Heat Equation with Axial Symmetry, Nonhomogeneous -...
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EqWorld http://eqworld.ipmnet.ru Exact Solutions > Linear Partial Differential Equations > Second-Order Parabolic Partial Differential Equations > Nonhomogeneous Heat Equation with Axial Symmetry 1.5. Heat Equation of the Form ∂w ∂t = a ‡ ∂ 2 w ∂r 2 + 1 r ∂w ∂r · + Φ(r, t) Nonhomogeneous heat (diffusion) equation with axial symmetry. 1.5-1. Solutions of boundary value problems in terms of the Green’s function. We consider boundary value problems for the nonhomogeneous heat equation with axial symmetry in domain 0≤ r ≤ R with the general initial condition w = f (r) at t =0 and various homogeneous boundary conditions (the solutions bounded at r =0 are sought). The solution can be represented in terms of the Green’s function as w(x, t) = Z R 0 f (ξ )G(r, ξ , t) dξ + Z t 0 Z R 0 Φ(ξ , τ )G(r, ξ , t - τ ) dξ dτ . 1.5-2. Domain: 0 ≤ r ≤ R. First boundary value problem for the heat equation. A boundary condition is prescribed: w =0 at r = R. Green’s function: G(r, ξ , t) = ∞ X n=1 2ξ R 2 J 2 1 (μ n ) J 0 μ n r R ¶ J 0 μ n ξ R ¶ exp - aμ 2 n t R 2 ¶ , where the μ n are positive zeros of the Bessel function, J 0 (μ) =0. Below are the numerical values of the first ten roots: μ 1 =2.4048, μ 2 =5.5201, μ 3 =8.6537, μ 4 = 11.7915, μ 5 = 14.9309, μ 6 = 18.0711, μ 7 = 21.2116, μ 8 = 24.3525, μ 9 = 27.4935, μ 10 = 30.6346. The zeroes of the Bessel function J 0 (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. 1.5-3. Domain: 0 ≤ r ≤ R. Second boundary value problem for the heat equation. A boundary condition is prescribed: ∂w ∂r =0 at r = R. Green’s function: G(r, ξ , t) = 2 R 2 ξ + 2 R 2 ∞ X n=1 ξ J 2 0 (μ n ) J 0 μ n r R ¶ J 0 μ n ξ R ¶ exp - aμ 2 n t R 2 ¶ , where the μ n are positive zeros of the first-order Bessel function, J 1 (μ) =0. Below are the numerical values of the first ten roots: μ 1 =3.8317, μ 2 =7.0156, μ 3 = 10.1735, μ 4 = 13.3237, μ 5 = 16.4706, μ 6 = 19.6159, μ 7 = 22.7601, μ 8 = 25.9037, μ 9 = 29.0468, μ 10 = 32.1897. As n →∞, we have μ n+1 - μ n → π. 1
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Exact Solutions > Linear Partial Differential Equations > Second-Order Parabolic Partial DifferentialEquations > Nonhomogeneous Heat Equation with Axial Symmetry
1.5. Heat Equation of the Form ∂w∂t
= a(
∂2w∂r2 + 1
r∂w∂r
)+ Φ(r, t)
Nonhomogeneous heat (diffusion) equation with axial symmetry.
1.5-1. Solutions of boundary value problems in terms of the Green’s function.
We consider boundary value problems for the nonhomogeneous heat equation with axial symmetryin domain0 ≤ r ≤ R with the general initial condition
w = f (r) at t = 0
and various homogeneous boundary conditions (the solutions bounded atr = 0 are sought). Thesolution can be represented in terms of the Green’s function as
w(x, t) =∫ R
0f (ξ)G(r, ξ, t) dξ +
∫ t
0
∫ R
0Φ(ξ, τ )G(r, ξ, t − τ ) dξ dτ .
1.5-2. Domain: 0≤ r ≤ R. First boundary value problem for the heat equation.
A boundary condition is prescribed:
w = 0 at r = R.
Green’s function:
G(r, ξ, t) =∞∑
n=1
2ξ
R2J21 (µn)
J0
(µn
r
R
)J0
(µn
ξ
R
)exp
(−
aµ2nt
R2
),
where theµn are positive zeros of the Bessel function,J0(µ) = 0. Below are the numerical valuesof 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 zeroes of the Bessel functionJ0(µ) may be approximated by the formula
µn = 2.4 + 3.13(n − 1) (n = 1, 2, 3, . . .),
which is accurate within 0.3%. Asn →∞, we haveµn+1 − µn → π.
1.5-3. Domain: 0≤ r ≤ R. Second boundary value problem for the heat equation.
A boundary condition is prescribed:
∂w
∂r= 0 at r = R.
Green’s function:
G(r, ξ, t) =2
R2 ξ +2
R2
∞∑
n=1
ξ
J20 (µn)
J0
(µnr
R
)J0
(µnξ
R
)exp
(−
aµ2nt
R2
),
where theµn are positive zeros of the first-order Bessel function,J1(µ) = 0. Below are the numericalvalues 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 haveµn+1 − µn → π.
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2 NONHOMOGENEOUSHEAT EQUATION WITH AXIAL SYMMETRY
ReferencesCarslaw, H. S. and Jaeger, J. C.,Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC,
2002.
Nonhomogeneous Heat Equation with Axial Symmetry
Copyright c© 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/lpde/lpde105.pdf
