Variational formulation for a fractional heat transfer

Applied Mathematical Modelling 40 (2016) 9675–9691

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Variational formulation for a fractional heat transfer model in firefighter protective clothing Yue Yu a, Dinghua Xu a,∗, Yongzhi Steve Xu b, Qifeng Zhang a a b

Department of Mathematics, College of Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, Zhejiang Province, PR China Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

a r t i c l e

i n f o

Article history: Received 2 July 2015 Revised 11 April 2016 Accepted 18 May 2016 Available online 28 May 2016 Keywords: Fractional heat transfer Superdiffusion Riemann–Liouville derivative Variational formulation Conditional well-posedness Galerkin approximation

a b s t r a c t We propose a fractional heat transfer model to simulate the situation with high temperature and high humidity. The variational formulation for the fractional model is first proposed, which eliminates the singularity of homogeneous problem by multiplying a simple function factor. To prove the existence of weak solution of the variational problem, Galerkin approximation is utilized. The corresponding stability and uniqueness are also derived. To verify the reliability of the proposed model, numerical experiment is carried out, which indicates that our fractional model is appropriate for the heat transfer in firefighter protective clothing. © 2016 Elsevier Inc. All rights reserved.

1. Introduction We consider the heat transfer in firefighter protective clothing during a flash fire exposure [1–3]. The practical experience has taught us that this process will be very different from the case under low temperature, viz. the high heat and moisture make the transmission process much faster than the classical case under low temperature. In recent years, anomalous diffusion which deviates the classical Fickian diffusion has gained considerable attention, due mainly to its successful applications in science and engineering [4–6]. Anomalous diffusion is characterised through the power law form

x2 (t ) ∼ Kα t α , α = 1

(1)

and can be modeled by fractional partial differential equations, where x2 (t) denotes the mean squared displacement. According to the value of the diffusion coefficient α , transport process is distinguished as following relationship

subdiffusion/dispersive,

x2 (t ) ∼ t α normal diffusion, superdiffusion,

0 < α < 1, α = 1, α > 1.

Inspired by the model proposed by J.T. Fan et al. and the faster transmission of superdiffusion [4,7], we present a spatial fractional heat transfer model to describe the faster transmission process instead of the classical Fourier’s law. The Fan’s model has been considered extensively in [8–15], but yet we haven’t found the corresponding spatial fractional model. In ∗

Corresponding author. Tel.: +86 13666695270; fax: +86 57186843082. E-mail addresses: [email protected], [email protected] (D. Xu).

http://dx.doi.org/10.1016/j.apm.2016.05.035 0307-904X/© 2016 Elsevier Inc. All rights reserved.

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Y. Yu et al. / Applied Mathematical Modelling 40 (2016) 9675–9691

Nomenclature Cv FL FR T L

effective volumetric heat capacity of the fibrous batting (kJ · m−3 · K−1 ) total thermal radiation incident traveling to the left (kJ · m−2 · s−1 ) total thermal radiation incident traveling to the right (kJ · m−2 · s−1 ) temperature in fabrics (K) thickness of firefighter protective clothing (m)

Greek symbols κγ thermal conductivity of textile ξi surface emissivity of the inner and outer covering fabrics, i = 1, 2 β0 radiative sorption constant of the fibers (m−1 ) σ Boltzmann constant (kJ · m−2 · K−4 · s−1 ) ε0 porosity of the fibrous batting τ0 effective tortuosity of the fibrous batting the numerical simulation, the probable cause will be given, that is, the fractional model goes against the real situation under low temperature because of the faster transmission. The paper is organized as follows. In Section 2, a new model of dynamic heat transfer within textiles is introduced. The γ model replaces the second derivative by the famous Riemann–Liouville derivative D0+ with 1 < γ < 2. In Section 3, a related fractional ODE model which has been considered in [16,17] and some useful lemmas are presented. In Section 4, conditional well-posedness for the fractional heat transfer model is given, in which we first propose the variational formulation which has a similar formulation to the ODE model but with a skilled processing to eliminate the singularity of final equation. One can find that this definition of weak solution is also valid for Caputo derivative. To prove the existence of weak solution, the commonly used Galerkin approximation is applied. In the paper, we first only obtain the local stability on a subinterval [ε 1/2 , 1] ⊂ [0, 1], where ε is the level of perturbation in initial data or boundary value. The problem is Ho¨lder stable in L∞ (0, T; L2 (ε 1/2 , 1)) with the estimate

u1 (x, t ) − u2 (x, t ) 2L∞ (0,T;L2 (ε1/2 ,1)) ≤ C ε 2−γ , 1 < γ < 2, where u1 (x, t) and u2 (x, t) are exact and noised solutions, respectively. In fact, the stability is extended to the remaining interval [0, ε 1/2 ] later, which, however, gives a worse result than the one in the interval [ε 1/2 , 1] for the case of perturbation in boundary value. The uniqueness is deduced from stability result and holds in the whole interval [0, 1] almost everywhere. In Section 5, the shifted Gr u¨ nwald formula is adopted to approximate the Riemann–Liouville derivative and combines the so called implicit-explicit (IMEX) method, that is, we approximate the nonlinear source term (Tin+1 ) by replacing it with the corresponding value at previous time step (Tin ). This scheme is unconditionally stable and has convergence rate O(τ + h ) under some reasonable assumptions. Numerical simulation shows that the fractional model is appropriate to simulate the situation we considered. In the final section, some conclusions and prospects are given. 2. Fractional heat transfer model for firefighter protective clothing 2.1. Notations

2.2. The description of the heat transfer model We summarily adopt some assumptions presented in [1,2,8–12,15] for the dynamic heat transfer in the body-clothingenvironment system (Fig. 1). (A1) Protective can be treated as a porous medium and fibrous battings are isotropic in fiber arrangement and material properties; (A2) Volume changes of the fibers due to the variation of moisture and water content are neglected and effective tortuosity is a constant; (A3) Sorption and condensation are considered; (A4) Radiation absorption coefficients of different materials are regarded as constants; As illustrated in the introduction, the faster transmission process of the situation we considered in this paper forces us to introduce an extra superdiffusion assumption: (A5) Heat conduction within the porous batting are non-Fourier, and can be described by the superdiffusion model. According to CTRW (continuous time random walk) scheme, the following standard fractional differential equation [4,7] is derived

∂W μ = K μ −∞ Dx W (x, t ) ∂t

( 1 < μ < 2 ),

(2)


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Fig. 1. Schematic diagram of body-clothing-environment system.

μ

where W(x, t) is the pdf of being at a certain position x at time t (called the propagator) and −∞ Dx is Weyl operator which in one dimension is equivalent to the Riesz operator ∇ μ . μ Different from Eq. (2), the Weyl operator −∞ Dx is replaced by utilizing the Riemann–Liouville differential operator γ D0+ , 1 < γ < 2 in the following equations because we only consider the problem in a finite domain. In the numerical simulation of the fifth section, we observe that it is more appropriate to choose γ such that 1.5 < γ < 2 in the fractional differential equation and hence the heat transfer in firefighter protective clothing satisfies the superdiffusion law. Based on the physical and mathematical consideration, the temperature T(x, t) and heat radiation FL (x, t), FR (x, t) in firefighter protective clothing satisfy the following partial differential equations

⎧ ∂ FL (x, t ) ∂ FR (x, t ) γ ⎪ ⎪ Cv ∂∂Tt (x, t ) = κγ (D0+ T )(x, t ) + − + λ (x, t ) , (x, t ) ∈ × (0, T ), ⎪ ⎪

⎨ ∂ x ∂ x

heat conductivity

heat radiation

phase change

∂ FL (x,t ) ⎪ = β0 FL (x, t ) − β0 σ T 4 (x, t ), ⎪ ∂x ⎪ ⎪ ⎩ ∂ FR (x,t ) = −β0 FR (x, t ) + β0 σ T 4 (x, t ), ∂x

(3)

together with the initial condition

T (x, 0 ) = TI (x ),

x ∈ ,

(4)

the left boundary value conditions

T (0, t ) = T0 (t ),

t ∈ [0, T],

(1 − ξ1 )FL (0, t ) + ξ1 σ T (0, t ) = FR (0, t ), 4

(5)

and the right boundary value conditions

T (L, t ) = TL (t ),

t ∈ [0, T],

(1 − ξ2 )FR (L, t ) + ξ2 σ T (L, t ) = FL (L, t ), 4

(6)

where = (0, L ), L is the thickness of firefighter protective clothing, T is a preestablished time. Remark 2.1. It is obvious that the classical Fourier’s law is replaced by the fractional second constitutive relation

γ −1

q(x, t ) = −κγ D0+ T (x, t ), 1 < γ < 2, where q(x, t) is the heat flux due to conduction. Remark 2.2. Chitrphiromstri and Kuznetsov [1] presented a model of heat and moisture transport in firefighter protective clothing during a flash fire exposure, where heat radiation is modeled by Beer’s radiation attenuation model

qrad (x ) = qrad (0 )e−α x . Here, qrad (x) is the incident radiation heat flux from the flame onto the fabric and α is the extinction coefficient of the fabric. In contrast, the heat radiation in Fan’s model is simulated by two flux approximation. The two flux approximation is not limited to the optically thin or optically thick approximations and it is considered as the appropriate technique for the

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very thin fibrous insulation spacers in applications. The two flux approximation of the heat radiation modeled by Stefan– Boltzmann law is given by

∂ FL 4 ∂ x = β0 FL − β0 σ T (x, t ), ∂ FR 4 ∂ x = −β0 FR + β0 σ T (x, t )

with radiation boundary conditions

(1 − ξ1 )FL (0, t ) + ξ1 σ T 4 (0, t ) = FR (0, t ), 0 < t < T, (1 − ξ2 )FR (0, t ) + ξ2 σ T 4 (0, t ) = FL (0, t ), 0 < t < T.

The radiation conditions stand for the radiative heat transfer at the interface between the inner thin fabric and the fibrous batting and that between the outer thin fabric and the fibrous batting [9]. Remark 2.3. λ (x, t) in Eq. (3) describes the phase change in heat and moisture transfer, which is very complicated in Fan’s model. In Fan’s model, λ (x, t) is determined by an empirical equation under low temperature, which may not be valid for the situation we concern. In [1], the phase change is modeled by solid phase continuity equation and gas phase diffusivity equation. This thermodynamic process is much more complicated than Fan’s model and leads to strong coupling of heat and moisture. For simplicity, we only focus on the heat component in this paper. Therefore, in the following discussion, phase change, due to sorption and condensation, is neglected. The Riemann–Liouville fractional integrals and fractional derivatives are defined as follows. Definition 2.1 (Riemann–Liouville fractional integrals and fractional derivatives [18, pp. 69]). Let [a, b] be a finite interval α f of order α > 0 are defined by on the real axis R. The Riemann–Liouville fractional integrals Iaα+ f, Ib−

(Iaα+ f )(x ) =

1

(α )

f (s )

x

(x − s )1−α

a

ds

and α f )(x ) = (Ib−

1

(α )

b

x

f (s ) ds, (s − x )1−α

respectively. These integrals are called the left-sided and the right-sided fractional integrals. α The Riemann–Liouville fractional derivatives Dα a+ f, Db− f of order α ≥ 0 are defined by

(Dαa+ f )(x ) : = = and

=

n (Ian+−α f )(x ) = Dn (Ian+−α f )(x )

1

(n − α )

(Dαb− f )(x ) : =

d dx

d − dx

d dx

n

x

a

f (s ) ds ( n = [α ] + 1 ) (x − s )α−n+1

n n n −α n −α (Ib− f )(x ) = (−D ) (Ib− f )(x )

d 1 −

(n − α ) dx

n

b x

f (s ) ds ( n = [α ] + 1 ), (s − x )α−n+1

respectively.

3. Variational formulation for a fractional ODE model and some useful lemmas For any β ≥ 0, we denote Hβ (0, 1) to be the Sobolev space of order β on the unit interval (0, 1), and H˜ β (0, 1 ) to be the set of functions in Hβ (0, 1) whose extension by zero to R in H β (R ). The fractional derivatives are well-defined in fractional Sobolev spaces by the following lemma. β

β

Lemma 3.1 [17]. For any β ∈ (n − 1, n ), the operators D0+ u and D1− u defined for u ∈ C0∞ (0, 1 ) extend continuously to operators β

β

(still denoted by D0+ u and D1− u) from H˜ β (0, 1 ) to L2 (a, b):


9679

β β D0+ u, D1− u : H˜ β (0, 1 ) → L2 (0, 1 ),

that is,

β D0+ u 2

L ( 0,1 )

β D1− u 2

≤ C u H˜ β (0,1) ,

L ( 0,1 )

≤ C u H˜ β (0,1) .

B.T. Jin et al. [17] has considered the following model problem

−Dα0+ u(x ) = f (x ), x ∈ (0, 1 ),

(7)

u ( 0 ) = u ( 1 ) = 0, with α ∈ (1, 2), f ∈ L2 (0, 1). The deduced variational formulation is given by: To find u ∈ H˜ α /2 (0, 1 ) such that

A(u, v ) := − Dα0+/2 u, Dα1−/2 v = ( f, v ),

∀v ∈ H˜ α/2 (0, 1 ).

(8)

The following lemma shows that the bilinear form A(u, v ) is coercive on H˜ α /2 (0, 1 ). Lemma 3.2 [17]. Let α ∈ (1, 2), then there exists a positive constant c = c (α ) > 0 such that

A(v, v ) = − Dα0+/2 v, Dα1−/2 v ≥ c v 2H˜ α/2 (0,1) ,

∀v ∈ H˜ α/2 (0, 1 ).

Remark 3.1 [16]. For any real number s satisfying n − 1 ≤ s < n, if v ∈ C0∞ (R ), then

(−∞ Dsx v(x ), x Ds∞ v(x ))R = cos(π s ) −∞ Dsx v(x ) L2 (R) , 2

(−∞ Dsx v(x ), x Ds∞ v(x ))R = cos(π s ) x Ds∞ v(x ) L2 (R) . 2

The above equations yield

− Dα0+/2 v, Dα1−/2 v ≥ 0, 1 < α < 2,

∀v ∈ H˜ α/2 (0, 1 ).

α (L p ) defined for α To present the rules for fractional integration by parts, we use the spaces of functions Iaα+ (L p ) and Ib− > 0 and 1 ≤ p ≤ ∞ by

Iaα+ (L p ) = g(x ) : ∃ψ ∈ L p (a, b) s.t. g(x ) = (Iaα+ ψ )(x ) and

α (L p ) = g(x ) : ∃ψ ∈ L p (a, b) s.t. g(x ) = (Iα ψ )(x ) , Ib− b−

respectively. Lemma 3.3 (18, pp. 76). If α > 0, p, q ≥ 1, and

1 p

+

1 q

≤ 1 + α (p = 1 and q = 1 in the case

(a) If ϕ (x) ∈ Lp (a, b) and ψ (x) ∈ Lq (a, b), then

α ϕ . ϕ , Iaα+ ψ = ψ , Ib−

Here (·, ·) denotes the inner product in L2 (0, 1). α (L p ), g(x ) ∈ Iα (Lq ), then (b) If f (x ) ∈ Ib− a+

f, Dαa+ g = g, Dαb− f .

For 1 < α < 2, the space Hα /2 (0, 1) can be defined by [19]

H α /2 ( 0, 1 ) =

u ∈ L2 ( 0, 1 ) :

1 0

1 0

|u(x ) − u(y )|2 d x d y < ∞ , |x − y|1+α

which is endowed with the norm

1 / 2 u Hα/2 (0,1) = u 2L2 () + [u]22,α/2

and the inner product

(u, v )Hα/2 = (u, v )L2 +

1 0

1 0

(u(x ) − u(y ))(v(x ) − v(y )) dxdy, |x − y|1+α

where

[u]22,α /2 =

1 0

1 0

|u(x ) − u(y )|2 dxdy. |x − y|1+α

1 p

+

1 q

=1+α )

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For the space H˜ α /2 (0, 1 ), 1 < α < 2, the inner product can be presented as

(u, v )H˜ α/2 (0,1) =

1

0

1 0

(u(x ) − u(y ))(v(x ) − v(y )) dxdy. |x − y|1+α

The following lemma is important to construct an approximating sequence in H˜ α /2 (0, 1 ), 1 < α < 2. Lemma 3.4 [20–22]. There exists a smooth sequence {wk }∞ k=1 such that

˜ α /2 ( 0, 1 ), {w k }∞ k=1 is an orthogonal basis of H and 2 {w k }∞ k=1 is an orthogonal basis of L (0, 1 ).

This conclusion is commonly used in nonlocal analysis and fractional Laplacian. In fact, let

Aα u =

R\{0}

u (y ) − u (x ) dy, α ∈ (1, 2 ) |y − x|1+α

which is a special fractional Laplacian, then one can check that the operator Aα defined in H˜ α /2 (0, 1 ) is linear, symmetry and compact. Since H˜ α /2 (0, 1 ) is an infinite-dimensional Hilbert space, the Hilbert-schmidt theorem [23, pp. 72] indicates that all eigenvalues λj of Aα are real, and if they are ordered so that

|λn+1 | ≤ |λn |, one has

lim

n→∞

|λn | = 0.

Furthermore, the eigenvectors wj can be chosen such that they form an orthonormal basis for Range(Aα ). Note that Range(Aα ) = L2 (0, 1 ), hence the eigenvectors wj of Aα form an an orthonormal basis for L2 (0, 1). One can check that 0 is not an eigenvalue. Therefore, we have

and

−Aα wi , w j

L2

(w j (y ) − w j (x ))(wk (y ) − wk (x )) 1 dydx 2 R R |x − y|1+α 1 = (wi , w j )H˜ α/2 (0,1) 2 =

1

2 (wi , w j )H˜ α/2 (0,1) = 2 −Aα wi , w j L2 = 2 wi , w j = δi j , λi λi L2

where δ ij is the Kronecker symbol. 4. Conditional well-posedness for the fractional heat-transfer model 4.1. Existence of the weak solution Substituting the heat radiation functions FL and FR into the heat equations in (3), the problem (3)–(6) is decoupled to the following fractional equations with respect to the temperature T(x, t)

γ

Cv ∂∂Tt = κγ D0+ T + ∗ (T ), (x, t ) ∈ × (0, T ), ¯, T (x, 0 ) = TI (x ), x ∈ T (0, t ) = T0 (t ), T (L, t ) = TL (t ), t ∈ [0, T],

where

∗ (T ) = −β 2 σ eβ x c1 =

x 0

e−β y T 4 (y, t )dy + c2 + β 2 σ e−β x

ξ1 4 T ( 0 , t ) − ( 1 − ξ1 ) c 2 , β

x 0

eβ y T 4 (y, t )dy + c1 − 2βσ T 4 (x, t ),

L 1 −β L c2 = ( 1 − ξ ) β e eβ x T 4 (x, t )dx 2 ( 1 − ξ2 ) β ( 1 − ξ 1 ) e − β L − β e β L 0 L β L −β x 4 −β L 4 4 + βe e T (x, t )dx + (1 − ξ2 )ξ1 e T (0, t ) + ξ2 T (L, t ) . 0


9681

Noting that the source term consists of the following terms

T 4 (x, t ),

L

0

eβ y T 4 (y, t )dy,

L 0

e−β y T 4 (y, t )dy,

x 0

eβ y T 4 (y, t )dy,

0

x

e−β y T 4 (y, t )dy,

¯ × [0, T] ∩ L2 0, T; L2 () → L2 0, T; L2 () satisfies one can check that ∗ : C

xγ ∗ (T1 ) − xγ ∗ (T2 ) L2 (0,T;L2 ()) ≤ C xγ T1 − xγ T2 L2 (0,T;L2 ())

(9)

with a constant C > 0. 4.1.1. Variational formulation Without loss of generality, we set L = 1. To derive the variational formulation as the classical case, we should homogenize the boundary value condition to obtain

γ

ut = c2 D0+ u + (u ), (x, t ) ∈ × (0, T ), ¯, u(x, 0 ) = u0 (x ), x ∈ u(0, t ) = 0, u(L, t ) = 0, t ∈ [0, T].

(10)

This motivation will be confirmed later. Here,

u(x, t ) = T (x, t ) − h(x, t ), TL (t ) − T0 (t ) h(x, t ) = T0 (t ) + x, L TL (0 ) − T0 (0 ) u0 (x ) = TI (x ) − T0 (0 ) − x, L 2 c = κγ /Cv ,

(u ) =

T (t ) − T0 (t ) 1 ∗ 1 TL (t ) − T0 (t ) 1

(u + h ) − T0 (t ) − L x + c2 T0 (t ) x −γ + c 2 x 1 −γ , Cv L

(1 − γ ) L

(2 − γ )

where we have used the following equalities γ

D0+ 1 =

1

(1 − γ )

γ

x−γ , D0+ x =

1

(2 − γ )

x 1 −γ .

For simplicity, we set

T (t ) − T 0 (t ) 1 ∗

(u + h ), 2 = −T 0 (t ) − L x, Cv L 1 TL (t ) − T0 (t ) 1

3 = c2 T0 (t ) x − γ , 4 = c 2 x 1 −γ .

(1 − γ ) L

(2 − γ )

1 ( u ) =

γ γ γ Since 3 is not bounded at x = 0, it is desired to multiply 3 by xγ . Note that if u ∈ I0+ (L2 ), v ∈ I1− (L2 ), then (D0+ u, v ) =

(u, Dγ1− v ) (Lemma 3.3). It is obvious that γ

u(0, t ) = lim+ u(x, t ) = lim+ I0+ f = 0 x→0

x→0

and

v(1, t ) = lim− v(x, t ) = lim− I1γ− f = 0, x→1

x→1

which justifies the above motivation. We observe that the inner product ( (u), v) does not make sense since the less regularity of the nonlinear term. To γ obtain the variational formulation, we might as well assume that the test function makes it valid, for example v ∈ xγ I1− (L2 ), γ γ where xγ I1− (L2 ) = {xγ f : f ∈ I1− (L2 )}. From the following equality

(ut , v ) = c2 (Dγ0+ u, v ) + ( (u ), v ), ∀v ∈ xγ I1γ− (L2 ), we obtain

(ut , v ) = c2 (u, Dγ1− v ) + ( (u ), v ), ∀v ∈ xγ I1γ− (L2 ) or

1 0

γ

γ

ut v − c2 uD1− v − (u )v dx = 0, ∀v ∈ xγ I1− (L2 ).

γ For smooth test functions belonging to xγ I1− (L2 ), one can obtain γ

ut v = c2 uD1− v + (u )v

9682


γ

γ

by the arbitrariness of v. In fact, we can recover the equality (ut , v ) = c2 (D0+ u, v ) + ( (u ), v ) when u ∈ I0+ (L2 ), v ∈ γ xγ I

1−

( L2 ).

Hence for any ρ (x),

(ρ (x )ut , v) = c2 ρ (x )u, Dγ1− v + (ρ (x ) (u ), v).

Similarly, it follows that

(ρ (x )ut , v ) = c2 Dγ0+ (ρ (x )u ), v + (ρ (x ) (u ), v ) γ

γ

when ρ (x )u ∈ I0+ (L2 ), v ∈ I1− (L2 ).

γ γ Choosing ρ (x ) = xγ , one can check that xγ u ∈ I0+ (L2 ) when u ∈ I0+ (L2 ). Let u˜ (x, t ) = xγ u. We have

(u˜t , v ) = c2 Dγ0+ u˜, v + (xγ (u ), v).

We notice that the test function v in above equation can be chosen in a larger range due to the multiplier xγ . We thus deduce the following definition of weak solution in variational formulation from Eq. (8). Definition 4.1 (Weak solution). We call u(x, t) a weak solution of (10) if there exists u˜ = xγ u(x,

(u˜t , v ) = c2 Dγ0+/2 u˜, Dγ1−/2 v + (xγ (u ), v), u˜ (x, 0 ) = u˜0 (x )

· ) ∈ H˜ γ /2 (0, 1 ) such that

∀v(x ) ∈ H˜ γ /2 (0, 1 ),

where u˜0 (x ) = xγ u0 (x ). Remark 4.1. The weak solution u defined above requires u˜ = xγ u ∈ H˜ γ /2 (0, 1 ). We, however, can not deduce that u = x−γ u˜ ∈ H˜ γ /2 (0, 1 ). Consider the following steady problem [18, pp. 181]

3 2

(Dγ0+ y )(x ) = xy(x )2 + x−ν , 1 < γ < 2, γ = ,

(−2γ

where ν = 1 + 2γ . Obviously, the above equation has the similar formation to our problem. Let μ = (−γ )) . One can check that y(x ) = μx−(1+γ ) is a solution. Since y(x) ∈ L2 (0, 1), y(x ) ∈ / H˜ γ /2 (0, 1 ). Of course, one has y ∈ H˜ γ /2 (D ) for D ⊂ ⊂ (0, 1). Remark 4.2. The weak solution u defined above is also valid for Caputo derivative. In fact, the Caputo derivative can be transformed into Riemann–Liouville case by the following identity

(C Dγ0+ u )(x ) = (Dγ0+ u )(x ) −

u (0 ) u ( 0 ) x −γ − x 1 −γ , 1 < γ < 2 .

(1 − γ )

(2 − γ )

We now present the existence theorem. 2 Theorem 4.1 (Existence of the weak solution) . For

any γ ∈ (1, 2), let u0 (x) ∈ L (0, 1), then the problem (10)

has a weak solution such that u˜ ∈ L2 0, T; H˜ γ /2 (0, 1 ) , ddut˜ ∈ L2 0, T; H˜ −s (0, 1 ) , s ≥ γ /2, viz. for any v ∈ L2 0, T; H˜ γ /2 (0, 1 ) ,

du˜ ,v dt

γ /2

γ /2

= c2 D0+ u˜, D1−

v + (xγ (u ), v ).

(11)

Proof. We shall prove Theorem 4.1 by Galerkin approximation [24, pp. 353]. The proof is presented in three steps. Step 1: Construct an approximate solution ∞ 2 From Lemma 3.4, we can choose {wk }∞ k=1 such that {wk }k=1 is an orthogonal basis of L (0, 1) and is an orthogonal basis γ / 2 of H˜ (0, 1 ) as well. Fix now a positive integer n. We will look for a function uñ : (0, T ) → H˜ γ /2 (0, 1 ) of the form

uñ =

n

γ /2 un j (t )w j , span{w1 , . . . , wn } ⊂ H˜ 0 (0, 1 ),

(12)

j=1

where we hope to select the coefficients un j (t ) such that

un j (0 ) = (u˜0 , w j ), j = 1, 2, . . . , n and

duñ , wj dt

γ /2

γ /2

= c2 (D0+ uñ , D1− w j ) + (xγ (un ), w j ), 0 ≤ t ≤ T, j = 1, 2, . . . , n.

Here (·, ·) denotes the inner product in L2 (0, 1). Since

dun

(13)

γ /2

γ /2

duñ dt

, wj =

dun dt

j = c2 (D0+ uñ , D1− w j ) + (xγ (un ), w j ) , j = 1, 2, . . . , n. dt un j ( 0 ) = ( u0 , w j )

j

(14)

, (15)


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It is clear from the standard existence theory (Picard theorem and Peano theorem [25, pp. 64]) for

ordinary differential equations that there exists a unique absolutely continuous function un (t ) = un1 (t ), un2 (t ), . . . , unn (t ) for a.e. 0 ≤ t ≤ T. And then uñ defined by (12) solves (14) for a.e. 0 ≤ t ≤ T. Step 2: Estimates for the approximate solution Multiply (15) by un j , sum for j = 1, 2, . . . , n, and then recall (12) to find

duñ , uñ dt

γ /2

γ /2

= c2 D0+ (uñ ), D1− uñ + (xγ (un ), uñ )

for a.e. 0 ≤ t ≤ T. From the following relations (Lemma 3.2),

duñ , un dt

=

1 d uñ 2L2 (0,1) , 2 dt γ /2

γ /2

C1 uñ 2H˜ γ /2 (0,1) ≤ −c2 (D0+ uñ , D1− uñ ), and according to the Galerkin method, one can easily obtain that

uñ ∈ L∞ (0, T; L2 (0, 1 )). and

uñ ∈ L2 0, T; H˜ γ /2 (0, 1 ) . Next, we need to estimate the derivative. For any v ∈ H˜ γ /2 (0, 1 ), with v H˜ γ /2 (0,1 ) ≤ 1, and write v = v(1 ) + v(2 ) , where v(1 ) ∈ span{w1 , w2 , . . . , wn }, (v(2 ) , w j ) = 0( j = 1, . . . , n). Since the functions {w j }∞ are orthogonal in H˜ γ /2 (0, 1 ),

v ( 1 )

j=1

H˜ γ /2 (0,1 )

hence

≤ v H˜ γ /2 (0,1 ) ≤ 1. Collecting (14) for the coefficients of v(1) , we obtain for a.e. 0 ≤ t ≤ T that

duñ (1) ,v dt

γ /2 γ /2 = c2 (D0+ (uñ ), D1− v(1) ) + (xγ (un ), v(1) ),

duñ (1) = c2 (Dγ /2 uñ , Dγ /2 v(1) ) + (xγ (un ), v(1) ) , v 0+ 1− dt γ /2 γ /2 ( 1 ) D v 2 ≤ c2 D0+ uñ 2 + xγ (un ) L2 (0,1) v(1) 2 1− L ( 0,1 ) L ( 0,1 ) L ( 0,1 )

≤ c2 uñ H˜ γ /2 (0,1) v(1) ˜ γ /2 + C u n L 2 ( 0 , 1 ) + 1 v ( 1 ) 2 H ( 0,1 ) L ( 0,1 )

(1) ≤ C uñ H˜ γ /2 (0,1) + un H˜ γ /2 (0,1) + 1 v . H˜ γ /2 (0,1 )

Choosing s ≥ γ /2, we derive H˜ −γ /2 (0, 1 ) ⊂ H˜ −s (0, 1 ) from H˜ s (0, 1 ) ⊂ H˜ γ /2 (0, 1 ), where H˜ −s (0, 1 ) is the dual space of H˜ s (0, 1 ) [19, pp. 146]. By the definition of operator norm,

duñ dt

H˜ −s (0,1 )

duñ ≤ dt

H˜ −γ /2 (0,1 )

and hence

T 0

duñ 2 dt −s H˜

≤C

dt ≤ C ( 0,1 )

T

0

uñ H˜ γ /2 (0,1) + un H˜ γ /2 (0,1) + 1

uñ 2H˜ γ /2 (0,1) dt +

T 0

un 2H˜ γ /2 (0,1) dt + T .

As a consequence, we have obtained

uñ ∈ L2 0, T; H˜ γ /2 (0, 1 ) ,

duñ ∈ L2 0, T; H˜ −s (0, 1 ) , s ≥ γ /2. dt Step 3: Pass to weak limits The proof is similar to the classical case, we omit it. Hence, Theorem 4.1 is proved.

4.2. Stability and uniqueness of the weak solution 4.2.1. Stability with respect to the initial condition Consider the following problems

(u1 )t = c2 Dγ0+ u1 + (u1 ), u1 (x, 0 ) = u1 (0 ), u1 ( 0, t ) = 0, u1 ( 1, t ) = 0

and

(u2 )t = c2 Dγ0+ u2 + (u2 ), u2 (x, 0 ) = u2 (0 ), u2 ( 0, t ) = 0, u2 ( 1, t ) = 0.

Assume that the initial value u1 (0) is changed into u2 (0) after a perturbation such that u1 (0 ) − u2 (0 ) ∈ L2 (0, 1 ) and u1 (0 ) − u2 (0 ) 2L2 (0,1) ≤ ε 2 . Let u = u1 − u2 , then

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γ

ut = c2 D0+ u + (u1 ) − (u2 ), u(x, 0 ) = u1 (0 ) − u2 (0 ), u ( 0, t ) = 0, u ( 1, t ) = 0.

From Theorem 4.1, there exists a weak solution u such that xγ u = u˜ ∈ L2 0, T; H˜ γ /2 (0, 1 ) . Let v = u˜ = xγ u, we then obtain from (11) that

du˜ , u˜ dt

γ /2

γ /2

= c2 D0+ u˜, D1− u˜ + (xγ (u1 ) − xγ (u2 ), u˜ )

or

1 d u˜ 2L2 (0,1) = c2 Dγ0+/2 u˜, Dγ1−/2 u˜ + (xγ (u1 ) − xγ (u2 ), u˜ ). 2 dt Then we have

1 d u˜ 2L2 (0,1) + C1 u˜ 2H˜ γ /2 (0,1) ≤ (xγ (u1 ) − xγ (u2 ), u˜ ) 2 dt 1 1 2 ≤ xγ (u1 ) − xγ (u2 ) L2 (0,1) + u˜ 2L2 (0,1) 2 2 C γ 1 2 2 γ ≤ x u1 − x u2 L2 (0,1) + u˜ L2 (0,1) 2 2 C 1 2 2 = u˜ L2 (0,1) + u˜ L2 (0,1) , 2 2 that is,

d u˜ 2L2 (0,1) ≤ C u˜ 2L2 (0,1) . dt The Gronwall’s inequality yields

u˜ 2L2 (0,1) ≤ eCt u˜0 2L2 (0,1) ≤ C xγ u0 2L2 (0,1) ≤ C ε 2 , hence

1

0

or

1

x2γ u2 (x, t )dx ≤ C ε 2

x 2 γ u 2

0

Since u2

1

L∞ ( 0,T )

L∞ ( 0,T )

maintains the sign and x2γ is continuous, there exists ξ ∈ (0, 1) such that

x 2 γ u 2

0

dx ≤ C ε 2 .

L∞ ( 0,T )

d x = ξ 2γ

1 0

2 u ∞

L ( 0,T )

dx

in view of the mean value theorem of integrals. We therefore derive

u(x, t ) L∞ (0,T;L2 (0,1)) ≤ C ξ −2γ ε 2 . Considering the small ξ leads to large value of ξ −2γ ε 2 , it is necessary to narrow the interval. In fact, when confined in (ε 1/2 , 1), we arrive at

ξ 2γ

1

ε 1/2

2 u ∞

L ( 0,T )

dx =

1

ε 1/2

x 2 γ u 2

L∞ ( 0,T )

dx ≤ C ε 2 ,

ξ ∈ ( ε 1/2 , 1 )

which yields

u(x, t ) 2L∞ (0,T;L2 (ε1/2 ,1)) ≤ C ε 2−γ , 1 < γ < 2. Remark 4.3. Since x−γ vanishes in (u1 ) − (u2 ) =

1 ∗ Cv [ (u1 )

− ∗ (u2 )], we can obtain the stability result for the whole interval [0, 1] by repeating the above process when the perturbation u1 (0 ) − u2 (0 ) ∈ L2 (0, 1 ). According to the above discussion, we draw the conclusion as follows. Theorem 4.2. Assume that the initial value u1 (0) is changed into u2 (0) after a perturbation such that u1 (0 ) − u2 (0 ) ∈ L2 (0, 1 ) 2 and u1 (0 ) − u2 (0 ) L2 (0,1 ) ≤ ε 2 . Then the weak solution of (10) is stable in L∞ (0, T; L2 (0, 1)) with respect to perturbations in the initial value.


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4.2.2. Stability with respect to the boundary conditions We next add the perturbations into the boundary values, which results in the following problems

(u1 )t = c2 Dγ0+ u1 + (u1 ), and u1 (x, 0 ) = v(x ), u1 ( 0, t ) = 0, u1 ( 1, t ) = 0

(u2 )t = c2 Dγ0+ u2 + (u2 ), u2 (x, 0 ) = v(x ), u2 (0, t ) = ε0 (t ), u2 (1, t ) = ε1 (t ),

where εi (t ) H 1 (0,T ) ≤ ε , i = 0, 1. Let u = u2 − u1 , then

γ

ut = c2 D0+ u + (u2 ) − (u1 ), u(x, 0 ) = 0, u(0, t ) = ε0 (t ), u(1, t ) = ε1 (t ),

where (u2 ) − (u1 ) =

1 ∗ Cv [ (u2 )

− ∗ (u1 )] doesn’t contain x−γ . Homogenizing the boundary conditions, we obtain

γ

wt = c2 D0+ w + w , w(x, 0 ) = w0 (x ), w ( 0, t ) = 0, w ( 1, t ) = 0,

where

w(x, t ) = u(x, t ) − h(x, t ), h(x, t ) =

ε0 (t ) + (ε1 (t ) − ε0 (t ))x,

w0 (x ) = −ε0 (0 ) − (ε1 (0 ) − ε0 (0 ))x,

w = (u2 ) − (u1 ) − ε0 (t ) − (ε1 (t ) − ε0 (t ))x + c2 ε0 (t )

1

(1 − γ )

x−γ + c2 (ε1 (t ) − ε0 (t ))

1

(2 − γ )

x 1 −γ .

Since x−γ again appears in the source term, repeating the above process, we obtain

1 d w˜ 2L2 (0,1) + C1 w˜ 2H˜ γ /2 (0,1) ≤ (xγ (u1 ) − xγ (u2 ), w˜ ) − ε 0 (t )xγ + (ε 1 (t ) − ε 0 (t ))x1+γ , w˜ 2 dt 1 1 2 2 ˜ + c (ε1 (t ) − ε0 (t )) ˜ + c ε0 (t ) ,w x, w

(1 − γ )

(2 − γ ) ≤

1 γ 1 x (u1 ) − xγ (u2 ) 2L2 (0,1) + w˜ 2L2 (0,1) 2 2 2 1 1 γ ε 0 (t )x + (ε 1 (t ) − ε 0 (t ))x1+γ L2 (0,1) + w˜ 2L2 (0,1) + 2 2

+

2

1 1 c2 ε0 (t ) 2

(1 − γ )

+ L2

( 0,1 )

1 w˜ 2L2 (0,1) 2

2 1 2 1 + c ( ε ( t ) − ε ( t )) x 1 0 2

(2 − γ ) 2

L ( 0,1 )

+

1 w˜ 2L2 (0,1) 2

C 2 2 2 2 2 ˜ 2L2 (0,1) + C ε 0 (t ) + ε 1 (t ) + ε0 (t ) + ε1 (t ) ≤ x γ u 1 − x γ u 2 L 2 ( 0 , 1 ) + 2 w 2 C 2 2 2 2 ˜ 2L2 (0,1) + 2 w ˜ 2L2 (0,1) + C[ε 0 (t ) + ε 1 (t ) + ε0 (t ) + ε1 (t ) ], = w 2 that is,

d w˜ 2L2 (0,1) ≤ C w˜ 2L2 (0,1) + ε0 (t )2 + ε1 (t )2 + ε0 (t )2 + ε1 (t )2 . dt ˜ 2L2 (0,1 ) , ξ (t ) = ε 0 (t )2 + ε 1 (t )2 + ε0 (t )2 + ε1 (t )2 . The Gronwall inequality yields Let η (t ) = w

t η (t ) ≤ e η (0 ) + C ξ (s )ds , 0 ≤ t ≤ T. Ct

0

We therefore obtain

w˜ 2L2 (0,1) ≤ C w0 (x ) 2L2 (0,1) + C ε0 (t ) 2H1 (0,T) + ε1 (t ) 2H1 (0,T)

or

u˜ 2L2 (0,1) ≤ C w0 (x ) 2L2 (0,1) + C ε0 (t ) 2H1 (0,T) + ε1 (t ) 2H1 (0,T) + h(x, t ) 2L2 (0,1) .

Assume that εi (t ) H 1 (0,T ) ≤ ε 2 , i = 1, 2. We have 2

u˜ 2L2 (0,1) ≤ C ε 2 ,

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which results in

u(x, t ) 2L∞ (0,T;L2 (ε1/2 ,1)) ≤ C ε 2−γ , 1 < γ < 2. Theorem 4.3. Assume that the perturbations εi (t ), i = 0, 1 in boundary values satisfying εi (t ) H 1 (0,T ) ≤ ε , i = 0, 1. Then the weak solution of (10) is stable in L∞ (0, T; L2 (ε 1/2 , 1)).

Remark 4.4. There exists 0 < μ < 1/2 such that 0 < ξ = ε μ < ε 1/2 and

Cε2 ≥

ε 1/2 0

x2γ u2 L∞ (0,T ) dx = ξ 2γ

ε 1/2 0

u2 L∞ (0,T) dx = ε 2μγ u 2L∞ (0,T;L2 (0,ε1/2 )) ,

hence

u L∞ (0,T;L2 (0,ε1/2 )) ≤ C ε 2(1−μγ ) . Since 1 < γ < 2, 0 < μ < 1/2, we have 1 − μγ > 0. We therefore conclude that

u(x, t ) 2L∞ (0,T;L2 (0,1)) ≤ C (ε 2−γ + ε 2(1−μγ ) ), 1 < γ < 2, 0 < μ < 1/2, which indicates the stability in the whole interval [0, 1]. 1 With respect to the uniqueness, we have 0 x2γ u2 (x, t )dx = 0, which implies that u ≡ 0 holds for x ∈ [0, 1] almost everywhere. Corollary 4.1. The weak solution of problem (10) is unique. 5. Numerical simulation 5.1. Numerical algorithm The Riemann–Liouville fractional derivative of order γ (1 < γ ≤ 2) is defined by

γ

D0+ u(x, t ) =

1 ∂2

(2−γ ) ∂ x2 ∂2u , ∂ x2

x

u(s,t ) 0 (x−s )γ −1

ds,

1 < γ < 2,

γ = 2.

We adopt the shifted Gr u¨ nwald formula at all time levels for approximating the fractional derivative [26] i+1 1 γ D0+ T (xi , tn+1 ) = γ g j T (xi − ( j − 1 )h, tn+1 ) + O(h ). h j=0

Here the normalized Gr u¨ nwald weights are defined by

g0 = 1, g j = (−1 ) j

γ (γ − 1 ) · · · (γ − j + 1 ) j!

, j = 1, 2, 3, . . . .

Particularly, g0 = 1, g1 = −β , g2 = β (β2−1 ) . Thus we have

Cv

Tin+1 − Tin

τ

i+1 1 = κγ γ g j T (xi − ( j − 1 )h, tn+1 ) + ni +1 h j=0

1 = κγ γ h Let s =

κγ τ Cv hγ

i+1

, r = Cτv . We have

+1 g j Tin+1 + ni +1 , i = 1, 2, . . . , M − 1, n = 0, 1, . . . , N − 1. −j

j=0

+1 −s g0 Tin+1 + g1 Tin+1 + · · · + gi+1 T0n+1 = Tin + r ni +1

or

(1 − sg1 )T1n+1 − sg0 T2n+1 = T1n + r n1+1 + sg2 T0n+1 , +1 +1 − sgi T1n+1 − sgi−1 T2n+1 − · · · − sg2 Tin−1 + (1 − sg1 )Tin+1 − sg0 Tin+1 = Tin + r ni +1 + sgi+1 T0n+1 ,

i = 2, 3, . . . , M − 1, n = 0, 1, . . . , N − 1. The above equations are expressed in matrix form

AT n+1 = T n + r n+1 + sT0n+1 G + sg0 TMn+1 eM−1 .

(16)


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Here A = (Ai j ) is the matrix of coefficients such that

Ai j =

0, 1 − sg1 , −sgi− j+1 ,

when j > i + 1, when i = j, otherwise,

n T n = T1n , T1n , . . . , TM−1

T

, n = n1 , n1 , . . . , nM−1

T

T

,

T

G = [g2 , g3 , . . . , gM ] , eM−1 = [0, 0, . . . , 1] . Remark 5.1. Since the source term appears in nonlinear form, we can solve it by iteration methods. Instead, in the numerical simulation we approximate (Tin+1 ) by replacing it with the corresponding value at previous time step (Tin ), viz.

AT n+1 = T n + r n + sT0n+1 G + sg0 TMn+1 eM−1 , which is called an implicit-explicit (IMEX) method. One can prove that the above scheme is unconditionally stable and has convergence rate O(τ + h ) under some reasonable assumptions [26]. One can refer to [27] for improving the numerical accuracy, where a second-order accurate numerical approximation for a spatial fractional diffusion equation was proposed. The approach based on the classical Crank–Nicholson method combined with spatial extrapolation was used to obtain temporally and spatially second-order accurate numerical estimates. It was shown that the fractional Crank–Nicholson method based on the shifted Gr u¨ nwald formula is unconditionally stable. One can also obtain higher accuracy by applying predictor-corrector schemes. 5.2. Parameters and conditions in numerical process The fractional thermal conductivity κ γ of textiles will be approximated by the classical case κ = ε κa + (1 − ε )κ f (Note that they have different dimensions). In the simulation, we set Thermal conductivities: κa = 0.025 W · m−1 · K −1 , κ f = 0.1 W · m−1 · K −1 , Thickness: L = 2.5 × 10−3 m, Time: t ∈ [0, 10 h], Left boundary value condition: T0 = 500◦ C, Right boundary value condition: T1 = 37◦ C, Initial condition:

TI (x ) = −

T0 − T1 2 x + T0 , L2

The physical parameters in equations are given as follows [15]: ε0 = 0.084, τ0 = 1.2, β0 = 8 m−1 , Cv = 1715.0 kJ · m−3 · K−1 , σ = 5.672 × 10−8 kJ · m−2 · K−4 · s−1 , ξ1 = ξ2 = 0 . 9 . 5.3. Numerical result In this subsection, we present two numerical examples. The example 1 shows that the fractional model may not be appropriate for low temperature case, while the classical model gives reasonable results. The example 2 shows that in high temperature situation, the fractional model gives more realistic results than the classical heat equation. In example 1, we set T0 = 37◦ C and T1 = 10◦ C, which represents the low temperature case. The numerical results are presented in Fig. 2 with γ = 2.0 and γ = 1.8, respectively. We observe from Fig. 2 (b) that the temperature drops rapidly to −60◦ C (γ = 1.8), which goes against the real situation. The results show that fractional model reported in the paper is not valid under low temperature, while the classical model (γ = 2.0) provides a reasonable result in Fig. 2 (a). It may be for this reason that we haven’t found related spatial fractional models concerning heat transfer within textiles. In example 2, we set T0 = 500◦ C and T1 = 37◦ C. As was depicted in the introduction, the higher heat and humidity force us to use the superdiffusion model to simulate the heat transfer process for firefighter protective clothing during flash fire exposure. In the following, we will find that the superdiffusion model for this case conforms to the real situation. In Fig. 3 (a), the classical order γ = 2.0 is utilized, whilst the image with fractional order γ = 1.8 is shown on the right. It is obvious that the second case shows faster diffusion than the classical case. To clarify this result, we plot several curves in Fig. 4 with γ = 2.0, 1.9, 1.8, 1.7, 1.6, 1.5, respectively. As expected, the temperature T(x, t) for fixed t = t10 (M = 50, N = 50 ) drops faster and faster as γ is decreased when γ > 1.5, which is also valid for fixed x = x10 . It seems that γ = 1.5 is a critical value because the downward trend is no longer maintained when γ ≤ 1.5. Indeed, we observe that the trend is almost reversed in this case from Fig. 5, which holds true for the ODE example in Remark 4.2. Although the heat transfer of situation we considered spreads faster than the case under low temperature, natural law should not deviate from the ideal situation too much. The above results indicate that it is appropriate to choose the fractional order γ such that γ > 1.5, which remains to be studied.

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Fig. 2. Fractional model under low temperature.

Fig. 3. T(x, t) with γ = 2.0 and γ = 1.8 under high temperature.

Fig. 4. Temperature for fixed x and t when γ ≥ 1.5.

Remark 5.2. The superdiffusion model for the high temperature case gives more reasonable result. On the other hand, the model describes the faster propagation as we expected. We are focusing on the high temperature-humidity environment, where the firefighters put on protective clothing near the fire temperature higher than 500°C. It is hard to predict the lowest environmental temperature, before which the fractional model is no longer valid because of the complicated physical mechanism and different environmental parameters.


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Fig. 5. Temperature for fixed x and t when γ ≤ 1.5.

Fig. 6. Temperature distribution of measurement and numerical simulation.

Much luckily, in the case of example 2, we are sure to choose lowest temperature T0 ≥ 270°C, here the fractional model is valid according to the numerical simulation. Remark 5.3. As depicted in Remark 2.1, Chitrphiromstri et al. [1] proposed a useful but complicated heat and moisture transfer model for firefighter protective clothing during a flash fire exposure. We find that the numerical images in [1] and [2] are closely matching our numerical results calculated by the simplified fractional model on the same conditions. See Fig. 6. Hence, it is expected to apply the fractional model to simplify the coupling phenomena with fewer parameters. To test the validity of the proposed fractional model or an improved fractional model which will be considered further, we shall carry out the experimental test on the numerical results observed in this paper in the future. Remark 5.4. In [1] and [2], the authors considered the heat transfer in the skin. The tissue burn injury model is introduced to predict the thermal damage. It is said that thermal damage occurs when the temperature at the interface between the epidermis and dermis in the human skin rises above 44°C. In Fig. 7, the temperature distribution at the interface between the epidermis and dermis is presented by the new fractional model. It seems the fractional model gives a more reasonable description. 6. Conclusions and prospects In the paper, we proposed a fractional heat transfer model to simulate the situation with high temperature and high humidity, which only possesses a small change on the equation involved in Fan’s model by replacing the second derivative with Riemann–Liouville derivative. However, we haven’t found similar related fractional model in heat and moisture transfer

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Fig. 7. Temperature at the interface between the epidermis and dermis.

within textiles, which maybe result from the fact that the simulation results go against the real situation under low temperature. With respect to the fractional model, variational formulation was presented and the conditional well-posedness was fulfilled. There are several avenues for future research. (1) The proposed model is very simple. Some other factors should be further considered. For example, the phase change, due to sorption and condensation, ought to be included. It is necessary to consider the moisture transport in firefighter protective clothing because the phenomenon of coupled heat and moisture transfer in porous media is common. (2) We are interested in the inverse problems of heat and moisture transfer model. In this new context, the formulation of inverse problems will be very different from the classical model considered in [12–15] because sweat can not be neglected and we are more concerned with thermal damage in this case. (3) The regularity of the weak solution should be considered. One can refer to [17] for additional information. The finite element approximation for the variational formulation reported in the paper also deserves further consideration. Acknowledgments The research is supported by National Natural Science Foundation of China (grant no. 11471287 and 11071221), by which the paper is completed during the third-named author, Professor Yongzhi Steve Xu’s visit at Zhejiang Sci-Tech University. References [1] P. Chitrphiromstri, A.V. Kuznetsov, Modeling heat and moisture transfer in firefighter protective clothing during flash fire exposure, Heat Mass Transf. 41 (2005) 206–215. [2] G.W. Song, P. Chitrphiromstri, D. Ding, Numerical simulations of heat and moisture transport in thermal protective clothing under flash fire conditions, Int. J. Occup. Saf. Ergon. 14 (2008) 89–106. [3] A. Elgafy, S. Mishra, A heat transfer model for incorporating carbon foam fabrics in firefighters garment, Heat Mass Transf. 50 (2014) 545–557. [4] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (20 0 0) 1–77. [5] J. Klafter, I.M. Sokolov, Anomalous diffusion spreads its wings, Phys.World 8 (2005) 29–32. [6] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. Gen. 37 (2004) R161–R208. [7] R. Metzler, J. Klafter, I.M. Sokolov, Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended, Phys. Rev. E 58 (1998) 1621–1632. [8] N. Du, J.T. Fan, H.J. Wu, W.W. Sun, Optimal porosity distribution of fibrous insulation, Int. J. Heat Mass Transf. 52 (2009) 4350–4357. [9] J.T. Fan, X. Cheng, X. Wen, An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results, Int. J. Heat Mass Transf. 47 (2004) 2343–2352. [10] J.T. Fan, Z.X. Luo, Y. Li, Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation, Int. J. Heat Mass Transf. 43 (20 0 0) 2989–30 0 0. [11] H. Wu, J.T. Fan, Study of heat and moisture transfer within multi-layer clothing assemblies consisting of different types of battings, Int. J. Therm. Sci. 47 (2008) 641–647. [12] D.H. Xu, Mathematical Modeling of Heat and Moisture Transfer within Textiles and Corresponding Inverse Problems of Textile Material design, first ed., Science Press, Beijing, 2014. [13] D.H. Xu, Y.B. Chen, X.H. Zhou, An inverse problem of thickness design for single layer textile material under low temperature, J. Math Ind. 2 (2010) 582–590. [14] D.H. Xu, M.B. Ge, Thickness determination in textile material design: dynamic modeling and numerical algorithms, Inverse Probl. 28 (2012) 035011. [15] D.H. Xu, L. Wen, An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer, Appl. Anal: Int. J. 93 (2013) 445–465. [16] V. Ervin, J. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer.Methods Part. Diff. Equ. 22 (2006) 558–576. [17] B.T. Jin, R. Lazarov, J. Pasciak, Variational formulation of problems involving fractional order differential operators, Math. Comput. 84 (2015) 2665–2700. [18] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, first ed., Elsevier B.V, The Netherlands, 2006. [19] M.X. Wang, Sobolev Spaces, first ed., Higher Education Press, Beijing, 2013.


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[20] Y.X. Hua, X.H. Yu, On the ground state solution for a critical fractional Laplacian equation, Nonlinear Anal. 87 (2013) 116–125. [21] Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012) 667–696. [22] M. Bowles, M. Agueh, Weak solutions to a fractional Fokker–Planck equation via splitting and Wasserstein gradient flow, Appl. Math. Lett. 42 (2015) 30–35. [23] J.C. Robinson, Infinite-Dimensional Dynamical Systems, first ed., Cambrige University Press, UK, 2001. [24] L.C. Evans, Partial Differential Equations, second ed., American Mathematical Society, US, 2010. [25] T.R. Ding, C.Z. Li, Ordinary Differential Equations, second ed., Higher Education Press, Beijing, 2004. [26] F. Liu, P. Zhuang, V. Anh, I. Tuner, A fractional-order implicit difference approximation for the space-time fractional diffusion equation, Anziam J. 47 (2005) C48–C68. [27] C. Tadjeran, M.M. Meerschaert, H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006) 205–213.