ABSTRACT
Let the domain be defined in R2 as D = {(x, t) : x ∈ [0, b], t ∈ [0, t∗)} and t∗ denote the end of the process. On the boundary of the domain, three components are distributed as (Hetmaniok et al. 2012)
0 = {(x, 0) : x ∈ [0, b]}, 1 = {(0, t) : t ∈ [0, t∗)}, 2 = {(b, t) : t ∈ [0, t∗)},
where the initial and boundary conditions are given. On the boundary 0 the uncertain initial condition is defined as
u˜(x, 0) = φ˜(x), x ∈ [0, b]. (9.2) On the boundary 1 the uncertain Dirichlet boundary condition is assumed as
u˜(0, t) = ψ˜(t), t ∈ [0, t∗). (9.3) In the uncertain inverse problem, the temperature distribution u˜(x, t) in region D along with uncertain temperature θ˜(t) and uncertain heat flux q˜(t) on boundary 2
2 boundary conditions as
u˜(b, t) = θ˜(t), t ∈ [0, t∗), (9.4) −k ∂u˜(b, t)
∂x = q˜(t), t ∈ [0, t∗). (9.5)
As per the single parametric form, we may write the above fuzzy inverse heat conduction Equation 9.1 as
∂
∂t [u(x, t; r), (u˜(x, t; r)] = a ∂
∂x2 [u(x, t; r), (u(x, t; r)], (9.6)
subject to the fuzzy initial condition [u(x, 0; r), (u(x, 0; r)] = [φ(x; r), (φ(x; r)]. Using the double parametric form (as discussed in Chapter 1), Equation 9.6 can
be expressed as
β
( ∂
∂t u(x, t; r)− ∂
∂t u(x, t; r)
) + ∂
∂t u(x, t; r)
= a
⎧⎪⎪⎪⎨ ⎪⎪⎪⎩
β
( ∂2
∂x2 u(x, t; r)− ∂
∂x2 u(x, t; r)
)
+ ∂ 2
∂x2 u(x, t; r)
⎫⎪⎪⎪⎬ ⎪⎪⎪⎭
, (9.7)
subject to the fuzzy initial condition
β ( u(x, 0; r)− (u(x, 0; r))+ u(x, 0; r) = β(φ(x; r)−φ(x; r))+φ(x; r),
where r and β ∈ [0, 1]. Let us now denote
β
( ∂
∂t u(x, t; r)− ∂
∂t u(x, t; r)
) + ∂
∂t u(x, t; r) = ∂
∂t u˜(x, t; r),
β
( ∂2
∂x2 u(x, t; r)− ∂
∂x2 u(x, t; r)
) + ∂
∂x2 u(x, t; r) = ∂
∂x2 u˜(x, t; r),
β(u(x, 0; r)− u(x, 0; r))+ u(x, 0; r) = u˜(x, 0; r, β), β(φ(x; r)−φ(x; r))+φ(x; r) = φ˜(x; r, β).
