Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 131

Solution. We fir s t deve l op W ϕ ( x , t ) , the solution of PDS (3.36) at f ( x , t )= 0. To

satisfy the boundary condition at x = 0, consider an odd continuation of ϕ ( x ) ,

⎨

− ϕ ( − x ) , x < 0 ,

Φ ( x )= 0 , x = 0 ,

⎩

ϕ ( x ) , x > 0 .

For this initial condition in the infinite domain, the solution is

+ ∞

u ( x , t )= V ( x , ξ , t ) Φ ( ξ ) d ξ .

− ∞

Therefore,

+ ∞ 0

u = W ϕ ( x , t )= V ( x , ξ , t ) Φ ( ξ ) d ξ + V ( x , ξ , t ) Φ ( ξ ) d ξ

− ∞

0

+ ∞ + ∞

= V ( x , ξ , t ) ϕ ( ξ ) d ξ − V ( x ,− ξ , t ) ϕ ( ξ ) d ξ

0 0

+ ∞

= ϕ ( ξ )[ V ( x , ξ , t ) − V ( x ,− ξ , t )] d ξ .

0

The solution of PDS (3.36) with ϕ ( x )= 0 is thus, by the solution structure theorem,

t t + ∞

u = W f τ ( x , t − τ ) d τ = d τ f ( ξ , τ )[ V ( x , ξ , t − τ ) − V ( x ,− ξ , t − τ )] d ξ .

0 0 0

The solution of PDS (3.36) is, by the principle of superposition,

t

u = W ϕ ( x , t )+ W f τ ( x , t − τ ) d τ

0

+ ∞

= ϕ ( ξ )[ V ( x , ξ , t ) − V ( x ,− ξ , t )] d ξ

0

t + ∞

+ d τ f ( ξ , τ )[ V ( x , ξ , t − τ ) − V ( x ,− ξ , t − τ )] d ξ .

0 0

Remark 1. The solution of PDS (3.36) at ϕ = 0 can also be obtained by a continu-

ation of nonhomogeneous term f ( x , t ) .