Heat Equation with Source ❴ ❵ = ❜ ❝ 3 ( ❵ ❡ + ❞ , ❢ , ❣ , ❛ )
3.2. Heat Equation with Source ❴ ❵ = ❜ ❝ 3 ( ❵ ❡ + ❞ , ❢ , ❣ , ❛ )
3.2.1. Problems in Cartesian Coordinates ❛
In the Cartesian coordinate system, the three-dimensional heat equation with a volume source has the form
It describes three-dimensional unsteady thermal phenomena in quiescent media or solids with constant thermal diffusivity. A similar equation is used to study the corresponding three-dimensional mass transfer processes with constant diffusivity.
< < ✧ ,− ✧ < < ✧ ,− ✧ < ❪ < ✧ . Cauchy problem. An initial condition is prescribed:
3.2.1-1. Domain: − ♥
( , , ❪ ) at
References : A. G. Butkovskiy (1979).
3.2.1-2. Domain: 0 ≤ ♥ <
,− ➁ < < ➁ ,− ➁ < ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-4 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-5 with the additional term (1).
3 ➂ . The solution of the third boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-6 with the additional term (1). ❿❏➀
References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984).
3.2.1-3. Domain: − ♥
< < ➁ ,− ➁ < < ➁ ,0≤ ⑥ ≤ ➃ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for an infinite layer is given by the formula in Paragraph 3.1.1-7 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem for an infinite layer is given by the formula in Paragraph 3.1.1-8 with the additional term (2).
3 ➂ . The solution of the third boundary value problem for an infinite layer is given by the formula in Paragraph 3.1.1-9 with the additional term (2).
4 ➂ . The solution of a mixed boundary value problem for an infinite layer is given by the formula in Paragraph 3.1.1-10 with the additional term (2).
3.2.1-4. Domain: − ♥
< < ➁ ,0≤ < ➁ ,0≤ ⑥ ≤ ➃ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a semiinfinite layer is given by the formula in Paragraph 3.1.1-11 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem for a semiinfinite layer is given by the formula in Paragraph 3.1.1-12 with the additional term (3).
3 ➂ . The solution of the third boundary value problem for a semiinfinite layer is given by the formula in Paragraph 3.1.1-13 with the additional term (3).
4 ➂ . The solutions of mixed boundary value problems for a semiinfinite layer are given by the formulas in Paragraph 3.1.1-14 with additional terms of the form (3). ❿❏➀
References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984).
3.2.1-5. Domain: 0 ≤ ♥ <
,0≤ < ➁ ,0≤ ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for the first octant is given by the formula in Paragraph 3.1.1-15 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for the first octant is given by the formula in Paragraph 3.1.1-16 with the additional term (4).
3 ➂ . The solution of the third boundary value problem for the first octant is given by the formula in Paragraph 3.1.1-17 with the additional term (4).
4 ➂ . The solutions of mixed boundary value problems for the first octant are given by the formulas in Paragraph 3.1.1-18 with additional terms of the form (4).
3.2.1-6. Domain: 0 ≤ ♥ ≤
1 ,0≤ ≤ ➃ 2 ,− ➁ < ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem in an infinite rectangular domain is given by the formula in Paragraph 3.1.1-19 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem in an infinite rectangular domain is given by the formula in Paragraph 3.1.1-20 with the additional term (5).
3 ➂ . The solution of the third boundary value problem in an infinite rectangular domain is given by the formula in Paragraph 3.1.1-21 with the additional term (5).
4 ➂ . The solution of a mixed boundary value problem in an infinite rectangular domain is given by the formula in Paragraph 3.1.1-22 with the additional term (5).
3.2.1-7. Domain: 0 ≤ ♥ ≤
1 ,0≤ ≤ ➃ 2 ,0≤ ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem in a semiinfinite rectangular domain is given by the formula of Paragraph 3.1.1-23 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem in a semiinfinite rectangular domain is given by the formula in Paragraph 3.1.1-24 with the additional term (6).
3 ➂ . The solution of the third boundary value problem in a semiinfinite rectangular domain is given by the formula in Paragraph 3.1.1-25 with the additional term (6).
4 ➂ . The solutions of mixed boundary value problems in a semiinfinite rectangular domain are given by the formulas in Paragraph 3.1.1-26 with additional terms of the form (6).
r✂t
3.2.1-8. Domain: 0 ≤ ♥ ≤
1 ,0≤ ≤ ➃ 2 ,0≤ ⑥ ≤ ➃ 3 . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-27 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-28 with the additional term (7).
3 ➂ . The solution of the third boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-29 with the additional term (7).
4 ➂ . The solutions of mixed boundary value problems for a rectangular parallelepiped are given by ❿❏➀ the formulas in Paragraph 3.1.1-30 with additional terms of the form (7).
References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984).
3.2.2. Problems in Cylindrical Coordinates
In the cylindrical coordinate system, the heat equation with a volume source is written as ❤ ✐ ❤
2 + ( , , , ). This representation is used to describe nonsymmetric unsteady thermal (diffusion) processes in
quiescent media or solids bounded by cylindrical surfaces and planes.
3.2.2-1. Domain: 0 ≤ ➆ ≤
,0≤ ≤2 ❻ ,− ➁ < ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for an infinite circular cylinder is given by the formula in Paragraph 3.1.2-2 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem for an infinite circular cylinder is given by the formula in Paragraph 3.1.2-3 with the additional term (1).
3 ➂ . The solution of the third boundary value problem for an infinite circular cylinder is the sum of the solution presented in Paragraph 3.1.2-4 and expression (1).
3.2.2-2. Domain: 0 ≤ ➆ ≤
,0≤ ≤2 ❻ ,0≤ ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a semiinfinite circular cylinder is given by the formula in Paragraph 3.1.2-5 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a semiinfinite circular cylinder is given by the formula in Paragraph 3.1.2-6 with the additional term (2).
3 ➂ . The solution of the third boundary value problem for a semiinfinite circular cylinder is the sum of the solution presented in Paragraph 3.1.2-7 and expression (2).
4 ➂ . The solutions of mixed boundary value problems for a semiinfinite circular cylinder are given by the formulas in Paragraph 3.1.2-8 with additional terms of the form (2).
3.2.2-3. Domain: 0 ≤ ➆ ≤
,0≤ ≤2 ❻ ,0≤ ⑥ ≤ ➃ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a circular cylinder of finite length is given by the formula in Paragraph 3.1.2-9 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a circular cylinder of finite length is given by the formula in Paragraph 3.1.2-10 with the additional term (3).
3 ➂ . The solution of the third boundary value problem for a circular cylinder of finite length is the sum of the solution presented in Paragraph 3.1.2-11 and expression (3).
4 ➂ . The solutions of mixed boundary value problems for a circular cylinder of finite length are given by the formulas in Paragraph 3.1.2-12 with additional terms of the form (3).
3.2.2-4. Domain: ➆
1 ≤ ≤ ➇ 2 ,0≤ ≤2 ❻ ,− ➁ < ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for an infinite hollow cylinder is given by the formula in Paragraph 3.1.2-13 with the additional term
which allows for the equation’s nonhomogeneity. ➉
2 ➂ . The solution of the second boundary value problem for an infinite hollow cylinder is given by the formula in Paragraph 3.1.2-14 with the additional term (4).
3 ➂ . The solution of the third boundary value problem for an infinite hollow cylinder is the sum of the solution presented in Paragraph 3.1.2-15 and expression (4).
3.2.2-5. Domain: ➆
1 ≤ ≤ ➇ 2 ,0≤ ≤2 ❻ ,0≤ ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a semiinfinite hollow cylinder is given by the formula in Paragraph 3.1.2-16 with the additional term
which allows for the equation’s nonhomogeneity. ➉
2 ➂ . The solution of the second boundary value problem for a semiinfinite hollow cylinder is given by the formula in Paragraph 3.1.2-17 with the additional term (5).
3 ➂ . The solution of the third boundary value problem for a semiinfinite hollow cylinder is the sum of the solution presented in Paragraph 3.1.2-18 and expression (5).
4 ➂ . The solutions of mixed boundary value problems for a semiinfinite hollow cylinder are given by the formulas in Paragraph 3.1.2-19 with additional terms of the form (5).
r✂t
3.2.2-6. Domain: ➆
1 ≤ ≤ ➇ 2 ,0≤ ≤2 ❻ ,0≤ ⑥ ≤ ➃ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a hollow cylinder of finite length is given by the formula in Paragraph 3.1.2-20 with the additional term
which allows for the equation’s nonhomogeneity. ➉
2 ➂ . The solution of the second boundary value problem for a hollow cylinder of finite length is given by the formula in Paragraph 3.1.2-21 with the additional term (6).
3 ➂ . The solution of the third boundary value problem for a hollow cylinder of finite length is the sum of the solution specified in Paragraph 3.1.2-22 and expression (6).
4 ➂ . The solutions of mixed boundary value problems for a hollow cylinder of finite length are given by the formulas in Paragraph 3.1.2-23 with additional terms of the form (6).
3.2.2-7. Domain: 0 ≤ ➆ <
,0≤ ≤ 0 ,− ➁ < ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for an infinite wedge domain is given by the formula in Paragraph 3.1.2-24 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the second boundary value problem for an infinite wedge domain is given by the formula in Paragraph 3.1.2-25 with the additional term (7).
3.2.2-8. Domain: 0 ≤ ➆ <
,0≤ ≤ 0 ,0≤ ⑥ < ➁ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a semiinfinite wedge domain is given by the formula in Paragraph 3.1.2-26 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a semiinfinite wedge domain is given by the formula in Paragraph 3.1.2-27 with the additional term (8).
3 ➂ . The solutions of mixed boundary value problems for a semiinfinite wedge domain are given by the formulas in Paragraph 3.1.2-28 with additional terms of the form (8).
3.2.2-9. Domain: 0 ≤ ➆ <
,0≤ ≤ 0 ,0≤ ⑥ ≤ ➃ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a wedge domain of finite height is given by the formula in Paragraph 3.1.2-29 with the additional term
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a wedge domain of finite height is given by the formula in Paragraph 3.1.2-30 with the additional term (9).
3 ➂ . The solutions of mixed boundary value problems for a wedge domain of finite height are given by the formulas in Paragraph 3.1.2-31 with additional terms of the form (9).
3.2.2-10. Different boundary value problems for a cylindrical sector.
1 ➂ . The solution of the first boundary value problem for an unbounded cylindrical sector (0 ≤ ➆ ➆ ≤ ➇ , 0≤ ≤ 0 ,− ➁ < ⑥ < ➁ ) is given by the formula in Paragraph 3.1.2-32 with the additional term
which allows for the equation’s nonhomogeneity. ⑧
2 ➂ . The solution of the first boundary value problem for a semibounded cylindrical sector (0 ≤ ➆ ➆ ≤ ➇ , 0≤ ≤ 0 ,0≤ ⑥ < ➁ ) is given by the formula in Paragraph 3.1.2-33 with the additional term
which allows for the equation’s nonhomogeneity.
3 ➂ . The solution of the mixed boundary value problem for a semibounded cylindrical sector ➅ ➆ ➆
(0 ≤ ≤ ➇ ,0≤ ≤ 0 ,0≤ ⑥ < ➁ ) is given by the formula in Paragraph 3.1.2-34 with the additional term (10).
4 ➂ . The solution of the first boundary value problem for a cylindrical sector of finite height (0 ≤ ➆ ➆ ≤ ➇ , 0≤ ≤ 0 ,0≤ ⑥ ≤ ➃ ) is given by the formula in Paragraph 3.1.2-35 with the additional term
which allows for the equation’s nonhomogeneity.
3 ➂ . The solution of a mixed boundary value problem for a cylindrical sector of finite height is given by the formula in Paragraph 3.1.2-36 with the additional term (11).
3.2.3. Problems in Spherical Coordinates
In the spherical coordinate system, the heat equation with a volume source has the form
One-dimensional problems with central symmetry that have solutions of the form ❥ = ( , ) are discussed in Subsection 1.2.4.
3.2.3-1. Domain: 0 ≤ ➆ ≤
,0≤ ➋ ≤ ❻ ,0≤ ≤2 ❻ . Different boundary value problems.
1 ➂ . The solution of the first boundary value problem for a spherical domain is given by the formula in Paragraph 3.1.3-1 with the additional term
( 2 ⑨ , ⑩ , ❶ , ❺ ) ❷ ( , ➋ , , ⑨ , ⑩ , ❶ , − ❺ ) ⑨ sin ⑩ ❸ ⑨ ❸ ⑩ ❸ ❶ ❸ ❺ , (1)
which allows for the equation’s nonhomogeneity.
2 ➂ . The solution of the second boundary value problem for a spherical domain is given by the formula in Paragraph 3.1.3-2 with the additional term (1).
3 ➂ . The solution of the third boundary value problem for a spherical domain is the sum of the solution specified in Paragraph 3.1.3-3 and expression (1).
3.2.3-2. Domain: 1 ≤ ✁ ≤ 2 ,0≤ ✂ ≤ ✄ ,0≤ ☎ ≤2 ✄ . Different boundary value problems.
1 ✆ . The solution of the first boundary value problem for a spherical layer is given by the formula in Paragraph 3.1.3-4 with the additional term
2 ( ☛ , ☞ , ✌ , ✍ ) ✎ ( ✁ , ✂ , ☎ , ☛ , ☞ , ✌ , ✏ − ✍ 2 ) ☛ sin ☞ ✑ ☛ ✑ ☞ ✑ ✌ ✑ ✠ ✍ , (2)
which allows for the equation’s nonhomogeneity.
2 ✆ . The solution of the second boundary value problem for a spherical layer is given by the formula in Paragraph 3.1.3-5 with the additional term (2).
3 ✆ . The solution of the third boundary value problem for a spherical layer is the sum of the solution specified in Paragraph 3.1.3-6, and expression (2).
3.2.3-3. Domain: 0 ≤ ✁ < ✒ ,0≤ ✂ ≤ ✂ 0 ,0≤ ☎ ≤2 ✄ . First boundary value problem. The solution of the first boundary value problem for an infinite cone is given by the formula in
Paragraph 3.1.3-7 with the additional term
( ☛ , ☞ , ✌ , ✍ ) ✎ ( ✁ , ✂ , ☎ , ☛ , ☞ , ✌ , ✏ − ✍ ) ☛ 2 sin ☞ ✑ ☛ ✑ ☞ ✑ ✌ ✑ ✍ ,
which allows for the equation’s nonhomogeneity.