Well-posed coupled fractional hyperbolic problem of thermoelasticity

Abstract

This is an attempt to construct the well-posed hyperbolic heat conduction model based on the Caputo fractional derivative and to study the corresponding coupled thermoelastic problem. The continuous dependence on initial data and energy supply, and the uniqueness of the solutions are mathematically proved. The general closed-form solution of the time fractional conduction model for the initial Dirichlet boundary value problem is obtained analytically by applying the Laplace transform and finite Fourier sine transform in one-dimensional case. The application of theoretical study for heat propagation in the wire is considered. As a special case, two different examples have been discussed to study the analysis of the temperature distributions in the spatial geometry. The influence of the fractional orders on the speed of heat conductivity in the model is discussed. The physical behavior of the temperature distribution has been graphically represented for different fractional orders. Furthermore, the thermal stress analysis is studied using the coupled thermoelasticity theory. In the Laplace domain, the analytical solutions have been obtained. The Gaver–Stehfest technique was employed to numerically perform time domain inversions of the Laplace transforms, which satisfied Kuznetsov’s convergence theorem.

Keywords

Well-posed problem coupled thermoelasticity hyperbolic heat conduction model Caputo fractional derivative integral transform

1. Introduction

The Fourier approach to describe heat conduction shows the infinite speed of wave propagation. This situation is not generally regarded as a physical situation since it contradicts the causality principle. To remove this issue, Cattaneo [1] and Vernotte [2] improved Fourier law by incorporating a relaxation time parameter and introduced the new constitutive relation. Thermal lagging in wave theory was discussed by Joseph and Preziosi [3] and Ozisik and Tzou [4]. Zlamal [5] and Fulks and Guenther [6] studied an initial value problem (IVP) of the hyperbolic model. Nishihara [7] gave a review of the hyperbolic equation subject to the Cauchy problem.

Glass et al. [8] applied MacCormack’s predictor-corrector scheme; Kiwan et al. [9] applied the trial solution method; Chen [10] used methods of Laplace, Green’s function and the ϵ-algorithm acceleration; Ciegis [11] developed and applied explicit and implicit Euler schemes; Kalis and Buikis [12] applied the finite difference scheme and the difference scheme using the spectrum; Chen [13] used hybrid transform technique; Dassios and Font [14] applied means of variable separation method and Fourier series to obtain analytical and approximate solution of hyperbolic heat equation. Povstenko and Ostoja-Starzewski [15] discussed the hyperbolic equation with a harmonic source in the Cartesian domain. Coimbra et al. [16] studied temperature distribution in thermal lens spectroscopy through the hyperbolic conduction model.

Kaliski [17] and Lord and Shulman [18] developed generalized dynamic thermoelasticity based on hyperbolic heat equation. Chandrasekharaiah [19] and Chandrasekharaiah [20] gave a review of theories of generalized and hyperbolic thermoelasticity. Ignaczak and Ostoja-Starzewski [21] discussed the dynamic coupled thermoelasticity of second sound for solid materials. Povstenko [22] proposed a fractional heat equation for modeling thermoelasticity. Compte and Metzler [23] suggested different time fractional hyperbolic heat conduction models. Povstenko [24] developed theories for thermal stresses that depend on time fractional Cattaneo equations. Kudinov and Kudinov [25] studied dynamic stresses in an infinite plate through hyperbolic heat equation.

Atanackovic et al. [26] developed the space-time fractional Cattaneo heat equation. They obtained solutions of developed equation based on the Cauchy problem. Ghazizadeh et al. [27] used both explicit and implicit finite difference methods to obtain solutions of the fractional Cattaneo problem. Qi et al. [28] employed the Laplace transform method to solve the generalized fractional Cattaneo model for laser heating. The heat conduction model depends on the fractional single-phase-lagging approach and has been studied by Mishra and Rai [29]. Alikhanov [30] provided a priori estimations for solving fractional differential equations subject to different boundary problems.

The phrase well-posed problem was invented by Jacques Hadamard (1865–1963), and he thought that mathematical models for the description of physical phenomena should satisfy criteria of existence, uniqueness, and continuous dependence. When the problem is not well-posed, then the solutions have high sensitivity to changes. A small change in the data leads to a relevant difference in the behaviors of the solutions and causes numerical instabilities. Hence, well-posedness is a crucial step to examine the numerical aspects in the analysis of a problem.

In this paper, our aim is to show well-posedness of the problem and study the coupled thermoelasic problem of fractional hyperbolic model. The continuous dependence and the uniqueness are mathematically proved for hyperbolic model based on the fractional derivative subject to initial Dirichlet boundary conditions. The general closed-form solution of hyperbolic time fractional heat equation is obtained analytically by employing the Laplace transform and finite Fourier sine transform in one-dimensional case.

The application of theoretical study for heat propagation in the wire is considered. As a special case, two different examples have been discussed to study the analysis of the temperature distributions in the Cartesian domain. The physical behavior of the temperature variations has been graphically represented for different fractional orders. Furthermore, the thermal stress analysis is studied by considering the coupled theory of thermoelasticity.

2. Basic equations

The Cattaneo [1] and Vernotte [2] form is represented by

q + τ_{0} \frac{\partial q}{∂t} = - κ \nabla T, τ_{0} > 0

(1)

gives to hyperbolic model of heat conduction as

\frac{∂T}{∂t} + τ_{0} \frac{\partial^{2} T}{\partial t^{2}} = ã Δ T,

(2)

where $ã = \frac{κ}{ρc}$ denotes thermal diffusivity, κ represents thermal conductivity, c represents specific heat capacity, $τ_{0}$ represents relaxation time. Consider following generalization of equation (1) as

I^{1 - α} q + τ_{0} \frac{\partial^{α} q}{\partial t^{α}} = - κ \nabla T, 0 < α \leq 1 .

(3)

The fractional hyperbolic model of heat conduction using equation (3) is given by

\frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = ã Δ T, 0 < α \leq 1, 1 < β \leq 2,

(4)

where

\frac{\partial^{α} T}{\partial t^{α}} = {\begin{matrix} \frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{∂T}{∂ξ} \frac{dξ}{{(t - ξ)}^{α}}, & 0 < α < 1 \\ \frac{1}{Γ (2 - α)} \int_{0}^{t} \frac{\partial^{2} T}{\partial ξ^{2}} \frac{dξ}{{(t - ξ)}^{α - 1}}, & 1 < α < 2 \end{matrix}

(5)

denotes the Caputo [31] fractional derivatives.

Consider equation (4) in bounded domain B, some boundary and initial conditions are imposed on B whose boundary ∂B is a smooth border that allows the divergence theorem to be applied.

Suppose that the homogeneous boundary conditions are

T (x, t) = 0, x \in ∂B,

(6)

and introduce the initial conditions as

T (x, 0) = T_{0} (x) and \frac{∂T}{∂t} (x, 0) = T_{1} (x), x \in B,

(7)

where $T_{0}$ and $T_{1}$ are continuous functions.

The primary goal is to investigate the defined problem as determined by equations (4), (6), and (7).

3. Continuous dependence of the solution

This section represents proof of the continuous dependence for the above-defined problem. By considering initial data and supply term $f (x, t)$ , equation (4) becomes

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = ã Δ T + f (x, t), τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, ã > 0, \end{matrix} \end{matrix}

(8)

where f is function of material point x and time t.

Lemma 1. Following Alikhanov [30], for any absolutely continuous function $M (t)$ and fractional derivative of order $α \in (0, 1]$ ,

\begin{matrix} \begin{matrix} M \frac{\partial^{α} M}{\partial t^{α}} = \frac{1}{2} \frac{\partial^{α} M^{2}}{\partial t^{α}} + \frac{α}{2 Γ (1 - α)} \int_{0}^{t} \frac{ds}{{(t - s)}^{1 - α}} [\int_{0}^{s} {\frac{M^{'} (y) dy}{{(t - y)}^{α}}]}^{2}, \end{matrix} \end{matrix}

(9)

Theorem 1. If the energy function of equation (8) is given by

\begin{matrix} \begin{matrix} E (t) = \frac{1}{2} \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB + ã \int_{0}^{t} \int_{B} [| \nabla T (s) |^{2}] dBds, \end{matrix} \end{matrix}

(10)

Then

E (t) \leq E (0) + \int_{0}^{t} {{[E (s)]}^{\frac{1}{2}} {[g (s)]}^{\frac{1}{2}} + E (s)} ds,

(11)

where

g (t) = \int_{B} f^{2} (t) dB .

(12)

Proof. The main tool to show continuous dependence result. Consider equation (10) as

\begin{matrix} \begin{matrix} E (t) = \frac{1}{2} \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB + ã \int_{0}^{t} \int_{B} [| \nabla T (s) |^{2}] dBds . \end{matrix} \end{matrix}

(13)

□

From equation (9), one can write

M \frac{\partial^{α} M}{\partial t^{α}} \geq \frac{1}{2} \frac{\partial^{α} M^{2}}{\partial t^{α}}, 0 < α \leq 1 .

(14)

By taking the time derivative of equation (10) on both sides, one obtains

\begin{matrix} \begin{matrix} \frac{d}{dt} [E (t)] = \frac{1}{2} \frac{d}{dt} \int_{0}^{L} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB + ã \int_{B} | \nabla T (t) |^{2} dB - ã \int_{B} | \nabla T (0) |^{2} dB, \end{matrix} \end{matrix}

(15)

Using equation (14), one gets

\begin{matrix} \begin{matrix} \frac{d}{dt} [E (t)] \leq \frac{d}{dt} \int_{B} (T + τ_{0} T_{t}) \frac{\partial^{α} (T + τ_{0} T_{t})}{\partial t^{α}} dB + ã \int_{B} | \nabla T (t) |^{2} dB - ã \int_{B} | \nabla T (0) |^{2} dB, \end{matrix} \end{matrix}

(16)

By considering equation (8), one becomes

\begin{matrix} \begin{matrix} \frac{d}{dt} [E (t)] = \frac{d}{dt} \int_{B} (T + τ_{0} T_{t}) [ã Δ T (t) + f (x, t)] dB + ã \int_{B} | \nabla T (t) |^{2} dB - ã \int_{B} | \nabla T (0) |^{2} dB, \end{matrix} \end{matrix}

(17)

Using Holder’s result and the divergence theorem, equation (17) achieves

\begin{matrix} \begin{matrix} \frac{d}{dt} [E (t)] \leq \frac{d}{dt} {[\int_{B} {(T + τ_{0} T_{t})}^{2} dB]}^{\frac{1}{2}} {[\int_{B} f^{2} dB]}^{\frac{1}{2}} + ã \int_{B} | \nabla T (t) |^{2} dB, \end{matrix} \end{matrix}

(18)

By taking integration of equation (18) on both sides, one gets

E (t) \leq E (0) + \int_{0}^{t} {{[E (s)]}^{\frac{1}{2}} {[g (s)]}^{\frac{1}{2}} + E (s)} ds,

(19)

where

g (t) = \int_{B} f^{2} (t) dB .

(20)

Hence the proof.

4. Uniqueness of the solution

Lemma 2. For absolutely continuous function,

h (t) = \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB,

(21)

then

h (t_{1}) \leq h (t_{0}) \forall t_{1} \geq t_{0},

(22)

Proof. Using equation (14), one can write

\frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} \leq 2 (T + τ_{0} T_{t}) [\frac{\partial^{α} (T + τ_{0} T_{t})}{\partial t^{α}}] .

(23)

□

By taking equation (4), one gets

\frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} = 2 (T + τ_{0} T_{t}) [ã Δ T] .

(24)

On solving equation (24), one obtains

\frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} = (2 ã) T Δ T + (2 ã τ_{0}) T_{t} Δ T .

(25)

By taking integration with respect to material point over B, one becomes

\begin{matrix} \begin{matrix} \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB = (2 ã) \int_{B} T Δ T dB + (2 ã τ_{0}) \int_{B} T_{t} Δ T dB, \end{matrix} \end{matrix}

(26)

Using the divergence theorem and Green’s identity, one gets

\begin{matrix} \begin{matrix} \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB = - (2 ã) \int_{B} | \nabla T |^{2} dB - (2 ã τ_{0}) \int_{B} \nabla T_{t} \cdot \nabla T dB, \end{matrix} \end{matrix}

(27)

This is negative semi-definite because $ã > 0$ and $τ_{0} > 0$ .

The following holds with the help of equation (27)

\frac{d}{dt} [h (t)] \leq 0,

(28)

where

h (t) = \int_{B} \frac{\partial^{α} {(T + τ_{0} T_{t})}^{2}}{\partial t^{α}} dB .

(29)

By taking integration from $t_{0}$ to $t_{1}$ ( $t_{1}$ ≥ $t_{0}$ ) of equation (28), one obtains

h (t_{1}) \leq h (t_{0}) \forall t_{1} \geq t_{0} .

(30)

This gives the desired inequality.

Theorem 2. (Uniqueness) The fractional hyperbolic model of heat conduction

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = ã Δ T (x, t), ã > 0, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, \end{matrix} \end{matrix}

(31)

has a unique solution for given initial and boundary conditions.

Proof. Assume that there are two solutions $T_{1} (x, t)$ and $T_{2} (x, t)$ of fractional hyperbolic model of heat equation (31) that satisfy the same boundary condition (6) and the initial conditions (7).

Let $T^{*} (x, t)$ = $T_{1} (x, t) - T_{2} (x, t)$ . Taking $t_{1} = t$ , $t_{0} = 0$ , then equation (30) becomes

h (t) \leq h (0) \forall t \geq 0 .

(32)

□

Using equation (29), one obtains

\begin{matrix} \begin{matrix} \int_{B} {\frac{\partial^{α}}{\partial t^{α}} [{(T^{*} + τ_{0} T_{t}^{*})}^{2} (t)]} dB \leq \int_{B} {\frac{\partial^{α}}{\partial t^{α}} [{(T^{*} + τ_{0} T_{t}^{*})}^{2} (0)]} dB = 0, \end{matrix} \end{matrix}

(33)

This needs

T^{*} (x, t) = 0 .

(34)

T_{1} (x, t) = T_{2} (x, t) .

(35)

Hence the proof.

5. Mathematical model

In the one-dimensional case x = {x}, consider a wire of length L and the temperature of a wire is zero at both ends of the domain $x \in (0, L)$ . Assume that $f (x)$ and $g (x)$ denote the initial temperature distributions and the initial rate of temperature change in the bounded domain $0 < x < L$ , respectively.

The fractional hyperbolic model of heat conduction becomes

\frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = ã \frac{\partial^{2} T}{\partial x^{2}}, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2,

(36)

subject to

T (0, t) = 0 and T (L, t) = 0, t > 0,

(37)

and

T (x, 0) = f (x) and \frac{∂T}{∂t} (x, 0) = g (x), 0 < x < L,

(38)

where $f (x)$ and $g (x)$ are given continuous functions.

The main objective is to achieve the closed-form solution of the above-defined problem.

5.1. General solution of the problem

The solution to equations (36)–(38) is determined using the Laplace transform and the finite Fourier sine transform.

The nondimensional quantities listed below have been introduced in equation (36)

\bar{t} = \frac{ã^{\frac{1}{α}}}{L^{\frac{2}{α}}} t, \bar{x} = \frac{x}{L}, \bar{τ_{0}} = \frac{ã^{\frac{1}{α}}}{L^{\frac{2}{α}}} τ_{0}, \bar{T} = \frac{T}{T_{0}},

(39)

where L denotes the characteristic length.

In view of nondimensional quantities, the fractional hyperbolic model of heat conduction equation (36) can be converted into nondimensional form as (neglecting bar symbol)

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = \frac{\partial^{2} T}{\partial x^{2}}, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, 0 < x < 1 . \end{matrix} \end{matrix}

(40)

Applying the Laplace transform on both sides of equation (40), one obtains

\begin{matrix} \begin{matrix} [p^{α} \bar{T} - p^{α - 1} T (x, 0)] + τ_{0} [p^{β} \bar{T} - p^{β - 1} T (x, 0) - p^{β - 2} \frac{∂T}{∂t} (x, 0)] = \frac{\partial^{2} \bar{T}}{\partial x^{2}}, \end{matrix} \end{matrix}

(41)

where p represents the parameter of the Laplace transform, and overline denotes the Laplace transform.

Using equation (38), one obtains

\begin{matrix} \begin{matrix} \frac{\partial^{2}}{\partial x^{2}} \bar{T} (x, p) - (τ_{0} p^{β} + p^{α}) \bar{T} (x, p) = - (τ_{0} p^{β - 1} + p^{α - 1}) f (x) - τ_{0} p^{β - 2} g (x), \end{matrix} \end{matrix}

(42)

Applying the finite Fourier sine transform and using equation (37), one achieves

{\bar{T}}^{*} (k, p) = \frac{(τ_{0} p^{β - 1} + p^{α - 1}) f (k) + τ_{0} p^{β - 2} g (k)}{(τ_{0} p^{β} + p^{α} + k^{2} π^{2})},

(43)

where the asterisk denotes the finite Fourier sine transform, k represents parameter of the finite Fourier sine transform, $f (k)$ and $g (k)$ are the finite Fourier sine transform $f (x)$ and $g (x)$ , respectively.

To find a solution in the time domain, one have [32]

\begin{matrix} \begin{matrix} L^{- 1} (\frac{p^{γ - 1}}{p^{α} + a p^{β} + b}; t) = t^{α - β} \sum_{r = 0}^{\infty} {(- a)}^{r} t^{(α - β) r} E_{α, α + (α - β) r - γ + 1}^{r + 1} (- b t^{α}), \end{matrix} \end{matrix}

(44)

where $ℛ (α) > 0$ , $ℛ (β) > 0$ , $ℛ (γ) > 0$ , $| \frac{a p^{β}}{(p^{α} + b)} | < 1$ , and provided that the series in equation (44) is convergent.

Applying inverse Laplace transform and using equation (44), equation (43) becomes

\begin{matrix} \begin{matrix} T^{*} (k, t) = t^{α - β} \sum_{r = 0}^{\infty} {(- τ_{0})}^{r} t^{(α - β) r} {[f (k)] [τ_{0} E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α}) \\ + E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α})] + [g (k)] [τ_{0} E_{α, α + (α - β) r - β + 2}^{r + 1} (- k^{2} π^{2} t^{α})]}, \end{matrix} \end{matrix}

(45)

Applying inverse finite Fourier sine transform, one becomes

\begin{matrix} \begin{matrix} T (x, t) = 2 t^{α - β} \sum_{r = 0}^{\infty} {(- τ_{0})}^{r} t^{(α - β) r} {[\sum_{k = 1}^{\infty} f (k) \sin (kπx)] [τ_{0} E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α}) \\ + E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α})] + [\sum_{k = 1}^{\infty} g (k) \sin (kπx)] [τ_{0} E_{α, α + (α - β) r - β + 2}^{r + 1} (- k^{2} π^{2} t^{α})]} . \end{matrix} \end{matrix}

(46)

This is the required solution.

5.2. Illustrative examples

To study analysis of temperature distributions under the framework of equation (36) by considering the copper wire of length $L = 1 m$ , thermal conductivity $κ = 400 W ∕ mK$ , thermal diffusivity $ã = 1.1 \times 1 0^{- 4} m^{2} ∕ s$ . The temperature distribution in the wire is studied by considering quadratic and sinusoidal functions.

5.2.1. Example 1

Consider the nondimensional fractional hyperbolic model

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = \frac{\partial^{2} T}{\partial x^{2}}, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, 0 < x < 1 . \end{matrix} \end{matrix}

(47)

subject to

T (0, t) = 0 and T (1, t) = 0, t > 0,

(48)

and

T (x, 0) = x (x - 1) and \frac{∂T}{∂t} (x, 0) = 0, 0 < x < 1 .

(49)

The solution using equation (46) is represented by

\begin{matrix} \begin{matrix} T (x, t) = 2 t^{α - β} \sum_{r = 0}^{\infty} {(- 1)}^{r + 1} τ_{0}^{r} t^{(α - β) r} {[\sum_{k = 1}^{\infty} (\frac{\sin (k)}{k} + \frac{2 + 2 \cos (k)}{k^{3}}) \sin (kπx)] \\ [τ_{0} E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α}) + E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α})]} . \end{matrix} \end{matrix}

(50)

5.2.2. Example 2

Consider the nondimensional fractional hyperbolic model

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = \frac{\partial^{2} T}{\partial x^{2}}, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, 0 < x < 1 . \end{matrix} \end{matrix}

(51)

subject to

T (0, t) = 0 and T (1, t) = 0, t > 0,

(52)

and

T (x, 0) = \sin (πx) and \frac{∂T}{∂t} (x, 0) = 0, 0 < x < 1 .

(53)

The solution using equation (46) is represented by

\begin{matrix} \begin{matrix} T (x, t) = t^{α - β} \sum_{r = 0}^{\infty} {(- τ_{0})}^{r} t^{(α - β) r} {[\sum_{k = 1}^{\infty} (\frac{(π + k) \sin (π - k) - (π - k) \sin (π + k)}{k}) \sin (kπx)] \\ [τ_{0} E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α}) + E_{α, α + (α - β) r - β + 1}^{r + 1} (- k^{2} π^{2} t^{α})]} . \end{matrix} \end{matrix}

(54)

5.3. Numerical results and discussion

This section presents graphical representations of analytically derived solutions from the previous section for various fractional orders $α, β$ . We use graphical representations for equations (50) and (54) to demonstrate the physical aspects of the proposed model. We are primarily interested in the effect of fractional parameters, specifically memory effects α and wave-like thermal transport β.

Figures 1 and 2 represent the behavior of the obtained solutions of equations (50) and (54) with the use of three-dimensional (3D) plots and demonstrate the impact of fractional orders on temperature distributions along the wire under the framework of time fractional hyperbolic heat conduction equation. From both graphs, we can conclude that thermal energy is transported in wavelike patterns. The fractional parameters influence the propagation velocity of heat on temperature along the wire.

Figure 1.

3D surfaces of temperature $T (x, t)$ distribution along wire with respect to fractional orders (a) $α = 0.5$ , $β = 1.5$ ; (b) $α = 0.65$ , $β = 1.65$ ; (c) $α = 1$ , $β = 2$ .

Figure 2.

3D surfaces of temperature $T (x, t)$ distribution along wire with respect to fractional orders (a) $α = 0.5$ , $β = 1.5$ ; (b) $α = 0.65$ , $β = 1.65$ ; (c) $α = 1$ , $β = 2$ .

6. Thermal stress analysis

This section is to study thermal stresses in a wire of length $1 m$ using coupled theory of thermoelasticity.

The nondimensional form of the fractional hyperbolic model of heat conduction, equation of motion, and thermal stress equation are as follows:

\begin{matrix} \begin{matrix} \frac{\partial^{α} T}{\partial t^{α}} + τ_{0} \frac{\partial^{β} T}{\partial t^{β}} = \frac{\partial^{2} T}{\partial x^{2}}, τ_{0} > 0, 0 < α \leq 1, 1 < β \leq 2, 0 < x < 1, \end{matrix} \end{matrix}

(55)

\frac{∂e}{∂x} - \frac{∂T}{∂x} = \frac{\partial^{2} u}{\partial t^{2}},

(56)

σ_{xx} = e - T .

(57)

The initial conditions

T (x, 0) = x, \frac{∂T}{∂t} (x, 0) = 0, 0 < x < 1 .

(58)

The Dirichlet boundary conditions

T (0, t) = 0, T (1, t) = 0, t > 0 .

(59)

The mechanical boundary conditions

σ_{xx} (0, t) = 0, σ_{xx} (1, t) = 0, t > 0 .

(60)

The governing nondimensional equations (55)–(57) converted in the Laplace domain as

\begin{matrix} \begin{matrix} [p^{α} \bar{T} - p^{α - 1} T (x, 0)] + τ_{0} [p^{β} \bar{T} - p^{β - 1} T (x, 0) - p^{β - 2} \frac{∂T}{∂t} (x, 0)] = \frac{\partial^{2} \bar{T}}{\partial x^{2}}, \end{matrix} \end{matrix}

(61)

\frac{∂ē}{∂x} - \frac{\partial \bar{T}}{∂x} = p^{2} ū,

(62)

{\bar{σ}}_{xx} = ē - \bar{T} .

(63)

Equation (61) reduces to

\frac{\partial^{2} \bar{T}}{\partial x^{2}} - (τ_{0} p^{β} + p^{α}) \bar{T} = - (τ_{0} p^{β - 1} + p^{α - 1}) x .

(64)

By taking differentiation of equation (62) with respect to x, equation (62) becomes

\frac{\partial^{2} ē}{\partial x^{2}} - \frac{\partial^{2} \bar{T}}{\partial x^{2}} = p^{2} ē .

(65)

The temperature is obtained by solving equation (64)

\begin{matrix} \begin{matrix} \bar{T} (x, p) = \frac{(τ_{0} p^{β - 1} + p^{α - 1}) x}{(τ_{0} p^{β} + p^{α})} - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \sinh (\sqrt{τ_{0} p^{β} + p^{α}} x)}{(τ_{0} p^{β} + p^{α}) \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} . \end{matrix} \end{matrix}

(66)

Using equation (66), equation (65) becomes

\begin{matrix} \begin{matrix} ē (x, p) = - [\frac{(τ_{0} p^{β - 1} + p^{α - 1}) (\sqrt{τ_{0} p^{β} + p^{α}})}{p \sinh (p) (p^{2} - (τ_{0} p^{β} + p^{α}))}] \sinh (px) - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh (px)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} [\frac{\sinh ((p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p + \sqrt{τ_{0} p^{β} + p^{α}})} \\ - \frac{\sinh ((p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p - \sqrt{τ_{0} p^{β} + p^{α}})}] - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \sinh (px)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} [\frac{\cosh ((p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p + \sqrt{τ_{0} p^{β} + p^{α}})} - \frac{\cosh ((p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p - \sqrt{τ_{0} p^{β} + p^{α}})}] . \end{matrix} \end{matrix}

(67)

The displacement function is achieved using equations (66)–(67) to equation (62), one obtains

\begin{matrix} \begin{matrix} ū (x, p) = \frac{1}{p^{2}} {(τ_{0} p^{β - 1} + p^{α - 1}) (\sqrt{τ_{0} p^{β} + p^{α}}) \frac{\cosh ((\sqrt{τ_{0} p^{β} + p^{α}}) x)}{2 \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} - \frac{(τ_{0} p^{β - 1} + p^{α - 1})}{(τ_{0} p^{β} + p^{α})} \\ - [\frac{(τ_{0} p^{β - 1} + p^{α - 1}) (\sqrt{τ_{0} p^{β} + p^{α}})}{\sinh (p) (p^{2} - (\sqrt{τ_{0} p^{β} + p^{α}}))}] \cosh (px) - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh ((2 p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} \\ + \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh ((2 p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} + \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh ((2 p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{2 \sinh (\sqrt{τ_{0} p^{β} + p^{α}}) (p + \sqrt{τ_{0} p^{β} + p^{α}})} \\ + \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh ((2 p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{2 \sinh (\sqrt{τ_{0} p^{β} + p^{α}}) (p - \sqrt{τ_{0} p^{β} + p^{α}})}} . \end{matrix} \end{matrix}

(68)

The stress ${\bar{σ}}_{xx}$ is found by substituting equations (66) and (67) to equation (63), one obtains

\begin{matrix} \begin{matrix} {\bar{σ}}_{xx} (x, p) = - [\frac{(τ_{0} p^{β - 1} + p^{α - 1}) (\sqrt{τ_{0} p^{β} + p^{α}})}{p \sinh (p) (p^{2} - (τ_{0} p^{β} + p^{α}))}] \sinh (px) - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \cosh (px)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} [\frac{\sinh ((p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p + \sqrt{τ_{0} p^{β} + p^{α}})} \\ - \frac{\sinh ((p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p - \sqrt{τ_{0} p^{β} + p^{α}})}] - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \sinh (px)}{2 p \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} [\frac{\cosh ((p + \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p + \sqrt{τ_{0} p^{β} + p^{α}})} - \frac{\cosh ((p - \sqrt{τ_{0} p^{β} + p^{α}}) x)}{(p - \sqrt{τ_{0} p^{β} + p^{α}})}] \\ - \frac{(τ_{0} p^{β - 1} + p^{α - 1}) x}{(τ_{0} p^{β} + p^{α})} + \frac{(τ_{0} p^{β - 1} + p^{α - 1}) \sinh (\sqrt{τ_{0} p^{β} + p^{α}} x)}{(τ_{0} p^{β} + p^{α}) \sinh (\sqrt{τ_{0} p^{β} + p^{α}})} . \end{matrix} \end{matrix}

(69)

Equations (66)–(69) represent solutions in the Laplace domain.

6.1. Gaver–Stehfest method

Gaver [33] and Stehfest’s [34] method is used to approximate the time domain solution

\begin{matrix} \begin{matrix} g (t) \approx g_{n} (t) = \frac{ln (2)}{t} \sum_{j = 1}^{2 n} {{(- 1)}^{n + j} [\sum_{k = ⌊ \frac{j + 1}{2} ⌋}^{min (j, n)} \frac{k^{n + 1}}{n!} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} k \\ j - k \end{matrix})] \cdot G (\frac{j ln (2)}{t})}, \end{matrix} \end{matrix}

(70)

where $⌊ \frac{j + 1}{2} ⌋$ is the flooring function and $2 n$ is an even integer whose value depends on the word length of the computer used.

6.2. Kuznetsov’s convergence theorem

Following Kuznetsov [35], let $g : (0, \infty) \to ℝ$ is a locally integrable function such that its Laplace transform $G (p)$ exists for all $p > 0$ and the sequence $g_{n} (t)$ is defined by equation (70), then the convergence of sequence $g_{n} (t)$ depends on the values of the function g in the neighborhood of t. If the function g is of bounded variation in the neighborhood of t, then the sequence $g_{n} (t)$ → $\frac{g (t + 0) + g (t - 0)}{2}$ when n → ∞.

6.3. Results and discussion

Figure 3: It exhibits temperature $T (x, t)$ variation with respect to various values of $α, β$ . It is found that the temperature value increases in $0 < x < 0.4$ and then decreases in $0.5 < x < 1$ when parameters $α, β$ increase. The temperature values are expansive behavior along distance x. The temperature distribution shows large differences for different values of $α, β$ in $0.4 < x < 0.6$ .

Figure 3.

Temperature $T (x, t)$ for different fractional orders along the wire.

Figure 4: It exhibits displacement $u (x, t)$ variation with respect to various values of $α, β$ . The displacement values are compressive behavior in $0.4 < x < 0.6$ along distance x for different values of $α, β$ . Initially, the displacement values are zero in the range of $x \in (0, 0.4)$ and show large differences in $0.6 < x < 1 .$

Figure 4.

Displacement $u (x, t)$ for different fractional orders along the wire.

Figure 5: It exhibits stress $σ (x, t)$ variation with respect to various values of $α, β$ . The stress distribution shows large differences for different values of $α, β$ in $0.6 < x < 1$ . It is observed that the stress function attains minima at $x = 0.8$ . It begins at zero and then finishes at zero.

Figure 5.

Stress $σ (x, t)$ for different fractional orders along the wire.

7. Conclusion

The main concluding remarks are as follows:

The continuous dependence on the initial data and energy supply term to the fractional hyperbolic heat equation satisfying condition $τ_{0} > 0$ is mathematically proved. This guarantees stability in the sense of Hadamard.

The uniqueness of the solution to the Caputo fractional hyperbolic model of heat equation subject to the initial boundary problem is mathematically proved.

The initial Dirichlet boundary problem of hyperbolic heat equation based on the Caputo time fractional derivatives of orders α $\in (0, 1]$ and β $\in (1, 2]$ is presented in the one-dimensional spatial domain.

The general closed-form solution of fractional hyperbolic model is obtained analytically by employing the Laplace transform and finite Fourier sine transform.

The influence of changing the parameters of the fractional parameters on temperature variation has been studied. The solutions are computed for the temperature field satisfying the defined initial and boundary conditions.

The thermal stress investigation of the defined problem is studied using coupled theory of thermoelasticity. Due to the application of the relaxation time $τ_{0} > 0$ , heat wave propagates with finite speed, making the model more physically realistic.

The influence of changing the parameters of the fractional orders $α, β$ on different fields such as temperature $T (x, t)$ , displacement $u (x, t)$ , and stress $σ (x, t)$ has been studied. The results are computed for different fields satisfying the prescribed initial and boundary conditions of the proposed model.

According to numerical results, the fractional parameter evolved into a new measure of its ability to conduct thermal energy.

The obtained numerical results can be applied in the field of nano-electromechanical systems, heat transfer in rarefied media, laser-aided material processing, biological medicine, material flaw analysis, electromagnetic irradiation of a solid, and so on.

Footnotes

ORCID iD

Vinayak S Kulkarni

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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