# Analytic Solutions for Heat Conduction in Functionally Graded Circular Hollow Cylinders with Time-Dependent Boundary Conditions

Authors: Sen-Yung Lee; Chih-Cheng Huang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

An analytic solution method, without integral transformation, is developed to find the exact solutions for transient heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order regular singular differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. The exact solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analyses are given to illustrate the efficiency and the accuracy of the analysis.


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## Body

## 1. Introduction

The applications of heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions can be widely found in many engineering fields, such as cannon barrels, heat exchanger tubes, time varied heating on walls of circular structure, and heat treatment on hollow cylinders. As such, an accurate solution method is very helpful for relevant developments.The problem of heat conduction with time-dependent boundary conditions cannot be solved directly by the method of separation of variables. In most of the analyses, an integral transform has been used to remove the time-dependent term. For the problem of heat conduction in circular hollow uniform cylinders with time-dependent boundary conditions, the associated governing differential equation is a second-order Bessel differential equation with constant coefficients. After conducting a Laplace transformation, the analytical solution can be obtained and found in the book by Őzisik [1].When the structure is an FG circular hollow cylinder, the associated governing differential equation is a second-order regular singular differential equation with variable coefficients. For problems with time-independent boundary conditions, various numerical methods have been developed, such as the perturbation method [2], the finite difference method [3], and the finite element method [4]. Jabbari et al. [5, 6] derived analytical solutions for thermal stresses of FG hollow cylinders whose material properties vary with power law distribution through the thickness due to radially symmetric loads and nonaxisymmetric loads. By using the Laplace transformation and a series expansion of Bessel functions, Ootao and Tanigawa [7] analyzed one-dimensional transient thermoelastic problem with the material properties varying with the powerlaw form of the radial coordinate variable. Zhao et al. [8] analyzed the temperature change when the thermal and thermoelastic properties are assumed to vary exponentially in the radial direction. Hosseini et al. [9] considered the material properties to be nonlinear with a power law distribution through the thickness, while the temperature distribution was derived analytically using the Bessel functions.In the study of heat conduction in FG circular hollow cylinders with time-dependent boundary conditions, only limited studies can be found. Ootao et al. [10] studied the transient temperature and thermal stress distribution in an infinitely long nonhomogeneous hollow cylinder due to a moving heat source in the axial direction from the inner and/or outer surfaces using the layerwise theory in conjunction with the method of Fourier cosine and the Laplace transformations. Shao and Ma [11] employed the Laplace transform techniques and the series solving method to study thermomechanical stresses in FG circular hollow cylinders with linearly increasing boundary temperature. Jabbari et al. [12] developed an analytical solution for the one-dimensional temperature distribution, mechanical and thermal stresses in an infinitely long FG hollow cylinder under a moving heat source, which moves across the thickness of the cylinder. Asgari and Akhlaghi [13] employed the finite element method to study the transient thermal stresses in a thick hollow cylinder with finite length mad of 2D-FGM that its material properties are varied in the radial and axial directions with a power law function. The thermal boundary conditions at the inner and outer radiuses are time dependent. Singh et al. [14] applied the finite integral transform method and the separation of variables method to solve time-dependent heat conduction problem in a multilayer annulus. Malekzadeh and Heydarpour [15] used the differential quadrature method (DQM) in conjunction with the finite element method (FEM) to study the response of FG cylindrical shells under moving thermomechanical loads. Recently, Wang and Liu [16] have employed the method of separation of variables to develop the analytical solution of transient temperature fields for two-dimensional transient heat conduction in a fiber-reinforced multilayer cylindrical composite.In the study of thermal elastic response of FG cylinders without mechanical loading, the heat conduction problem is not incorporated with the elastic field and can be studied independently. However, the thermal field will be coupled with the temperature field. In this paper, one considers the heat conduction problem of FG cylinders only. A new solution method, which is a modification on the method developed by  Lee and Lin [17] and Chen et al. [18],  is used to develop the analytical solution for transient heat conduction in FG circular hollow cylinders with time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. The analytic solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analysis are given to illustrate the efficiency and accuracy of the solution method.


## 2. Mathematical Modeling

Consider the transient heat conduction in an FG circular hollow cylinder with time-dependent boundary condition at the inner and outer surfaces, as shown in Figure1. The governing differential equation of the system is
(1)1r1∂r[rk(r)∂T(r,t)∂r]+g˙(r,t)=ρc(r)∂T(r,t)∂t.a<r<b,t>0.
The boundary conditions are
(2)T(a,t)=T-1(t),T(b,t)=T-2(t),
and the initial condition is
(3)T(r,0)=T0(r).
Here, r is the space variable and t is the time variable, c(r) is the specific heat, k(r) is the thermal conductivity, T(r,t) is the temperature, ρ is the mass density, and g˙(r,t) is the heat source inside the circular hollow cylinder. T-1(t) and T-2(t) are the time-dependent temperatures at the inner and outer surfaces, respectively.Figure 1
FG circular hollow cylinder with time-dependent boundary conditions.In terms of the following dimensionless parameters(4)K(ξ)=k(r)k(a),C(ξ)=c(r)c(a),G(ξ,τ)=g˙(r,t)b2k(a)Tr,ξ=rb,r-=ab,θ=TTr,θ0=T0Tr,θ-i=T-iTr,τ=k(a)tc(a)ρb2,
where Tr is the reference temperature, the boundary value problem of heat conduction becomes
(5)1ξ1∂ξ[ξK(ξ)∂θ(ξ,τ)∂ξ]+G(ξ,τ)=C(ξ)∂θ(ξ,τ)∂τ,1ξ1∂ξ[ξK(ξ)∂θ(ξ,τ)∂ξ]+rrrrriiir-<ξ<1,τ>0,θ(r-,τ)-θ-1=0,θ(1,τ)-θ-2=0,θ(ξ,0)=θ0(ξ).


## 3. Solution Method

### 3.1. Change of Variables

To find the solution for the second-order differential equation with nonhomogeneous boundary conditions, the shifting variable method developed by  Lee and Lin [17] and Chen et al. [18]   was extended by taking
(6)θ(ξ,τ)=ν(ξ,τ)+f1(τ)g1(ξ)+f2(τ)g2(ξ),
where gi(ξ), i=1,2 are shifting functions to be specified and ν(ξ,τ) is the transformed function. Substituting (6) into (5) yields the following partial differential equation:
(7)1ξ∂∂ξ[ξK(ξ)∂ν(ξ,τ)∂ξ]-C(ξ)∂ν(ξ,τ)∂τ=-G(ξ,τ)+∑i=12{C(ξ)gi(ξ)dfi(τ)dτ-fi(τ)1ξddξ[ξK(ξ)dgi(ξ)dξ]},
the associated boundary conditions:
(8)ν(r-,τ)+f1(τ)g1(r-)+f2(τ)g2(r-)=f1(τ),ν(1,τ)+f1(τ)g1(1)+f2(τ)g2(1)=f2(τ),
and the associated initial condition
(9)ν(ξ,0)=θ(ξ,0)-f1(0)g1(ξ)-f2(0)g2(ξ).


### 3.2. Shifting Functions

To simplify the analysis, the shifting functions are specifically chosen such that they satisfy the following differential equations and the boundary conditions(10)d2gi(ξ)dξ2=0,i=1,2,g1(r-)=1,g1(1)=0,g2(r-)=0,g2(1)=1.
These two shifting functions can be easily determined as
(11)g1(ξ)=11-r-(1-ξ),g2(ξ)=11-r-(-r-+ξ).


### 3.3. Reduced Homogenous Problem

With these two shifting functions, (11), the governing differential equation (7) becomes
(12)1ξ∂∂ξ[ξK(ξ)∂ν(ξ,τ)∂ξ]-C(ξ)∂ν(ξ,τ)∂τ=F(ξ,τ),
where
(13)F(ξ,τ)=-G(ξ,τ)=+∑i=12{C(ξ)gi(ξ)dfi(τ)dτ-fi(τ)ξddξ[ξK(ξ)]dgi(ξ)dξ}.
The two nonhomogenous boundary conditions, (8), for the transformed function ν(ξ,τ) are reduced to homogenous ones:
(14)ν(r-,τ)=0,ν(1,τ)=0.
The transformed initial condition is
(15)ν(ξ,0)=θ0(ξ)-f1(0)g1(ξ)-f2(0)g2(ξ)=ν0(ξ).


### 3.4. Solution of Transformed Variable

#### 3.4.1. Characteristic Solution

To find the solutionν(ξ,τ), we use the eigenfunction expansion method and assume the solution to be in the form
(16)ν(ξ,τ)=ϕ(ξ)B(τ).
The separation equation for the dimensionless time variable B(τ) is
(17)dB(τ)dτ=-λ2B(τ),
and the dimensionless space variable ϕ(ξ) satisfies the following self-adjoin operator:
(18)1ξddξ[ξK(ξ)dϕ(ξ)dξ]+λ2C(ξ)ϕ(ξ)=0,ξ∈(r-,1),(19)ϕ(r-)=0,(20)ϕ(1)=0.
Let Xj(ξ), j=1,2, be the two linearly independent fundamental solutions of the system; then, the solution of the differential equation (18) can be expressed as
(21)ϕ(ξ)=C1X1(ξ)-C2X2(ξ),
where C1 and C2 are constants to be determined from the boundary conditions, (19)-(20).After substituting solutions, (21), into the boundary conditions, (19)-(20), we obtain the following characteristic equation
(22)X1(1)X2(r-)-X1(r-)X2(1)=0.
Consequently, the eigenvalues λn, n=1,2,3,… can be determined. The associated nth eigenfunction ϕn(ξ) is determined as
(23)ϕn(ξ)=X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ),
where Xn,1(ξ) and Xn,2(ξ) are, respectively, the fundamental solutions X1(λn,ξ) and X2(λn,ξ) associated with eigenvalues λn, n=1,2,3,….  They are defined as Xn,1(ξ)=X1(λn,ξ) and Xn,2(ξ)=X2(λn,ξ). The eigenfunctions ϕn(ξ) constitute an orthogonal set in the interval r-≤ξ≤1, with respect to a weighting function ξC(ξ):
(24)∫r-1ξC(ξ)ϕm(ξ)ϕn(ξ)dξ={0,form≠nδn,form=n,
where
(25)δn=∫r-1ξC(ξ)ϕn2(ξ)dξ.In terms of eigenfunctions, the transformed variableν(ξ,τ) can be expressed as
(26)ν(ξ,τ)=∑n=1∞ϕn(ξ)Bn(τ).
Substituting (26) into (12), multiplying it by ξϕm, and integrating from r- to 1, we obtain
(27)dBn(τ)dτ+λn2Bn(τ)=-γn(τ),
where
(28)γn(τ)=1δn∫r-1ξϕn(ξ)F(ξ,τ)dξ.
The general solution of (27) is
(29)Bn(τ)=e-λn2τ[αn-∫0τeλn2χγn(χ)dχ],αn is determined from the initial condition (15), and
(30)αn=Bn(0)=1δn∫r-1ξC(ξ)ϕn(ξ)ν0(ξ)dξ.
After substituting the solution of the transformed variable (26) and the shifting functions (11) back to (6), the exact solution for the general system is obtained.


## 3.1. Change of Variables

To find the solution for the second-order differential equation with nonhomogeneous boundary conditions, the shifting variable method developed by  Lee and Lin [17] and Chen et al. [18]   was extended by taking
(6)θ(ξ,τ)=ν(ξ,τ)+f1(τ)g1(ξ)+f2(τ)g2(ξ),
where gi(ξ), i=1,2 are shifting functions to be specified and ν(ξ,τ) is the transformed function. Substituting (6) into (5) yields the following partial differential equation:
(7)1ξ∂∂ξ[ξK(ξ)∂ν(ξ,τ)∂ξ]-C(ξ)∂ν(ξ,τ)∂τ=-G(ξ,τ)+∑i=12{C(ξ)gi(ξ)dfi(τ)dτ-fi(τ)1ξddξ[ξK(ξ)dgi(ξ)dξ]},
the associated boundary conditions:
(8)ν(r-,τ)+f1(τ)g1(r-)+f2(τ)g2(r-)=f1(τ),ν(1,τ)+f1(τ)g1(1)+f2(τ)g2(1)=f2(τ),
and the associated initial condition
(9)ν(ξ,0)=θ(ξ,0)-f1(0)g1(ξ)-f2(0)g2(ξ).


## 3.2. Shifting Functions

To simplify the analysis, the shifting functions are specifically chosen such that they satisfy the following differential equations and the boundary conditions(10)d2gi(ξ)dξ2=0,i=1,2,g1(r-)=1,g1(1)=0,g2(r-)=0,g2(1)=1.
These two shifting functions can be easily determined as
(11)g1(ξ)=11-r-(1-ξ),g2(ξ)=11-r-(-r-+ξ).


## 3.3. Reduced Homogenous Problem

With these two shifting functions, (11), the governing differential equation (7) becomes
(12)1ξ∂∂ξ[ξK(ξ)∂ν(ξ,τ)∂ξ]-C(ξ)∂ν(ξ,τ)∂τ=F(ξ,τ),
where
(13)F(ξ,τ)=-G(ξ,τ)=+∑i=12{C(ξ)gi(ξ)dfi(τ)dτ-fi(τ)ξddξ[ξK(ξ)]dgi(ξ)dξ}.
The two nonhomogenous boundary conditions, (8), for the transformed function ν(ξ,τ) are reduced to homogenous ones:
(14)ν(r-,τ)=0,ν(1,τ)=0.
The transformed initial condition is
(15)ν(ξ,0)=θ0(ξ)-f1(0)g1(ξ)-f2(0)g2(ξ)=ν0(ξ).


## 3.4. Solution of Transformed Variable

### 3.4.1. Characteristic Solution

To find the solutionν(ξ,τ), we use the eigenfunction expansion method and assume the solution to be in the form
(16)ν(ξ,τ)=ϕ(ξ)B(τ).
The separation equation for the dimensionless time variable B(τ) is
(17)dB(τ)dτ=-λ2B(τ),
and the dimensionless space variable ϕ(ξ) satisfies the following self-adjoin operator:
(18)1ξddξ[ξK(ξ)dϕ(ξ)dξ]+λ2C(ξ)ϕ(ξ)=0,ξ∈(r-,1),(19)ϕ(r-)=0,(20)ϕ(1)=0.
Let Xj(ξ), j=1,2, be the two linearly independent fundamental solutions of the system; then, the solution of the differential equation (18) can be expressed as
(21)ϕ(ξ)=C1X1(ξ)-C2X2(ξ),
where C1 and C2 are constants to be determined from the boundary conditions, (19)-(20).After substituting solutions, (21), into the boundary conditions, (19)-(20), we obtain the following characteristic equation
(22)X1(1)X2(r-)-X1(r-)X2(1)=0.
Consequently, the eigenvalues λn, n=1,2,3,… can be determined. The associated nth eigenfunction ϕn(ξ) is determined as
(23)ϕn(ξ)=X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ),
where Xn,1(ξ) and Xn,2(ξ) are, respectively, the fundamental solutions X1(λn,ξ) and X2(λn,ξ) associated with eigenvalues λn, n=1,2,3,….  They are defined as Xn,1(ξ)=X1(λn,ξ) and Xn,2(ξ)=X2(λn,ξ). The eigenfunctions ϕn(ξ) constitute an orthogonal set in the interval r-≤ξ≤1, with respect to a weighting function ξC(ξ):
(24)∫r-1ξC(ξ)ϕm(ξ)ϕn(ξ)dξ={0,form≠nδn,form=n,
where
(25)δn=∫r-1ξC(ξ)ϕn2(ξ)dξ.In terms of eigenfunctions, the transformed variableν(ξ,τ) can be expressed as
(26)ν(ξ,τ)=∑n=1∞ϕn(ξ)Bn(τ).
Substituting (26) into (12), multiplying it by ξϕm, and integrating from r- to 1, we obtain
(27)dBn(τ)dτ+λn2Bn(τ)=-γn(τ),
where
(28)γn(τ)=1δn∫r-1ξϕn(ξ)F(ξ,τ)dξ.
The general solution of (27) is
(29)Bn(τ)=e-λn2τ[αn-∫0τeλn2χγn(χ)dχ],αn is determined from the initial condition (15), and
(30)αn=Bn(0)=1δn∫r-1ξC(ξ)ϕn(ξ)ν0(ξ)dξ.
After substituting the solution of the transformed variable (26) and the shifting functions (11) back to (6), the exact solution for the general system is obtained.


## 3.4.1. Characteristic Solution

To find the solutionν(ξ,τ), we use the eigenfunction expansion method and assume the solution to be in the form
(16)ν(ξ,τ)=ϕ(ξ)B(τ).
The separation equation for the dimensionless time variable B(τ) is
(17)dB(τ)dτ=-λ2B(τ),
and the dimensionless space variable ϕ(ξ) satisfies the following self-adjoin operator:
(18)1ξddξ[ξK(ξ)dϕ(ξ)dξ]+λ2C(ξ)ϕ(ξ)=0,ξ∈(r-,1),(19)ϕ(r-)=0,(20)ϕ(1)=0.
Let Xj(ξ), j=1,2, be the two linearly independent fundamental solutions of the system; then, the solution of the differential equation (18) can be expressed as
(21)ϕ(ξ)=C1X1(ξ)-C2X2(ξ),
where C1 and C2 are constants to be determined from the boundary conditions, (19)-(20).After substituting solutions, (21), into the boundary conditions, (19)-(20), we obtain the following characteristic equation
(22)X1(1)X2(r-)-X1(r-)X2(1)=0.
Consequently, the eigenvalues λn, n=1,2,3,… can be determined. The associated nth eigenfunction ϕn(ξ) is determined as
(23)ϕn(ξ)=X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ),
where Xn,1(ξ) and Xn,2(ξ) are, respectively, the fundamental solutions X1(λn,ξ) and X2(λn,ξ) associated with eigenvalues λn, n=1,2,3,….  They are defined as Xn,1(ξ)=X1(λn,ξ) and Xn,2(ξ)=X2(λn,ξ). The eigenfunctions ϕn(ξ) constitute an orthogonal set in the interval r-≤ξ≤1, with respect to a weighting function ξC(ξ):
(24)∫r-1ξC(ξ)ϕm(ξ)ϕn(ξ)dξ={0,form≠nδn,form=n,
where
(25)δn=∫r-1ξC(ξ)ϕn2(ξ)dξ.In terms of eigenfunctions, the transformed variableν(ξ,τ) can be expressed as
(26)ν(ξ,τ)=∑n=1∞ϕn(ξ)Bn(τ).
Substituting (26) into (12), multiplying it by ξϕm, and integrating from r- to 1, we obtain
(27)dBn(τ)dτ+λn2Bn(τ)=-γn(τ),
where
(28)γn(τ)=1δn∫r-1ξϕn(ξ)F(ξ,τ)dξ.
The general solution of (27) is
(29)Bn(τ)=e-λn2τ[αn-∫0τeλn2χγn(χ)dχ],αn is determined from the initial condition (15), and
(30)αn=Bn(0)=1δn∫r-1ξC(ξ)ϕn(ξ)ν0(ξ)dξ.
After substituting the solution of the transformed variable (26) and the shifting functions (11) back to (6), the exact solution for the general system is obtained.


## 4. Verification and Examples

To illustrate the previous analysis, the following examples and limiting cases are given.Example 1.
Consider the heat conduction in an uniform circular hollow cylinder with time-dependent boundary conditions. The boundary value problem of the heat conduction in dimension-less form is(31)1ξ1∂ξ[ξ∂θ(ξ,τ)∂ξ]=∂θ(ξ,τ)∂τ,r-<ξ<1,τ>0,θ(r-,τ)=ψ1(τ)Tr,θ(1,τ)=ψ2(τ)Tr,θ(ξ,0)=θ-0(ξ)Tr.
In this case,K(ξ)=1, C(ξ)=1. The two shifting functions are
(32)g1(ξ)=11-r-(1-ξ),g2(ξ)=11-r-(ξ-r-).
The two linearly independent fundamental solutions are
(33)X1(λn,ξ)=J0(λnξ),X2(λn,ξ)=Y0(λnξ).
This leads to
(34)F(ξ,τ)=1(1-r-)Tr[ψ2(τ)ξ(1-ξ)ψ1′(τ)+(-r-+ξ)ψ2′(τ)2+ψ1(τ)ξ-ψ2(τ)ξ],
where the characteristic equation is
(35)X1(1)X2(r-)-X1(r-)X2(1)=0.
The associated nth eigenfunction ϕn(ξ) is determined as
(36)ϕn(ξ)=J0(λnξ)J0(λn)-Y0(λnξ)Y0(λn).
The eigenvalues λn and the associated eigenfunctions ϕn(ξ) are obtained from (35) and (36). The two coefficients in (28)-(29) are derived as
(37)γn(τ)=1δn1(1-r-)Trγn(τ)=×∫r-1{ϕn(ξ)[(ξ-ξ2)ψ1′(τ)+(-r-ξ+ξ2)ψ2′(τ)i+ψ1(τ)-ψ2(τ)(ξ-ξ26)](ξ-ξ26)}dξBn(τ)=e-λn2τ[αn-∫0τeλn2χγn(χ)dχ],
where
(38)δn=[∫r-1ξϕn2(ξ)dξ],αn=1δnTr∫r-1ξϕn(ξ)θ-0(ξ)dξ.
As a result, the analytic closed solution of the system in dimensionless form is(39)θ(ξ,τ)=∑n=1∞[ϕn(ξ)Bn(τ)]θ(ξ,τ)=+(1-ξ1-r-)ψ1(τ)Tr+(-r-+ξ1-r-)ψ2(τ)Tr,
when
(40)ψ1(τ)=ψ2(τ)=0.
The analytic closed solution, in dimensionless form, is reduced to
(41)θ(ξ,τ)=∑n=1∞[e-λn2τϕn(ξ)αn].
The solution is exactly the same as the one given by Őzisik [1].Example 2.
Consider the heat conduction in an FG circular hollow cylinder with time-dependent boundary conditions. The coefficients of thermal conductivity and the specific heat areK(ξ)=kmξβ1 and C(ξ)=cmξβ2, respectively. The boundary value problem of the heat conduction in dimensionless form is
(42)1ξ∂∂ξ[ξK(ξ)∂θ(ξ,τ)∂ξ]-C(ξ)∂θ(ξ,τ)∂τ=0,1ξ∂∂ξ[ξK(ξ)∂θ(ξ,τ)∂ξ]-iiir-<ξ<1,τ>0,(43)θ(r-,τ)=0,(44)θ(1,τ)=(1-e-C0τ)θ-2,(45)θ(ξ,0)=0.
In this case, two shifting functions are(46)g1(ξ)=11-r-(1-ξ),g2(ξ)=11-r-(ξ-r-).
The route to two independent fundamental solutions of of (42) lies in the use of the Frobenius method which can be represented in terms of the Bessel functions:
(47)X1(λn,ξ)=ξ-β1/2Jν(ηnξ(2+β2-β1)/2),X2(λn,ξ)=ξ-β1/2Yν(ηnξ(2+β2-β1)/2),
where
(48)ηn=cmkm(2λnβ1-β2-2),ν=β1β1-β2-2.
Now,(49)F(ξ,τ)=e-C0τθ-2[cmξβ2C0(a*+b*ξ)+b*(β1+1)kmξβ1-1]=ii-b*[θ-2(β1+1)kmξβ1-1],
where
(50)a*=-r-1-r-,b*=11-r-.
The characteristic equation is
(51)X1(1)X2(r-)-X1(r-)X2(1)=0.
The associated nth eigenfunction ϕn(ξ) is determined as
(52)ϕn(ξ)=X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ).
The eigenvalues λn and the associated eigenfunctions ϕn(ξ) are obtained from (51) and (52) by numerical analysis. Two coefficients in (28)–(30) are derived as
(53)γn(τ)=γ-n1(ξ)e-C0τ+γ-n2(ξ),Bn(τ)=[αn+γ-n1(ξ)λn2-C0+γ-n2(ξ)λn2]e-λn2τBn(τ)=-γ-n1(ξ)λn2-C0e-C0τ-γ-n2(ξ)λn2,
where
(54)δn=[∫r-1cmξβ2+1ϕn2(ξ)dξ],γ-n1(ξ)=θ-2δn{∫r-1ϕn(ξ)[cmξβ2C0(a*+b*ξ)i+b*(β1+1)kmξβ1-1]dξ∫21},γ-n2(ξ)=-b*θ-2δn[∫r-1ϕn(ξ)(β1+1)kmξβ1-1dξ],αn=0.
Consequently, the analytic closed solution for the system can be derived as(55)θ(ξ,τ)=(1-e-C0τ)θ-21-r-(ξ-r-)=+∑n=1∞{X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ)}==×{[γ-n1(ξ)λn2-C0+γ-n2(ξ)λn2]e-λn2τ-γ-n1(ξ)λn2-C0e-c0τ-γ-n2(ξ)λn2}.
In Figure2, the temperature variation of FG circular hollow cylinders with various parameters of β1, β2, and C0 at ξ=0.75 is shown. It can be found that when C0 is a positive constant, the temperature parameter of the mediums at ξ=0.75 increases then reaches the associated constant temperatures over time. The temperature increase rate for the system with a higher value of C0 is greater than that of one with a lower value of C0. Figures 3 and 4 show the temperature distribution of FG circular hollow cylinders with various parameters of β1, β2, and C0 at τ=0.2. From these figures, it can be observed that with constant parameter β1, the temperature of the mediums increases as parameter β2 is increased. With constant parameter β2, the temperature of the mediums increases as parameter β1 is increased.Figure 2
Temperature variation of FG circular hollow cylinders with constant value ofβ1 and various values of β2 and C0 at ξ=0.75, [km=1, cm=1, θ(1,τ)=(1-e-C0τ)θ-2:θ-2=4].Figure 3
Temperature distribution of FG circular hollow cylinders with constant value ofβ1 and various values of β2 and C0 at τ=0.2, [km=1, cm=1, θ(1,τ)=(1-e-C0τ)θ-2:θ-2=4].Figure 4
Temperature distribution of FG circular hollow cylinders with constant value ofβ2 and various values of β1 and C0 at τ=0.2, [km=1, cm=1, θ(1,τ)=(1-e-C0τ)θ-2:θ-2=4].Example 3.
Consider the same physical system as discussed in Example2. In this case, the time dependent boundary condition at ξ=1, (44), is changed to the form
(56)θ(1,τ)=(C0sinωτ)θ¯2.
In this case, the eigenvalues and eigenfunctions are the same as those given in Example 2.
Now,(57)F(ξ,τ)=-θ-2C01-r-[(β1+1)kmξβ1sinωτi-(ξ-r-)ωcmξβ2cosωτ].
Following the same solution procedures as shown, the exact solution for the general system can be derived as(58)θ(ξ,τ)=C0sinωτ1-r-θ-2(ξ-r-)θ(ξ,τ)=+∑n=1∞[X2(1)Xn,1(ξ)-X1(1)Xn,2(ξ)]Bn(τ),
where
(59)Bn(τ)=1ω[(λn2/ω)2+1]Bn(τ)=×{(λn2ωcosωτ+sinωτ-λn2ωe-λn2τ)γ-n1(ξ)-(λn2ωsinωτ-cosωτ-e-λn2τ)γ-n2(ξ)},δn=[∫r-1cmξβ2+1ϕn2(ξ)dξ],γ-n1(ξ)=-θ-2C0(1-r-)δn[∫01ϕn(ξ)(ξ-r-)ωcmξβ2+1dξ],γ-n2(ξ)=-θ-2C0(1-r-)δn[∫01ϕn(ξ)(β1+1)kmξβ1+1dξ],αn=0.
Figure5 shows the harmonic temperature variation of FG circular hollow cylinders with constant value of β1, various parameters of β2 at ξ=0.75, C0=10, and ω=2.5. It can be observed that with constant value of β1, the amplitude of temperature oscillation for the system with a higher value of β2 will be more than that of one with a lower value of β2.
In Table1, the temperature variations of FG circular hollow cylinders with constant value of β2 and various values of β1 and C0 at τ=0.5 are given. It can be observed that with constant parameter β2, the temperature of the mediums will decrease as parameter β1 is decreased.Table 1
Temperature distribution of FG circular hollow cylinders with constant value ofβ2 and various parameters of β1 and C0 at τ=0.5, [km=1, cm=1, θ(1,τ)=(C0sinωτ)θ¯2:θ¯2=4 and ω=2.5].
C
0
β
1
β
2
ξ
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0.75
1
0
4.595
2.259
1.845
1.632
0.380
1
0
4.592
2.285
1.990
1.820
0.380
1.25
0
4.486
2.279
2.133
1.975
0.380
1
0.75
1
0
45.952
22.589
18.453
16.320
3.796
1
0
45.920
22.848
19.895
18.202
3.796
1.25
0
44.865
22.788
21.329
19.752
3.796
10
0.75
1
0
459.520
225.888
184.526
163.201
37.959
1
0
459.196
228.480
198.950
182.017
37.959
1.25
0
448.647
227.881
213.290
197.520
37.959Figure 5
Temperature variation of FG circular hollow cylinders with constant value ofβ1 and various values of β2 at ξ=0.75, [km=1, cm=1, θ(1,τ)=(C0sinωτ)θ¯2:θ¯2=4, C0=10 and ω=2.5].


## 5. Conclusions

The problem of heat conduction with general time-dependent boundary conditions cannot be solved directly by the method of separation of variables. In most of the analyses, an integral transform was used to remove the time-dependent term. In this paper, a new analytic solution method is developed to find the analytic closed solutions for the transient heat conduction in FG circular hollow cylinders with general time-dependent boundary conditions. The developed solution method is free of any kind of integral transformation.By introducing suitable shifting functions, the governing second-order regular singular differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. The analytic solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analyses are given to illustrate the efficiency and the accuracy of the analysis. The proposed solution method can also be extended to the problems with various kinds of FG materials and time-dependent boundary conditions.



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