Heat Equation with Axial or Central Symmetry and

1.2. Heat Equation with Axial or Central Symmetry and

Related Equations

1.2.1. Equation of the Form ✳ ✴ = ✶ P ✳ ✴ 2 +1 ✳ ✴

This is a sourceless heat equation that describes one-dimensional unsteady thermal processes having ❘ axial symmetry. It is often represented in the equivalent form ❯

A similar equation is used for the analysis of the corresponding one-dimensional unsteady diffusion ✜ processes.

1.2.1-1. Particular solutions ( ✲

, ❯ , and ✰ are arbitrary constants).

where r is an arbitrary positive integer, ♥ 0 ( ❦ ) and ♦ 0 ( ❦ ) are the Bessel functions, and 0 ( ❦ ) and

0 ( ❦ ) are the modified Bessel functions. ❜ ❜

Suppose = ( ❝ , ❞ ) is a solution of the original equation. Then the functions

, ✉ , ✇ , , and t are arbitrary constants, are also solutions of this equation. The second ✈

④✧⑤ formula usually may be encountered with = 1, ③ = −1, and = = 0.

Reference : H. S. Carslaw and J. C. Jaeger (1984), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

1.2.1-2. Particular solutions in the form of an infinite series.

A solution containing an arbitrary function of the space variable:

where ⑥ ( ❝ ) is any infinitely differentiable function. This solution satisfies the initial condition

( ❝ , 0) = ⑥ ( ❝ ). The sum is finite if ⑥ ( ❝ ) is a polynomial that contains only even powers.

A solution containing an arbitrary function of time:

where ⑨ ( ❞ ) is any infinitely differentiable function. This solution is bounded at ❝ = 0 and possesses the properties

1.2.1-3. Domain: 0 ≤ ❝ ≤ ❸ . First boundary value problem. The following conditions are prescribed:

= ⑥ ( ❝ ) at ❞ =0

(initial condition),

= ⑨ ( ❞ ) at ❝ = ❸

(boundary condition),

| |≠ ❹ at ❝ =0 (boundedness condition). Solution:

are positive zeros of the Bessel function, ♥ 0 ( ♠ ) = 0. Below are the numerical values of the first ten roots:

where the ❺ ♠

6 = 18.0711, ♠ 7 = 21.2116, ♠ 8 = 24.3525, ♠ 9 = 27.4935, ♠ 10 = 30.6346. The zeroes of the Bessel function ♥ 0 ( ♠ ) may be approximated by the formula

= 2.4 + 3.13( r − 1)

( r = 1, 2, 3, ❨❃❨❃❨ ),

which is accurate within 0.3%. As r ➁ ❹ , we have ♠ ❬ +1 − ♠ ❬ ➁ ➂ .

Example 1. The initial temperature of the cylinder is uniform, ➃ ( ➄ )= ➅ 0 , and its lateral surface is maintained all the time at a constant temperature, ➆ ( ➇ )= ➅ ➈ . Solution:

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

1.2.1-4. Domain: 0 ≤ ❝ ≤ ❸ . Second boundary value problem. The following conditions are prescribed:

= ⑥ ( ❝ ) at ❞ =0

(initial condition),

= ⑨ ( ❞ ) at ❝ = ❸

(boundary condition),

| ❜ |≠ ❹ at ❝ =0 (boundedness condition). Solution:

where the are positive zeros of the first-order Bessel function, ➛ 1 ( ) = 0. Below are the numerical

values of the first ten roots: ↔

, we have +1 −

Example 2. The initial temperature of the cylinder is uniform, ↔ ↔ ➃ ( ➄ )= ➅ 0 . The lateral surface is maintained at constant thermal flux, ➆ ( ➇ )= ➆➝➈ . Solution:

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

1.2.1-5. Domain: 0 ≤ ≤ ➤ . Third boundary value problem. The following conditions are prescribed: ➔

( ➔ ) at =0

(initial condition),

= ➔ ➫ ( ) at = ➤

(boundary condition),

=0 (boundedness condition). Solution:

where the are positive roots of the transcendental equation

The numerical values of the first six roots ↔ can be found in Carslaw and Jaeger (1984).

Example 3. The initial temperature of the cylinder is uniform, ↔ ➃ ( ➽ )= ➾ 0 . The temperature of the environment is also uniform and is equal to ➾ ➚ , which corresponds to ➪ ( ➶ )= ➹ ➾ ➚ .

References : A. V. Lykov (1967), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-6. Domain: ➤ 1 ≤ ❰ ≤ ➤ 2 . First boundary value problem.

The following conditions are prescribed: ➙

= ➙ ➦ ( ❰ ) at

(initial condition),

= ➫ ➙ 1 ( ) at ❰ = ➤ 1 (boundary condition),

= ➫ 2 ( ) at ❰ = ➤ 2 (boundary condition).

) and Ú 0 ( Þ ) are the Bessel functions; the Õ Õ are positive roots of the transcendental equation

where ➲ 0 (

The numerical values of the first five roots

( Ö ) range in the interval 1.4 ≤ Ö ≤ 4.0 and can be found in Carslaw and Jaeger (1984). See also Abramowitz and Stegun (1964). Ô Ô

Example 4. The initial temperature of the hollow cylinder is zero, and its interior and exterior surfaces are held all the

time at constant temperatures, ➪ 1 ( ➶ )= ➾ 1 and ➪ 2 ( ➶ )= ➾ 2 .

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

1.2.1-7. Domain: ➤ 1 ≤ ❰ ≤ ➤ 2 . Second boundary value problem.

The following conditions are prescribed:

= ➦ ( ❰ ) at Ð =0

(initial condition),

= ➫ 1 ( Ð ) at ❰ = ➤ 1 (boundary condition),

= ➫ 2 ( Ð ) at ❰ = ➤ 2 (boundary condition). Solution:

) and Ú ( Þ ) are the Bessel functions ( ➩ Õ Õ = 0, 1), and the are positive roots of the transcendental equation

The numerical values of the first five roots Ô Ô = Ô ( Ö ) can be found in Abramowitz and Stegun Ô

References Ô : A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-8. Domain: ➤ 1 ≤ ❰ ≤ ➤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ➦ ( ❰ ) at Ð =0

(initial condition),

− ➥ ➩ 1 = ➫ 1 ( Ð ) at ❰ = ➤

1 (boundary condition),

+ ➥ 2 = ➫ 2 ( Ð ) at ❰ = ➤ 2 (boundary condition). Solution:

Here, ➲

( ➳ ) exp(− Ñ 2 Ð ),

1 ( ã ➤ 1 )] Ú ã 0 ( ❰ ), and the Õ are positive roots of the transcendental equation Õ ã ã ã ã ã

References : A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984). ã ã ã

1.2.1-9. Domain: 0 ≤ ❰ < ➭ . Cauchy type problem. This problem is encountered in the theory of diffusion wake behind a drop or a solid particle.

Given the initial condition

= ➦ ( ❰ ) at Ð = 0,

the equation has the following bounded solution:

exp Ø −

where ➡✧➢ 0 ( ➳ ) is the modified Bessel function.

References æ : W. G. L. Sutton (1943), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985).

1.2.1-10. Domain: ➤ ≤ ❰ < ➭ . Third boundary value problem. The following conditions are prescribed:

= ➦ ( ❰ ) at Ð =0

(initial condition),

− ➥ = ➫ ( Ð ) at ❰ = ➤ (boundary condition),

(boundedness condition). Solution:

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

1.2.2. Equation of the Form 2

This equation is encountered in plane problems of heat conduction with heat release (the function ø is proportional to the amount of heat released per unit time in the volume under consideration). The ú

equation describes one-dimensional unsteady thermal processes having axial symmetry.

1.2.2-1. Domain: 0 ≤ ❰ ≤ í . First boundary value problem. The following conditions are prescribed:

= ü ( ❰ ) at ý =0

(initial condition),

= þ ( ý ) at ❰ = í

(boundary condition),

| |≠ ÿ at ❰ =0 (boundedness condition). Solution:

are positive zeros of the Bessel function, ✍ 0 ( ✎ ) = 0. The numerical values of the first ten roots ✎

Here, the ✁

are given in Paragraph 1.2.1-3. 1.2.2-2. Domain: 0 ≤ ✌

≤ . Second boundary value problem.

The following conditions are prescribed:

= ü ( ✒ ) at ý =0

(initial condition),

= þ ( ý ) at ✒ =

(boundary condition),

| û |≠ ÿ at ✒ =0 (boundedness condition). Solution:

where the ✎ are positive zeros of the first-order Bessel function, ✍ 1 ( ✎ ) = 0. The numerical values of the first ten roots ✘✚✙ ✎

can be found in Paragraph 1.2.1-4.

References : A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

1.2.2-3. Domain: 0 ≤ ✌

≤ . Third boundary value problem.

The following conditions are prescribed:

= ü ( ✒ ) at ý =0

(initial condition),

+ ✛ û = þ ( ✓ ý ) at ✒ = (boundary condition),

at ✒ =0 (boundedness condition). Solution:

where the ✎ are positive roots of the transcendental equation

The numerical values of the first six roots ✘✚✙ ✎ can be found in Carslaw and Jaeger (1984).

References : A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

1.2.2-4. Domain: ✌ 1 ≤

≤ 2 . First boundary value problem.

The following conditions are prescribed:

= ü ( ✒ ) at ý =0

(initial condition),

1 ( ý ) at ✒ = 1 (boundary condition),

2 ( ý ) at ✒ = 2 (boundary condition).

where ✍ 0 ( ✪ ) and ✩ 0 ( ✪ ) are the Bessel functions, the ✎ are positive roots of the transcendental equation

The numerical values of the first five roots ✎ = ✎ ( ✥ ) can be found in Carslaw and Jaeger (1984).

1.2.2-5. Domain: ✌ 1 ≤

≤ 2 . Second boundary value problem.

The following conditions are prescribed:

= ü ( ✒ ) at ý =0

(initial condition),

= 1 ( ) at

= 1 (boundary condition),

= ✗ þ 2 ( ý ) at ✒ = 2 (boundary condition). Solution:

where ✍ ✮ ( ✪ ) and ✩ ✮ ( ✪ ) are the Bessel functions of order ✛ = 0, 1 and the ✎ are positive roots of the transcendental equation

The numerical values of the first five roots ✎ = ✎ ( ✥ ) can be found in Abramowitz and Stegun (1964).

1.2.2-6. Domain: ✌ 1 ≤

≤ 2 . Third boundary value problem.

The following conditions are prescribed: ✯

= ✰ ( ✒ ) at ✬ =0

(initial condition),

− 1 = 1 ( ) at

= 1 (boundary condition),

+ ✛ 2 = 2 ( ✬ ) at ✒ = 2 (boundary condition). The solution is given by the formula from Paragraph 1.2.1-8 with the additional term ✭

which takes into account the nonhomogeneity of the equation. ✁

1.2.2-7. Domain: 0 ≤ ✒ < ✱ . Cauchy type problem. The bounded solution of this equation subject to the initial condition ✯

= ✰ ( ✒ ) at ✬ =0

is given by the relations ✯

where ✲ ✘✚✙ 0 ( ✪ ) is the modified Bessel function.

References : W. G. L. Sutton (1943), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

1.2.2-8. Domain: ✌ ≤

< ✱ . Third boundary value problem.

The following conditions are prescribed: ✯

= ✰ ( ✒ ) at ✬ =0

(initial condition),

− ✛ ✯ = ( ✬ ) at ✒ = (boundary condition),

(boundedness condition). The solution is given by the formula from Paragraph 1.2.1-10 with the additional term

at ✒ ✳ ✱

which takes into account the nonhomogeneity of the equation. ✁

1.2.3. Equation of the Form ✴ ✵ = ✷ ✸ ✴ ✵ 2 +2 ✴ ✵

This is a sourceless heat equation that describes unsteady heat processes with central symmetry. It ✺ is often represented in the equivalent form ✯

A similar equation is used for the analysis of the corresponding one-dimensional unsteady diffusion ✓ processes.

1.2.3-1. Particular solutions ( ✯ , ✻ , and ✎ are arbitrary constants).

( ✒ , ✬ )= ✒ −1 exp(− ✆ ❅ 2 ✬ ) cos( ❅ ✒ )+ ✻ ,

( ✜ , )= ✒ −1 exp(− ✆ ❅ 2 ✬ ) sin( ❅ ✒ )+ ✻ ,

( ✜ ✒ , ✬ )= ✒ −1 exp(− ❅ ✒ ) cos( ❅ ✒ −2 ✆ ❅ 2 ✬ )+ ✻ ,

( ✒ , ✬ )= ✒ −1 exp(− ❅ ✒ ) sin( ❅ ✒ −2 ✆ ❅ 2 ✬ )+ ✻ ,

where ❁ is an arbitrary positive integer.

1.2.3-2. Reduction to a constant coefficient equation. Some formulas. ✯

1 ❊ . The substitution ❋ ( ✒ , ✬ )= ✒ ( ✒ , ✬ ) brings the original equation with variable coefficients to the constant coefficient equation

which is discussed in Subsection 1.1.1 in detail. ✓

2 ❊ . Suppose ✯ = ✯ ( ✒ , ✬ ) is a solution of the original equation. Then the functions

where ◆ , ❍ , ❏ , ■ , and ● are arbitrary constants, are also solutions of this equation. The second formula may usually be encountered with ❏ = 1, ▲ = −1, and ■ = ● = 0.

1.2.3-3. Infinite series particular solutions.

A solution containing an arbitrary function of the space variable:

A solution containing an arbitrary function of time:

where ❯ ( ❚ ) is any infinitely differentiable function. This solution is bounded at ❑ = 0 and possesses the properties

1.2.3-4. Domain: 0 ≤ ❑ ≤ ❳ . First boundary value problem. The following conditions are prescribed:

= ❘ ( ❑ ) at ❚ =0

(initial condition),

= ❯ ( ❚ ) at ❑ = ❳

(boundary condition),

| |≠ ❨ at ❑ =0 (boundedness condition). Solution:

( ❚ )= ❭ ❪ ❘ ( ) sin ❈ ❩ ❫ − (−1) ❴ ✆ ❁ ❭ ✝ ❯ ( ❵ ) exp ❈ ❩

Using the relation [see Prudnikov, Brychkov, and Marichev (1986)] ♠

we rewrite the solution in the form

) exp(− ✇❃✈ 2 s ) ①

( s )= ② ③ ( ) sin( ✈

( s ) exp( ✇❃✈ 2 ♠ s ) − (−1) ✇⑥♣ ♥ ② ⑧ ✈ t ( ⑨ ) exp( ✇❃✈ 2 ⑨ ) ⑦✦⑨

Example 1. The initial temperature is uniform,

( r )= 0 , and the surface of the sphere is maintained at constant .

temperature, t ( s )= q

The average temperature ③ q depends on time

as follows:

where ✉ is the volume of the sphere of radius .

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

1.2.3-5. Domain: 0 ≤ ❾ ≤ ❿ . Second boundary value problem. The following conditions are prescribed:

= ➁ ( ❾ ) at ➂ =0

(initial condition),

= ➅ ( ➂ ) at ❾ = ❿

(boundary condition),

| ➀ |≠ ➆ at ❾ =0 (boundedness condition). Solution:

Here, the

are positive roots of the transcendental equation tan − = 0. The first five roots are

Example 2. The initial temperature of the sphere is uniform, ➏ ➏ ➏ ( ➒ )= ➏ ➓ 0 . The thermal flux at the sphere surface is a ➏

maintained constant, ➔ ( → )= ➔↔➣ . Solution: ⑤

References : A. V. Lykov (1967), A. V. Bitsadze and D. F. Kalinichenko (1985).

1.2.3-6. Domain: 0 ≤ ➦ ≤ ➧ . Third boundary value problem. The following conditions are prescribed:

= ➩ ( ➦ ) at ➫ =0

(initial condition),

+ ➲ ➨ = ➳ ( ➫ ) at ➦ = ➧

(boundary condition),

at ➦ =0 (boundedness condition). Solution:

where the are positive roots of the transcendental equation ➘

cot ➹ + ➲ ➧ − 1 = 0.

The numerical values of the first six roots ➘ ➘ can be found in Carslaw and Jaeger (1984).

Example 3. The initial temperature of the sphere is uniform, ➘ ( ➒ )= ➓ 0 . The temperature of the ambient medium is zero, ➔ ( → ) = 0.

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

1.2.3-7. Domain: ➧ 1 ≤ ➦ ≤ ➧ 2 . First boundary value problem.

The following conditions are prescribed:

= ➩ ( ➦ ) at ➫ =0

(initial condition),

= ➳ 1 ( ➫ ) at ➦ = ➧ 1 (boundary condition),

= ➳ 2 ( ➫ ) at ➦ = ➧ 2 (boundary condition).

Example 4. The initial temperature is zero. The temperatures of the interior and exterior surfaces of the spherical layer

are maintained constants, ❰ 1 ( Ï )= Ð 1 and ❰ 2 ( Ï )= Ð 2 .

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

1.2.3-8. Domain: ➧ 1 ≤ ➦ ≤ ➧ 2 . Second boundary value problem.

The following conditions are prescribed:

= ➩ ( ➦ ) at ➫ =0

(initial condition),

= ➭ ➳ ➯ 1 ( ➫ ) at ➦ = ➧ 1 (boundary condition),

= ➳ 2 ( ➫ ) at ➦ = ➧ 2 (boundary condition). Solution:

( ➦ ) = sin[ Ù ( ➦ − ➧ 1 )] + ➧ 1 Ù cos[ Ù ( ➦ − ➧ 1 )],

where the Ù are positive roots of the transcendental equation ( 2 Ù ➧ 1 ➧ 2 + 1) tan[ Ù ( ➧ 2 − ➧ 1 )] − Ù ( ➧ 2 − ➧ 1 ) = 0.

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

1.2.3-9. Domain: ➧ 1 ≤ ➦ ≤ ➧ 2 . Third boundary value problem.

The following conditions are prescribed:

= ➩ ( ➦ ) at ➯ ➫ =0 (initial condition),

1 = ➯ ➳ 1 ( ➫ ) at ➦ = ➧ 1 (boundary condition),

+ ➨ ➲ 2 = ➳ 2 ( ➫ ) at ➦ = ➧ 2 (boundary condition). Solution:

( ➦ ➹ )= Ý 1 sin[ Ù ( ➦ − ➧ 1 )] + ➧ 1 Ù cos[ Ù ( ➦ − ➧ 1 )], Ý 1 = ➲ 1 ➧ 1 + 1, Ý 2 = ➲ 2 ➧ 2 − 1, where the Ù are positive roots of the transcendental equation ( Ý Ý − ➧ ➧

) sin[ Ù ( ➧ 2 − ➧ 1 )] + Ù ( ➧ 1 Ý 2 + ➧ 2 Ý 1 ) cos[ Ù ( ➧ 2 − ➧ 1 )] = 0.

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

1.2.3-10. Domain: 0 ≤ ➦ < ➵ . Cauchy type problem. The bounded solution of this equation subject to the initial condition

= ➩ ( ➦ ) at ➫ =0

has the form

Reference ➱ : A. G. Butkovskiy (1979).

1.2.3-11. Domain: ➧ ≤ ➦ < ➵ . First boundary value problem. The following conditions are prescribed:

= ➩ ( ➦ ) at ➫ =0

(initial condition),

= ➳ ( ➫ ) at ➦ = ➧

(boundary condition).

Example 5. The temperature of the ambient medium is uniform at the initial instant Ï = 0 and the boundary of the

domain is held at constant temperature, that is, å ( Ñ )= Ð 0 and ❰ ( Ï )= Ð æ .

Solution:

= Ò erfc ç Ñ

where erfc ê = 2 è Ô ë ì

exp(− í 2 ) î✦í is the error function.

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

1.2.4. Equation of the Form ñ ò = ô õ ñ ò 2 +2 ñ ò

This equation is encountered in heat conduction problems with heat release; the function ÷

is proportional to the amount of heat released per unit time in the volume under consideration. The equation describes one-dimensional unsteady thermal processes having central symmetry.

The substitution ú ( û , ä )= û ü ( û , ä ) brings the original nonhomogeneous equation with variable coefficients to the nonhomogeneous constant coefficient equation

which is considered in Subsection 1.1.2.

1.2.4-1. Domain: 0 ≤ û ≤ þ . First boundary value problem. The following conditions are prescribed:

= ÿ ( û ) at ä =0

(initial condition),

=( ä ) at û = þ

(boundary condition),

at û =0 (boundedness condition). Solution:

1.2.4-2. Domain: 0 ≤ û ≤ þ . Second boundary value problem. The following conditions are prescribed:

= ÿ ( û ) at ✝ =0

(initial condition),

=( ✝ ) at û = þ

(boundary condition),

at û =0 (boundedness condition). Solution:

Here, the ✌ are positive roots of the transcendental equation tan

− = 0. The values of the first five roots

can be found in Paragraph 1.2.3-5.

References ✗

: A. V. Lykov (1967), A. V. Bitsadze and D. F. Kalinichenko (1985).

1.2.4-3. Domain: 0 ≤ û ≤ þ . Third boundary value problem. The following conditions are prescribed:

= ÿ ( û ) at ✝ =0

(initial condition),

+ ✛ ü =( ✝ ) at û = þ

(boundary condition),

at û =0 (boundedness condition). Solution: ✂ ✄

where the ✘✚✙

are positive roots of the transcendental equation ✗ cot + ✛ þ − 1 = 0.

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

1.2.4-4. Domain: þ 1 ≤ û ≤ þ 2 . First boundary value problem.

The following conditions are prescribed:

= ÿ ( û ) at ✝ =0

(initial condition),

= 1 ( ✝ ) at û = þ 1 (boundary condition),

= 2 ( ✝ ) at û = þ 2 (boundary condition). Solution:

= 2 Reference : H. S. Carslaw and J. C. Jaeger (1984).

1.2.4-5. Domain: þ 1 ≤ û ≤ þ 2 . Second boundary value problem.

The following conditions are prescribed:

= ÿ ( û ) at ✝ =0

(initial condition),

= 1 ( ✝ ) at û = þ 1 (boundary condition),

= 2 ( ✝ ) at û = þ 2 (boundary condition). Solution:

✘✚✙ where the function ✆ ( û , ☎ , ✝ ) is the same as in Paragraph 1.2.3-8.

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

1.2.4-6. Domain: 1 ≤ ✁ ≤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ✄ ( ✁ ) at ☎ =0

(initial condition),

− ✂ ✞ 1 = ✟ 1 ( ☎ ) at ✁ = 1 (boundary condition),

+ ✞ ✂ 2 = ✟ 2 ( ☎ ) at ✁ = 2 (boundary condition). The solution is given by the formula of Paragraph 1.2.3-9 with the additional term

which takes into account the nonhomogeneity of the equation.

1.2.4-7. Domain: 0 ≤ ✁ < ✑ . Cauchy type problem. The bounded solution of this equation subject to the initial condition

= ✄ ( ✁ ) at ☎ =0

is given by

Reference : A. G. Butkovskiy (1979).

1.2.4-8. Domain: ≤ ✁ < ✑ . First boundary value problem. The following conditions are prescribed:

= ✄ ( ✁ ) at ☎ =0

(initial condition),

= ✟ ( ☎ ) at ✁ =

(boundary condition).

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

1.2.5. Equation of the Form ✦ ✧ = ✦ ✧

This dimensionless equation is encountered in problems of the diffusion boundary layer. For ✩

= 1 2 , or ✫ =− 1 2 , see the equations of Subsections 1.2.1, 1.1.1, or 1.2.3, respectively.

1.2.5-1. Particular solutions ( ✬ , ✭ , and ✮ are arbitrary constants).

where ❁ is an arbitrary positive integer, ( , ) is the incomplete gamma function, ( ) and ✰ ( ❅ ) are the Bessel functions, and ✰ ✚✜✛ ❂ ( ❅ ) and ❃ ✰ ( ❅ ) are the modified Bessel functions.

References : W. G. L. Sutton (1943), A. D. Polyanin (2001a).

1.2.5-2. Infinite series solutions.

A solution containing an arbitrary function of the space variable:

where ✄ ( ✯ ) is any infinitely differentiable function. This solution satisfies the initial condition

( ✯ , 0) = ✄ ( ✯ ). The sum is finite if ✄ ( ✯ ) is a polynomial that contains only even powers.

A solution containing an arbitrary function of time:

where ✟ ( ☎ ) any infinitely differentiable function. This solution is bounded at ✯ = 0 and possesses the properties

1.2.5-3. Formulas and transformations for constructing particular solutions.

Suppose ✂ = ( ✯ , ☎ ) is a solution of the original equation. Then the functions

where ✬ , ❊ , ✕ , ❋ , and ● are arbitrary constants, are also solutions of this equation. The second formula usually may be encountered with ✕ = ✞ = 0, ❋ = 1, and = −1.

) brings the equation with parameter ● ✫ to an equation of the same type with parameter − ✫ :

The substitution ✂ = ✯ 2 ✰ ❍ ( ✯ , ☎

1.2.5-4. Domain: 0 ≤ ✯ < ✑ . First boundary value problem. The following conditions are prescribed:

= ✄ ( ✯ ) at ☎ =0 (initial condition),

at ✯ =0 (boundary condition).

Solution for 0 < ✫ < 1:

Example. For ❏ ( ❑ )= ▲ 0 and ▼ ( ◆ )= ▲ 1 , where ❏ ( ❑ ) = const and ▼ ( ◆ ) = const, we have

Here, ( P , ❚

) is the incomplete gamma function and ❖ ( P ( P , ❭ ) is the gamma function.

Reference : W. G. L. Sutton (1943).

1.2.5-5. Domain: 0 ≤ ✯ < ❴ . Second boundary value problem. The following conditions are prescribed: ❵

= ❛ ( ❜ ) at ❝ =0 (initial condition),

= ❤ ( ❝ ) at ❜ =0 (boundary condition). Solution for 0 < ✐ ❵ < 1:

exp −

( ✍ ) exp s

− ✍ ) t ( ❝ − ✍ ) 1− ❡

1.2.5-6. Domain: 0 ≤ ❜ < ❴ . Third boundary value problem. The following conditions are prescribed: ❵

=0 at ❝ =0 (initial condition),

+ ✇ ( 0 − ) ① =0 at ❜ =0 (boundary condition). Solution for 0 < ❵ ✐ < 1: ❵

where the function ( ❝ ) is given as the series

which is convergent for any ❶✜❷ ⑥ .

Reference : W. G. L. Sutton (1943).

1.2.6. Equation of the Form

This equation is encountered in problems of a diffusion boundary layer with sources/sinks of ❻ substance. For ⑩ = 0, ⑩ = 1 2 , or ⑩ =− 1 2 , see the equations of Subsections 1.2.2, 1.1.2, or 1.2.4,

respectively. 1.2.6-1. Domain: 0 ≤ ❾ < ❿ . First boundary value problem.

The following conditions are prescribed:

= ➁ ( ❾ ) at ⑥ =0 (initial condition),

at ❾ =0 (boundary condition).

Solution for 0 < ⑩ < 1:

1.2.6-2. Domain: 0 ≤ ❾ < ❿ . Second boundary value problem. The following conditions are prescribed:

= ➁ ( ❾ ) at ⑥ =0 (initial condition),

= ➂ ( ⑥ ) at ❾ =0 (boundary condition). Solution for 0 < ⑩ < 1:

Reference for Subsection

1.2.6: W. G. L. Sutton (1943).