Problems in Spherical Coordinates ✼

3.1.3. Problems in Spherical Coordinates ✼

The heat equation in the spherical coordinate system has the form

This representation is convenient to describe three-dimensional heat and mass exchange phenomena in domains bounded by coordinate surfaces of the spherical coordinate system.

One-dimensional problems with central symmetry that have solutions of the form ★ = ( ✟ , ✍ ) are discussed in Subsection 1.2.3.

3.1.3-1. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . First boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

= ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸

(boundary condition).

× ❃ ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ),

Here, the ✾ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽

) are the associated Legendre functions ✘

expressed in terms of the Legendre polynomials ❃ ❃●✜ ❈ ( ✽ ✘ ) as follows: ✘

and the ❇ ✻ ❍❏■ are positive roots of the transcendental equation +1 2 ( ❇ ✸ ) = 0.

References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984). ✼

3.1.3-2. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Second boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

= ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸

(boundary condition).

× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ), for ❉ = 0,

Here, the ✼ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre functions (see Paragraph 3.1.3-1), and the ❇ ✻ ✼ are positive roots of the transcendental equation ✘ ✜ ✘ ✜

3.1.3-3. Domain: 0 ≤ ✟ ≤ ✸ ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Third boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

+ ❉ ★ = ✫ ( ✿ , ✠ , ✍ ) at ✟ = ✸ (boundary condition). The solution ★ ( ✟

, ✘ ✿ , ✠ , ✍ ) is determined by the formula in Paragraph 3.1.3-2 where

× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ◆ ( ✠ − ✌ )] exp(− ❇ ✻ ✑ ✍ ),

Here, the ✘ +1 ✼ 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre functions (see Paragraph 3.1.3-1), and the ❇ ✻ ✼ ✘ are positive roots of the transcendental equation ✜ ✘ ✜

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). ✼ ✾ ✼

3.1.3-4. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . First boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

= ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),

= ✫ 2 ( ✿ , ✛ ✠ ✛ , ✍ ) at ✟ = ✸ 2 (boundary condition). Solution:

× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ).

where the ✘ +1 2 (

) are the Bessel functions, the ❃ ❈ ✼ ( ✽ ) are the associated Legendre functions ✘ ✼

expressed in terms of the Legendre polynomials ❃ ❃●✜ ❈

( ) as follows: ✘ ✘

are positive roots of the transcendental equation ❙ ✘ ✜

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

3.1.3-5. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Second boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

= ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),

= 2 (boundary condition). Solution:

= ✛ 2 ( , , ) at

( 2 ☛ , ☞ , ✌ ) ( ✟ , ✿ , ✠ , ☛ , ☞ , ✌ , ✍ ) ☛ sin ☞ ✭ ☛ ✭ ☞ ✭ ✌

1 ( ☞ , ✌ , ✯ ) ( ✟ , ✿ , ✠ , ✸ 1 , ☞ , ✌ , ✍ − ✯ ) sin ☞ ✭ ☞ ✭ ✌ ✭ ✯

2 ( ☞ , ✌ , ✯ ) ( ✟ , ✿ , ✠ , ✸ 2 , ☞ , ✌ , ✍ − ✯ ) sin ☞ ✭ ☞ ✭ ✌ ✭ ✯ ,

× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ❉ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ). Here,

) and ❚ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ✼ ) are the associated Legendre functions (see Paragraph 3.1.3-4), and the ❙ ✼ ✘ ✜ ❇ ✻ are positive roots of the transcendental equation ❙ ✘ ✜

where the ✘ +1 2 (

can be expressed in terms of the Bessel functions and their derivatives; see Budak, Samarskii, and Tikhonov (1980).

The integrals that determine the coefficients ✾

3.1.3-6. Domain: ✸ 1 ≤ ✟ ≤ ✸ 2 ,0≤ ✿ ≤ ✏ ,0≤ ✠ ≤2 ✏ . Third boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

− ★ ❉ 1 = ✫ 1 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 1 (boundary condition),

+ ★ ❉ 2 = ✫ 2 ( ✿ , ✠ , ✍ ) at ✟ = ✸ 2 (boundary condition).

The solution ✘ (

, ✿ , ✠ , ✍ ) is determined by the formula in Paragraph 3.1.3-5 where ▲ ❙ ✘ ✜ ✘ ❙ ✘ ✜ ✘

× ❈ (cos ✿ ) ❈ (cos ☞ ) cos[ ◆ ( ✠ − ✌ )] exp(− ❇ 2 ✻ ✑ ✍ ). Here,

where the ✾ +1 2 ( ✟ ) and ❚ +1 2 ( ✟ ) are the Bessel functions, the ❈ ( ✽ ) are the associated Legendre ✼

functions (see Paragraph 3.1.3-4), and the ❙ ✘ ❇ ✻ ✼ ✜ are positive roots of the transcendental equation ❙ ✘ ✜

can be expressed in terms of the Bessel functions and their derivatives. ❍❏■

The integrals that determine the coefficients ✾

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

3.1.3-7. Domain: 0 ≤ ✟ < ✧ ,0≤ ✿ ≤ ✿ 0 ,0≤ ✠ ≤2 ✏ . First boundary value problem.

A cone is considered. The following conditions are prescribed:

= ✩ ( ✟ , ✿ , ✠ ) at ✍ =0

(initial condition),

= ✫ ( ✟ , ✠ , ✍ ) at ✿ = ✿ 0 (boundary condition). Solution: ✛

− ❆ ✻ (cos ✿ ) ❈ − ✻ (cos ☞ ) cos[ ❲ ( ✠ − ✌

( ✽ ) is the modified Legendre function expressed as ✜

, ❬ , ❭ ; ✽ ) is the Gaussian hypergeometric function and ( ❪ ) is the gamma function. The ✻

summation with respect to ❱ is performed over all roots of the equation ❈ − (cos ✿ 0 ) = 0 that are greater than −1 ❫ 2.

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