For solutions of various boundary value problems, see Subsection T8.1.5 withΦr, t≡ 0.. Solutions of boundary value problems in terms of the Green’s function.. Second boundary value probl
Trang 1T8.1.2-8 Domain: 0 ≤x≤l Third boundary value problem (k1 >0, k2>0).
Boundary conditions are prescribed:
∂ x w – k1w=0 at x=0, ∂ x w + k2w=0 at x = l.
Green’s function:
G(x, ξ, t) =
∞
n=1
1
y n 2y n (x)y n (ξ) exp(–aμ2n t),
y n (x) = cos(μ n x) + μ k1
n sin(μ n x), y n 2= k2
2μ2
n
μ2
n + k21
μ2
n + k22
+ k1
2μ2
n
+ l 2
1+ k12
μ2
n
,
where μ nare positive roots of the transcendental equation tan(μl)
μ = k1+ k2
μ2– k1k2.
T8.1.3 Equation of the Form ∂w
∂t = a ∂
2w
∂x2 + b ∂w
∂x +cw+ Φ(x, t)
The substitution
w(x, t) = exp(βt + μx)u(x, t), β = c – b
2
4a, μ= – b
2a
leads to the nonhomogeneous heat equation
∂u
∂t = a ∂2u
∂x2 + exp(–βt – μx) Φ(x, t),
which is considered in Subsections T8.1.1 and T8.1.2
T8.1.4 Heat Equation with Axial Symmetry ∂w
∂t = a
∂2w
∂r2 + 1
r
∂w
∂r
This is a heat (diffusion) equation with axial symmetry, where r =
x2+ y2 is the radial
coordinate
T8.1.4-1 Particular solutions
w(r) = A + B ln r,
w(r, t) = A + B(r2+4at),
w(r, t) = A + B(r4+16atr2+32a2t2),
w(r, t) = A + B
r2n+n
k=1
4k [n(n –1) (n – k +1)]2
k r2n–2k
,
w(r, t) = A + B 4at ln r + r2ln r – r2
,
Trang 2w(r, t) = A + B exp(–aμ2t)J0(μr),
w(r, t) = A + B exp(–aμ2t)Y0(μr),
w(r, t) = A + B
t exp
–r
2+ μ2
4t
I0
μr
2t
,
w(r, t) = A + B
t exp
–r
2+ μ2
4t
K0
μr
2t
,
where A, B, and μ are arbitrary constants, n is an arbitrary positive integer, J0(z) and Y0(z) are Bessel functions, and I0(z) and K0(z) are modified Bessel functions.
T8.1.4-2 Formulas allowing the construction of particular solutions
Suppose w = w(r, t) is a solution of the original equation Then the functions
w1 = Aw( λr, λ2t + C) + B,
w2 = A
δ + βtexp
– βr
2
4a(δ + βt)
w
r
δ + βt,
γ + λt
δ + βt
, λδ – βγ =1,
where A, B, C, β, δ, and λ are arbitrary constants, are also solutions of this equation The second formula usually may be encountered with β =1, γ = –1, and δ = λ =0
T8.1.4-3 Boundary value problems
For solutions of various boundary value problems, see Subsection T8.1.5 withΦ(r, t)≡ 0
T8.1.5 Equation of the Form ∂w
∂t = a
∂2w
∂r2 + 1
r
∂w
∂r
+ Φ(r, t)
T8.1.5-1 Solutions of boundary value problems in terms of the Green’s function
We consider boundary value problems in domain0 ≤r≤Rwith the general initial condition
w = f (r) at t = 0 (T8.1.5.1)
and various homogeneous boundary conditions (the solutions bounded at r =0are sought) The solution can be represented in terms of the Green’s function as
w(x, t) =
R
0 f (ξ)G(r, ξ, t) dξ +
t 0
R
0 Φ(ξ, τ)G(r, ξ, t – τ) dξ dτ. (T8.1.5.2)
T8.1.5-2 Domain: 0 ≤r≤R First boundary value problem.
A boundary condition is prescribed:
w=0 at r = R.
Trang 3Green’s function:
G(r, ξ, t) =
∞
n=1
2ξ
R2J2
1(μ n)J0
μ n R r
J0
μ n R ξ
exp
–aμ2n t
R2
,
where μ n are positive zeros of the Bessel function, J0(μ) =0 Below are the numerical values of the first ten roots:
μ1 =2.4048, μ2=5.5201, μ3=8.6537, μ4=11.7915, μ5=14.9309,
μ6 =18.0711, μ7=21.2116, μ8=24.3525, μ9=27.4935, μ10=30.6346
The zeros of the Bessel function J0(μ) may be approximated by the formula
μ n=2.4+3.13(n –1) (n =1, 2, 3, ), which is accurate within 0.3% As n → ∞, we have μ n+1– μ n → π.
T8.1.5-3 Domain: 0 ≤r≤R Second boundary value problem.
A boundary condition is prescribed:
∂ r w=0 at r = R.
Green’s function:
G(r, ξ, t) = 2
R2ξ+
2
R2
∞
n=1
ξ
J2
0(μ n)J0
μ n r R
J0
μ n ξ R
exp
–aμ
2
n t
R2
,
where μ n are positive zeros of the first-order Bessel function, J1(μ) =0 Below are the numerical values of the first ten roots:
μ1 =3.8317, μ2=7.0156, μ3=10.1735, μ4=13.3237, μ5=16.4706,
μ6 =19.6159, μ7=22.7601, μ8=25.9037, μ9=29.0468, μ10=32.1897
As n → ∞, we have μ n+1– μ n → π.
T8.1.6 Heat Equation with Central Symmetry
∂w
∂t = a
∂2w
∂r2 +
2
r
∂w
∂r
This is the heat (diffusion) equation with central symmetry; r =
x2+ y2+ z2 is the radial
coordinate
T8.1.6-1 Particular solutions
w(r) = A + Br–1,
Trang 4w(r, t) = A + B
r2n+n
k=1
(2n+1)(2n) (2n–2k+2)
k r2n–2k
,
w(r, t) = A +2aBtr–1+ Br,
w(r, t) = Ar–1exp(aμ2t μr) + B,
w(r, t) = A + B
t3 2 exp
– r
2
4at
,
w(r, t) = A + B
r √
texp
– r2
4at
,
w(r, t) = Ar–1exp(–aμ2t) cos(μr + B) + C,
w(r, t) = Ar–1exp(–μr) cos(μr –2aμ2t + B) + C,
w(r, t) = A
r erf
r
2√ at
+ B, where A, B, C, and μ are arbitrary constants, and n is an arbitrary positive integer.
T8.1.6-2 Reduction to a constant coefficient equation Some formulas
1◦ The substitution u(r, t) = rw(r, t) brings the original equation with variable coefficients
to the constant coefficient equation ∂ t u = a∂ ww, which is discussed in Subsection T8.1.1
2◦ Suppose w = w(r, t) is a solution of the original equation Then the functions
w1= Aw( λr, λ2t + C) + B,
w2= |δ + βt A|3 2 exp
– βr
2
4a(δ + βt)
w
r
δ + βt,
γ + λt
δ + βt
, λδ – βγ =1,
where A, B, C, β, δ, and λ are arbitrary constants, are also solutions of this equation The second formula may usually be encountered with β =1, γ = –1, and δ = λ =0
T8.1.6-3 Boundary value problems
For solutions of various boundary value problems, see Subsection T8.1.7 withΦ(r, t)≡ 0
T8.1.7 Equation of the Form ∂w
∂t = a
∂2w
∂r2 + 2
r
∂w
∂r
+ Φ(r, t)
T8.1.7-1 Solutions of boundary value problems in terms of the Green’s function Solutions to boundary value problems on an interval 0 ≤ x ≤ R with the general initial condition (T8.1.5.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.1.5.2)
T8.1.7-2 Domain: 0 ≤r≤R First boundary value problem.
A boundary condition is prescribed:
w=0 at r = R.
Trang 5Green’s function:
G(r, ξ, t) = 2ξ
Rr
∞
n=1
sin
nπr R
sin
nπξ R
exp
–an
2π2t
R2
T8.1.7-3 Domain: 0 ≤r≤R Second boundary value problem.
A boundary condition is prescribed:
∂ r w=0 at r = R.
Green’s function:
G(r, ξ, t) = 3ξ2
R3 +
2ξ Rr
∞
n=1
μ2
n+1
μ2
n sin
μ n r
R
sin
μ n ξ
R
exp
–aμ
2
n t
R2
,
where μ n are positive roots of the transcendental equation tan μ – μ =0 The first five roots are
μ1=4.4934, μ2=7.7253, μ3 =10.9041, μ4=14.0662, μ5=17.2208
T8.1.8 Equation of the Form ∂w
∂t =
∂2w
∂x2 + 1 – 2β
x
∂w
∂x
This dimensionless equation is encountered in problems of the diffusion boundary layer
For β =0, β = 12, or β = –12, see the equations in Subsections T8.1.4, T8.1.1, or T8.1.6, respectively
T8.1.8-1 Particular solutions
w(x) = A + Bx2β,
w(x, t) = A +4(1– β)Bt + Bx2,
w(x, t) = A +16(2– β)(1– β)Bt2+8(2– β)Btx2+ Bx4,
w(x, t) = x2n+
n
p=1
4p
p! s n,p s n–β,p t
p x2 (n–p), s
q,p = q(q –1) (q – p +1),
w(x, t) = A +4(1+ β)Btx2β + Bx2β+2,
w(x, t) = A + Bt β–1exp
–x
2
4t
,
w(x, t) = A + B x
2β
t β+1 exp
–x
2
4t
,
w(x, t) = A + Bγ
β, x
2
4t
,
w(x, t) = A + B exp(–μ2t)x β J β (μx),
w(x, t) = A + B exp(–μ2t)x β Y β (μx),
w(x, t) = A + B x β
t exp
–x
2+ μ2
4t
Iβ
μx
2t
,
Trang 6where A, B, and μ are arbitrary constants, n is an arbitrary positive number, γ(β, z) is the incomplete gamma function, J β (z) and Y β (z) are Bessel functions, and I β (z) and K β (z)
are modified Bessel functions
T8.1.8-2 Formulas and transformations for constructing particular solutions
1◦ Suppose w = w(x, t) is a solution of the original equation Then the functions
w1 = Aw( λx, λ2t + C),
w2 = A|a + bt|β–1exp
– bx
2
4(a + bt)
w
x
a + bt,
c + kt
a + bt
, ak – bc =1,
where A, C, a, b, and c are arbitrary constants, are also solutions of this equation The second formula usually may be encountered with a = k =0, b =1, and c = –1
2◦ The substitution w = x2β u(x, t) brings the equation with parameter β to an equation
of the same type with parameter –β:
∂u
∂t = ∂
2u
∂x2 +
1+2β x
∂u
∂x
T8.1.8-3 Domain: 0 ≤x<∞ First boundary value problem.
The following initial and boundary conditions are prescribed:
w = f (x) at t =0, w = g(t) at x=0 Solution for0< β <1:
w(x, t) = x β
2t
∞
0 f (ξ)ξ
1 –βexp
–x
2+ ξ2
4t
I β
ξx
2t
dξ
2β
22β+1Γ(β +1)
t
0 g(τ ) exp
– x
2
4(t – τ )
dτ (t – τ )1+β.
T8.1.8-4 Domain: 0 ≤x<∞ Second boundary value problem.
The following initial and boundary conditions are prescribed:
w = f (x) at t=0, (x1–2β ∂ x w) = g(t) at x=0 Solution for0< β <1:
w(x, t) = x β
2t
∞
0 f (ξ)ξ
1 –βexp
–x
2+ ξ2
4t
I–β
ξx
2t
dξ
– 22β–1 Γ(1– β)
t
0 g(τ ) exp
– x
2
4(t – τ )
dτ (t – τ )1–β.
Trang 7T8.1.9 Equations of the Diffusion (Thermal) Boundary Layer
1. f (x) ∂w
∂x + g(x)y ∂w
∂y = ∂
2w
∂y2
This equation is encountered in diffusion boundary layer problems (mass exchange of drops and bubbles with flow)
The transformation (A and B are any numbers)
t=
h2(x)
f (x) dx + A, z = yh(x), where h(x) = B exp
–
g(x)
f (x) dx
,
leads to a constant coefficient equation, ∂ t w = ∂ zz w, which is considered in Subsection
T8.1.1
2. f (x)y n–1 ∂w
∂x + g(x)y n ∂w
∂y = ∂
2w
∂y2
This equation is encountered in diffusion boundary layer problems (mass exchange of solid particles, drops, and bubbles with flow)
The transformation (A and B are any numbers)
t= 1
4(n +1)2
h2(x)
f (x) dx + A, z = h(x)y
n+1
2 , where h(x) = B exp
–n+1 2
g(x)
f (x) dx
, leads to the simpler equation
∂w
∂t = ∂
2w
∂z2 +
1–2k z
∂w
∂z , k= 1
n+1, which is considered in Subsection T8.1.8
T8.1.10 Schr ¨ odinger Equation ih ∂w
∂t = –
h2
2m
∂2w
∂x2 +U (x)w
T8.1.10-1 Eigenvalue problem Cauchy problem
Schr¨odinger’s equation is the basic equation of quantum mechanics; w is the wave function,
i2= –1, is Planck’s constant, m is the mass of the particle, and U(x) is the potential energy
of the particle in the force field
1◦ In discrete spectrum problems, the particular solutions are sought in the form
w(x, t) = exp
–iE n
t
ψ n (x),
where the eigenfunctions ψ n and the respective energies E n have to be determined by solving the eigenvalue problem
d2ψ
n
dx2 +
2m
2
E n – U (x)
ψ n=0,
ψ n →0at x → ∞,
∞
|ψ n|2dx=1
(T8.1.10.1)
... functions, and I β (z) and K β (z)are modified Bessel functions
T8.1.8-2 Formulas and transformations for constructing particular solutions
1◦... encountered in diffusion boundary layer problems (mass exchange of solid particles, drops, and bubbles with flow)
The transformation (A and B are any numbers)
t=
4(n... basic equation of quantum mechanics; w is the wave function,
i2= –1, is Planck’s constant, m is the mass of the particle, and U(x) is the potential