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Exact Solutions > Linear Partial Differential Equations >Second-Order Parabolic Partial Differential Equations > Nonhomogeneous Heat (Diffusion) Equation

1.2. Nonhomogeneous Heat Equation ∂w∂t

= a∂2w∂x2 + Φ(x, t)

1.2-1. Domain: –∞ < x < ∞. Cauchy problem for the nonhomogeneous heat equation.

An initial condition is prescribed:w = f (x) at t = 0.

Solution:

w(x, t) =∫ ∞

−∞f (ξ)G(x, ξ, t) dξ +

∫ t

0

∫ ∞

−∞Φ(ξ, τ )G(x, ξ, t − τ ) dξ dτ ,

where

G(x, ξ, t) =1

2√

πatexp

[−

(x − ξ)2

4at

].

1.2-2. Solutions of boundary value problems in terms of the Green’s function.

We consider boundary value problems for the heat equation* on an interval0 ≤ x ≤ l with the generalinitial condition

w = f (x) at t = 0

and various homogeneous boundary conditions. The solution can be represented in terms of theGreen’s function as

w(x, t) =∫ l

0f (ξ)G(x, ξ, t) dξ +

∫ t

0

∫ l

0Φ(ξ, τ )G(x, ξ, t − τ ) dξ dτ .

1.2-3. Domain: 0≤ x < ∞. First boundary value problem for the heat equation.

A boundary condition is prescribed:

w = 0 at x = 0.

Green’s function:

G(x, ξ, t) =1

2√

πat

exp

[−

(x − ξ)2

4at

]− exp

[−

(x + ξ)2

4at

].

1.2-4. Domain: 0≤ x < ∞. Second boundary value problem for the heat equation.

A boundary condition is prescribed:

∂w

∂x= 0 at x = 0.

Green’s function:

G(x, ξ, t) =1

2√

πat

exp

[−

(x − ξ)2

4at

]+ exp

[−

(x + ξ)2

4at

].

* Hereinafter we shell used the term “heat equation” to mean “nonhomogeneous heat equation”.

1

2 NONHOMOGENEOUSHEAT EQUATION

1.2-5. Domain: 0≤ x < ∞. Third boundary value problem for the heat equation.

A boundary condition is prescribed:

∂w

∂x− kw = 0 at x = 0.

Green’s function:

G(x, ξ, t) =1

2√

πat

exp

[−

(x − ξ)2

4at

]+ exp

[−

(x + ξ)2

4at

]− 2k

∫ ∞

0exp

[−

(x + ξ + η)2

4at− kη

]dη

.

1.2-6. Domain: 0≤ x ≤ l. First boundary value problem for the heat equation.

Boundary conditions are prescribed:

w = 0 at x = 0, w = 0 at x = l.

Two forms of representation of the Green’s function:

G(x, ξ, t) =2l

∞∑

n=1

sin

(nπx

l

)sin

(nπξ

l

)exp

(−

an2π2t

l2

)

=1

2√

πat

∞∑n=−∞

exp

[−

(x − ξ + 2nl)2

4at

]− exp

[−

(x + ξ + 2nl)2

4at

].

The first series converges rapidly at larget and the second series at smallt.

1.2-7. Domain: 0≤ x ≤ l. Second boundary value problem for the heat equation.

Boundary conditions are prescribed:

∂w

∂x= 0 at x = 0,

∂w

∂x= 0 at x = l.

Two forms of representation of the Green’s function:

G(x, ξ, t) =1l

+2l

∞∑

n=1

cos

(nπx

l

)cos

(nπξ

l

)exp

(−

an2π2t

l2

)

=1

2√

πat

∞∑n=−∞

exp

[−

(x − ξ + 2nl)2

4at

]+ exp

[−

(x + ξ + 2nl)2

4at

].

The first series converges rapidly at larget and the second series at smallt.

ReferencesBudak, B. M., Samarskii, A. A., and Tikhonov, A. N., Collection of Problems on Mathematical Physics[in Russian],

Nauka, Moscow, 1980.Butkovskiy, A. G., Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York,

1982.Carslaw, H. S. and Jaeger, J. C.,Conduction of Heat in Solids, Clarendon Press, Oxford, 1984.Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publ., New York, 1990.Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC,

2002.

Nonhomogeneous Heat (Diffusion) Equation

Copyright c© 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/lpde/lpde102.pdf