Equations with ß Space Variables

8.5. Equations with ß Space Variables

8.5.1. Laplace Equation à á â =0

The ➴ -dimensional Laplace equation in the rectangular Cartesian system of coordinates ã 1 , Ñ✘Ñ✘Ñ , ã q

has the form

2 q = 0.

For ➴ = 2 and ➴ = 3, see Subsections 7.1.1 and 8.1.1.

A regular solution of the Laplace equation is called a harmonic function. In what follows we use the notation: x = { ã 1 , Ñ✘Ñ✘Ñ , ã q } and |x| = ä ã 2 1 + ➶✘➶✘➶ + ã 2 q .

8.5.1-1. Particular solutions.

1 ➪ . Fundamental solution: å å

2 ➪ . Solution containing arbitrary functions of ➴ − 1 variables:

2 q r +1

( ã 1 , Ñ✘Ñ✘Ñ ã q

, r )= (−1) Ø ( ã 1 , Ñ✘Ñ✘Ñ , ã q

( ì ã 1 , Ñ✘Ñ✘Ñ , ã q −1 ) and ( ã 1 , Ñ✘Ñ✘Ñ , ã q −1 ) are arbitrary infinitely differentiable functions.

. Let ( ã 1 , Ñ✘Ñ✘Ñ , ã q ) be a harmonic function. Then the functions Ö Ö

( î ➥ ã 1 + ï 1 , Ñ✘Ñ✘Ñ , î ð ã q + ï q ),

2 , Ñ✘Ñ✘Ñ , 2 ò | , x| | x|

| q x| −2

are also harmonic functions everywhere they are defined; Ö Ò , ï 1 , Ñ✘Ñ✘Ñ , ï q , and ð are arbitrary constants. The signs at ó★ô ð in the expression of 1 can be taken independently of one another.

References : A. V. Bitsadze and D. F. Kalinichenko (1985), R. Courant and D. Hilbert (1989).

8.5.1-2. Domain: − õ < ã 1 < õ , ö✘ö✘ö ,− õ < ã q −1 < õ ,0≤ ã q < õ .

The first boundary value problem for an ÷ -dimensional half-space is considered. A boundary condition is prescribed:

= Ø ( ã 1 , ö✘ö✘ö , ã q −1 ) at ã q = 0.

( û 1 , ö✘ö✘ö , û q −1 ) û 1 ö✘ö✘ö✁ û q −1 ,

where ó★ô ✂ ( ✄ ) is the gamma function. ù ù

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

8.5.1-3. Domain: |x| ≤ 1. First boundary value problem.

A sphere of unit radius in the ÷ -dimensional space is considered. A boundary condition is prescribed:

= ÿ (x) for | x| = 1.

Solution (Poisson integral):

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

8.5.2. Other Equations

This is the Poisson equation in ÷ independent variables. For ÷ = 2 and ÷ = 3, see Sections 7.2 and 8.2.

−2 . ( ü 1 − û 1 ) 2 + ✓✔✓✔✓ +( ü q − û q ) 2 ✕ 2

Reference : S. G. Krein (1972).

2 ✍ . Domain: 0 ≤ ü r ≤ ✖ r ; ✗ = 1, ö✘ö✘ö , ÷ . First boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

= ÿ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = 0,

= ✘ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = ✖ r . Green’s function:

) sin( r ) ( 1 ,

sin( ✛ r 1 ü 1 ) sin( ✛ r 1 û 1 ) ö✘ö✘ö sin( ✛ r ✏ ü q

1 ✛ =1 r ✏ =1 r 1 + ✓✔✓✔✓ + ✛ r ✏

, ö✘ö✘ö , ✛ r ✏ =

3 ✍ . Domain: 0 ≤ ü r ≤ ✖ r ; ✗ = 1, ö✘ö✘ö , ✜ . Mixed boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

, −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = 0,

= ✘ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = ✖ r . Green’s function:

sin( ✛ r 1 ü 1 ) sin( ✛ r 1 û 1 ) ö✘ö✘ö sin( ✛ r ✏ ü q ) sin( ✛ r ✏ û q

( ü 1 , ö✘ö✘ö , ü q , û 1 , ö✘ö✘ö , û q

1 r 1 =0

=0 ✛ r 2 1 + ✓✔✓✔✓ + ✛ 2 r ✏

This is the Helmholtz equation in ✜ independent variables. For ✜ = 2 and ✜ = 3, see Sections 7.3 and 8.3.

1 ✍ . Fundamental solution for ★ = ✗ 2 > 0:

for even ✜ ,

where ✮ ✯ ( ✄ ) and ✫ ✯ ( ✄ ) are the Bessel functions.

2 ✍ . Domain: 0 ≤ ü r ≤ ✖ r ; ✗ = 1, ö✘ö✘ö , ✜ . First boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

= ÿ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = 0,

= ✘ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = ✖ r . Green’s function:

sin( ✛ r 1 ü 1 ) sin( ✛ r 1 û 1 ) ö✘ö✘ö sin( ✛ r ✏ ü q ) sin( ✛ r ✏ û q ) ( 1 ,

3 ✍ . Domain: 0 ≤ ü r ≤ ✖ r ; ✗ = 1, ö✘ö✘ö , ✜ . Second boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

= ÿ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = 0,

= ✘ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = ✖ r . Green’s function:

cos( 1 1 ) cos( 1 û 1 ) ö✘ö✘ö cos( ✛ r ✏ ü q ) cos( ✛ r ✏ û q ) ( 1 ,

, ✖ ✛ r 2 = ✖ , ö✘ö✘ö , ✛ r ✏ =

1 for = 0,

1 r 2 ✱✲✱✲✱ r

✖ ö✘ö✘ö✥✖ q

2 for ✶ ≠ 0.

4 ✍ . Domain: 0 ≤ ü r ≤ ✖ r ; ✗ = 1, ö✘ö✘ö , ✜ . Mixed boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed:

= ÿ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = 0,

= ✘ r ( ü 1 , ö✘ö✘ö , ü r −1 , ü r +1 , ö✘ö✘ö , ü q ) at ü r = ✖ r . Green’s function:

2 sin( r 1 ü 1 ) sin( ✛ r 1 û 1 ) ö✘ö✘ö sin( ✛ r ✏ ü q ) sin( ✛ r ✏ û q ) ( 1 ,

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

1 ✽ , ö✘ö✘ö It is assumed that for any real numbers ✽ û , û q the relation ✼

holds, where

is some positive constant. Fundamental solution:

1 , ö✘ö✘ö , ü q , û 1 , ö✘ö✘ö û q

✷★ô where is the determinant of the matrix A = { ✾ } and the ✾ are the entries of the inverse of A.

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

= 0 is set on the entire boundary of the domain. Eigenvalues and eigenfunctions:

1 ✍ . Case 0 < ❊

< 1. First boundary value problem. The condition ☎

where ❋ ✯ is the ✶

th positive root of the equation ✮ ✯ ( ) = 0, 1− ❊

2 ❚ . Case 1 ≤ ❊ < 2. Boundary conditions: the solution must be bounded at P ■ = 0, and the condition

= 0 must hold on the rest of the boundary of the domain. The eigenvalues and eigenfunctions of this problem are given by the relations of Item 1 ❚ with

Reference : M. M. Smirnov (1975).

Chapter 9

Higher-Order Partial Differential Equations