condition

= ñ ( ) at

is set at the boundary of the circle. The function ñ ( ) must satisfy the solvability condition

1 ß . Solution of the inner problem ( ≤ ù ): ð

where ô is an arbitrary constant; this formula is known as the Dini integral. Series solution of the inner problem:

where ô is an arbitrary constant. ð

2 ✌ . Solution of the outer problem ( ≥ ù ):

where ô is an arbitrary constant. Series solution of the outer problem:

where the coefficient ì and ☛❁ì are defined by the same relations as in the inner problem, and ô is an arbitrary constant. ✎❭❬

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

7.1.2-4. Domain: 0 ≤ ≤ ù or ù ≤ < ú . Third boundary value problem. The condition

= ( ) at

is set at the circle boundary; ☞ ( ) is a given function.

. Solution of the inner problem ( ≤ ù ):

2 ✌ . Solution of the outer problem ( ≥ ✜ ):

where the coefficient

0 , ✢ , and are defined by the same relations as in the inner problem.

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

7.1.2-5. Domain: ✜ 1 ≤ ≤ ✜ 2 . First boundary value problem.

An annular domain is considered. Boundary conditions are prescribed: ð ð

where the coefficient ✧ 0 , ★ 0 , ✧ ✙ , ★ ✙ , ✩ ✙ , and ✪ ✙ are expressed as

and ☛ ( ✙ Here, the ✫ ) ñ ( ✬ = 1, 2) are the coefficients of the Fourier series expansions of the functions

Reference : M. M. Smirnov (1975).

7.1.2-6. Domain: ✜ 1 ≤ ≤ ✜ 2 . Second boundary value problem.

An annular domain is considered. Boundary conditions are prescribed: ✏ ✯ ð ✯ ð

Here, the coefficients ★ , ✧ ✙ , ★ ✙ , ✩ ✙ , and ✪ ✙ are expressed as

( ✜ 2 2 − ✜ 2 1 ) where the constants ✫ ( ✫ ) ☛ ( ) ✬

and ✙ ( = 1, 2) are defined by the same relations as in the first boundary value problem; ✰ ✱ ✲✴✳ ✵ ✶✸✷ ✹ is an arbitrary constant.

0 2 must hold; this relation is a consequence of the solvability condition for the problem,

Note that the condition (2)

TABLE 21 Two-dimensional Laplace operator in some curvilinear orthogonal systems of coordinates

Coordinates

Laplace operator, ✽ Transformation ( ✍ 2

2 Parabolic coordinates ✍ = , =

2 ✍ Elliptic coordinates ❄ ✼

= ✍ ✼ sinh sin ❇ 1 2 , ❇

= cosh cos ❇ , ❁

2 (sinh ❅ 2 + sin 2 ❇ ) ❃ ❄ 2 ❄ ❇ 2

2 Bipolar coordinates ✍

sinh ❈ sin

(cosh ❉ − cos ) 2 , ❉

cosh ❈ − cos ✛

cosh − cos

7.1.2-7. Domain: ✜ 1 ≤ ❊ ≤ ✜ 2 . Mixed boundary value problem.

An annular domain is considered. Boundary conditions are prescribed:

1 ✍ ( ● ) at ❊ = ✜ 1 , = ❋ 2 ( ● ) at ❊ = ✜ 2 . Solution: ❄

Here, the coefficients ✧ ✙ , ★ ✙ , ✩ ✙ , and ✪ ✙ are expressed as

( ✜ 2 + ✜ 1 ) where the constants ( ✙ ✫ )

and ■ ( ✙ ) ( ✬ = 1, 2) are defined by the same formulas as in the first boundary value problem. ❏✮✭

Reference : M. M. Smirnov (1975).

7.1.3. Other Coordinate Systems. Conformal Mappings Method

7.1.3-1. Parabolic, elliptic, and bipolar coordinate systems. In a number of applications, it is convenient to solve the Laplace equation in other orthogonal

systems of coordinates. Some of those commonly encountered are displayed in Table 21 . In all the coordinate systems presented, the Laplace equation ✍ ✽ 2 = 0 is reduced to the equation considered

in Paragraph 7.1.1-1 in detail (particular solutions and solutions to boundary value problems are given there).

The orthogonal transformations presented in Table 21 can be written in the language of complex variables as follows: + ✾ ✬❑❁ =− 1

2 ✬▲✼ ( + ✬❑✿ ) 2 (parabolic coordinates),

= ✼ cosh( + ✬❑❇ )

(elliptic coordinates),

+ ✬❑❁ = ✬▲✼ cot ▼ 1 2 ❈ ( + ✬❑❉ ) ◆

(bipolar coordinates).

The real parts, as well as the imaginary parts, in both sides of these relations must be equated to each other ( ✬ 2 = −1). Example. Plane hydrodynamic problems of potential flows of ideal (inviscid) incompressible fluid are reduced to the

Laplace equation for the stream function. In particular, the motion of an elliptic cylinder with semiaxes ◗ and ❘ at a velocity in the direction parallel to the major semiaxis ( ◗ > ❘ ) in ideal fluid is described by the stream function

❏✮❭ where ❙ and ❚ are the elliptic coordinates. References : G. Lamb (1945), J. Happel and H. Brenner (1965), G. Korn and T. Korn (1968).

7.1.3-2. Domain of arbitrary shape. Method of conformal mappings.

1 ✌ . Let ❪ = ❪ ( ❫ ) be an analytic function that defines a conformal mapping from the complex plane

= ❴ + ❵❑❛ into a complex plane ❪ = ❜ + ❵❑❝ , where ❜ = ❜ ( ❴ , ❛ ) and ❝ = ❝ ( ❴ , ❛ ) are new independent variables. With reference to the fact that the real and imaginary parts of an analytic function satisfy the Cauchy–Riemann conditions, we have ❜ = ❝ and ❜ =− ❝ , and hence

Therefore, the Laplace equation in the ❄ ❄ ❴ ❛ -plane transforms under a conformal mapping into the ❄ ❄

Laplace equation in the ❜ ❝ -plane.

2 ❥ . Any simply connected domain ❦ in the ❴ ❛ -plane with a piecewise smooth boundary can be mapped, with appropriate conformal mappings, onto the upper half-plane or into a unit circle in the

-plane. Consequently, a first and a second boundary value problem for the Laplace equation in ❦

can be reduced, respectively, to a first and a second boundary value problem for the upper half-space or a circle; such problems are considered in Subsections 7.1.1 and 7.1.2.

Subsection 7.2.4 presents conformal mappings of some domains onto the upper half-plane or

a unit circle. Moreover, examples of solving specific boundary value problems for the Poisson equation by the conformal mappings method are given there; the Green’s functions for a semicircle and a quadrant of a circle are obtained.

A large number of conformal mappings of various domains can be found, for example, in the references cited below. ❏✮❭

References : V. I. Lavrik and V. N. Savenkov (1970), M. A. Lavrent’ev and B. V. Shabat (1973), V. I. Ivanov and M. K. Trubetskov (1994).

7.1.3-3. Reduction of the two-dimensional Neumann problem to the Dirichlet problem. Let the position of any point ( ❴ ❧ , ❛ ❧ ) located on the boundary ♠

of a domain ❦

be specified by

a parameter ♥ , so that ❴ ❧ = ❴ ❧ ( ♥ ) and ❛ ❧ = ❛ ❧ ( ♥ ). Then a function of two variables, ❋ ( ❴ , ❛ ), is determined on ♠ by the parameter ♥ as well, ❋ ( ❴ , ❛ ) ❣♣♦ = ❋ ( ❴ ❧ ( ♥ ), ❛ ❧ ( ♥ )) = ❋ ❧ ( ♥ ). The solution of the two-dimensional Neumann problem for the Laplace equation q ❢ 2 = 0 in ❦

with the boundary condition of the second kind

= ❋ ❧ ( ♥ ) for r

can be expressed in terms of the solution of the two-dimensional Dirichlet problem for the Laplace ❄ r

equation q 2 ❜ = 0 in ❦ with the boundary condition of the first kind

= t ❧ ( ♥ ) for r s ♠ ,

where t ❧ ( ♥ )= ✉ ✈ ❧ ( ♥ ) ✇ ♥ , as follows:

Here, ( ❏✮❭ ❴ 0 , ❛ 0 ) are the coordinates of any point in ❞ ① ❦ , and ❡ ③ ① is an arbitrary constant.

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