Equations of Special Form

1.9. Equations of Special Form

1.9.1. Equations of the Diffusion (Thermal) Boundary Layer

This equation is encountered in diffusion boundary layer problems (mass exchange of drops and ⑨ bubbles with flow).

The transformation ( ❸ and ❹ are any numbers)

where ❻ ( ❼ )= ❹ exp ❿ − ❺

leads to a constant coefficient equation, ➂✼➂ = , which is considered in Subsection 1.1.1.

References : V. G. Levich (1962), A. D. Polyanin and V. V. Dilman (1994), A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, ♥ ♥

and D. A. Kazenin (2001).

This equation is encountered in diffusion boundary layer problems with a first-order volume chemical ⑨ reaction (usually ❻ ≡ const).

The transformation ( ❸ and ❹ are any numbers)

, leads to a constant coefficient equation, = , which is

considered in Subsection 1.1.1.

Reference : Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985).

This equation is encountered in diffusion boundary layer problems (mass exchange of solid particles, ⑨ drops, and bubbles with flow).

The transformation ( ❸ and ❹ are any numbers)

, leads to a simpler equation of the form 1.3.6.6:

where ❻ ( )= ❹ exp ❿ − ❺ ❽

References : V. G. Levich (1962), A. D. Polyanin and V. V. Dilman (1994), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998), A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin (2001).

4. ⑦ s ➆

This is a generalization of the problem of thermal boundary layer on a flat plate. ⑨

1 ➇ . By passing from ❼ , ➈ to the new variables ♣ = ln ❼ , ➉ = ➈ ➊ ➋ ❼ , we arrive at the separable equation

There are particular solutions of the form

( → ➉ ), where the function ( ➉ ) satisfies the ordinary differential equation → ↔↕↔ → ↔ →

2 ➇ . The solution of the original equation with the boundary conditions

( 0 ➍ and 1 ➍ are some constants) is given by − 0 exp ➏ − ➠ ( ➉ ) ➑ ➍ ➉ ➍

= ➈ ➊ ➋ ❼ . It is assumed that the inequality ➉ ( ➉ )>2 ♦ ( ➉ ) holds for ➉ > 0.

3 ➇ . The equation of thermal boundary layer on a flat plate corresponds to ↔ ↔

( ➉ ) is Blasius’ solution in the problem of translational flow past a flat plate and Pr is the Prandtl number ( ❼ is the coordinate measured along the plate and ➈ is the transverse coordinate to the plate surface). In this case the formulas in Item 2 ➇ transform into Polhausen’s solution. See Schlichting (1981) for details. ➤⑥➥

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

For ➨

= 1, equations of this sort govern the concentration distribution in the internal region of the diffusion wake behind a moving particle or drop. The transformation ( ➳ and ➵ are any numbers)

where ➺ ( )= ➵ exp ➼ − ➸ ❽

leads to an equation of the form 1.2.5:

References ➌ : Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

The substitution ➍ ( ❼ , ➈

)= ➚ ( ❼ , ➈ ) exp ➼➪➸

leads to an equation of the form 1.9.1.5:

Reference : A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

The transformation ( ➨

and ➵ are any numbers)

, leads to the equation

which is considered in Subsection 1.2.5 (see also equations in Subsections 1.2.1 and 1.2.3).

1.9.2. One-Dimensional Schr ¨odinger Equation

1.9.2-1. Eigenvalue problem. Cauchy problem. Schr¨odinger’s equation is the basic equation of quantum mechanics;

is the wave function, ❰ Ï 2 = −1,

is Planck’s constant, is the mass of the particle, and ( ) is the potential energy of the particle in the force field.

1 Ò . In discrete spectrum problems, the particular solutions are sought in the form

where the eigenfunctions ❛ and the respective energies

have to be determined by solving the eigenvalue problem

The last relation is the normalizing condition for Ø .

. In the cases where the eigenfunctions ( Ù ) form an orthonormal basis in Ø à 2 ( á ), the solution of the Cauchy problem for Schr ¨odinger’s equation with the initial condition Ø

(2) is given by Ö

( Ù ) are considered below and particular solutions of the boundary value problem (1) or the Cauchy problem for Schr ¨odinger’s equation are presented. In some cases, ❛ ❛

Various potentials å

nonnormalized eigenfunctions ë ( Ù ) are given instead of normalized eigenfunctions ( Ù ); the former differ from the latter by a constant multiplier.

1.9.2-2. Free particle: ê ( Ù ) = 0. The solution of the Cauchy problem with the initial condition (2) is given by

Reference : W. Miller, Jr. (1977). ß

1.9.2-3. Linear potential (motion in a uniform external field): ê ( Ù )= Ù ó . Solution of the Cauchy problem with the initial condition (2):

See also Miller, Jr. (1977). ➺

2 Ù 1.9.2-4. Linear harmonic oscillator: 2 . Eigenvalues:

Normalized eigenfunctions:

) are the Hermite polynomials. ❛

The functions ✝ ✠ ( ) form an orthonormal basis in ✡ 2 ( ☛

References : S. G. Krein (1964), W. Miller, Jr. (1977), A. N. Tikhonov and A. A. Samarskii (1990).

1.9.2-5. Isotropic free particle: 2 ê ( )=

Here, the variable ✝ ≥ 0 plays the role of the radial coordinate, and

> 0. The equation with ( )= ó ☞

results from Schr ¨odinger’s equation for a free particle with ✁ space coordinates if one passes to spherical (cylindrical) coordinates and separates the angular variables. The solution of Schr ¨odinger’s equation satisfying the initial condition (2) has the form

exp − 2 ( + 1) sign

where ( ✚ ) is the Bessel function.

Reference ✖ ✗ : W. Miller, Jr. (1977).

1.9.2-6. Isotropic harmonic oscillator: ê ✝

Here, the variable ✝ ≥ 0 plays the role of the radial coordinate, and > 0. The equation with this

( ó ) results from Schr ¨odinger’s equation for a harmonic oscillator with ✁ space coordinates if one passes to spherical (cylindrical) coordinates and separates the angular variables.

Normalized eigenfunctions:

th generalized Laguerre polynomial with parameter . The norm | ✠ ( )| refers to the semiaxis ✗ ≥ 0.

where ✝ ✡ 2 ( ✤ ) is the ✁ ✎

The functions ❛

( ✝ ) form an orthonormal basis in ✡ 2 ( ☛ +

Reference : W. Miller, Jr. (1977).

− ✪ 1.9.2-7. Morse potential: ✫✭✬ ( )= 0 ( −2 ).

, ✚ ý , ) is the degenerate hypergeometric function. In this case the number of eigenvalues (energy levels)

and eigenfunctions is finite:

References : S. G. Krein (1964), L. D. Landau and E. M. Lifshitz (1974).

1.9.2-8. Potential with a hyperbolic function: ★ ( ✶ )=− ★

for even ✁ ,

, , − sinh ✵ for odd ✁ ,

where ✯ ( ,

, , ) is the hypergeometric function and =

The number of eigenvalues (energy levels) ❉

and eigenfunctions

is finite in this case:

= 0, 1, ✄☎✄☎✄ , ✁ max .

Reference : S. G. Krein (1964).

). Eigenvalues: ✵

1.9.2-9. Potential with a trigonometric function: 2 ★ ( ✶ )= ★ 0 cot ( ❊ ✶ ✼

for even ✁ ,

where ( ❍ , ❏ , ❇ , ✚ ) is the hypergeometric function. In particular, if ❍ = ❊ ■ ✳ ✼

2 , ★ 0 = 2, and ✁ = ❑ − 1, we have

References : S. G. Krein (1964), L. D. Landau and E. M. Lifshitz (1974).

Chapter 2

Parabolic Equations with Two Space Variables