Local fractional variational iteration algorithm II for non-homogeneous model associated with the non-differentiable heat flow

Abstract

In this article, we begin with the non-homogeneous model for the non-differentiable heat flow, which is described using the local fractional vector calculus, from the first law of thermodynamics in fractal media point view. We employ the local fractional variational iteration algorithm II to solve the fractal heat equations. The obtained results show the non-differentiable behaviors of temperature fields of fractal heat flow defined on Cantor sets.

Introduction

The model for the non-differentiable heat flow involving the local fractional vector calculus was discussed in Yang ¹ and Cattani et al. ² Several methods for solving the heat flow problem have been suggested and applied in the literature such as cylindrical-coordinate method of Cantor-type (CCMCT), ³ Laplace variational iteration method (LVIM),^4,5 Laplace transform (LT) method, ⁶ and variational iteration method (VIM).^7,8 There are also new methods for dealing with the partial differential equations (PDEs) within local fractional time and space derivatives (LFDE). For example, the decomposition method (DM) was used to solve the Burgers’ equation, ⁹ diffusion equation, ¹⁰ and the Laplace equation ¹¹ involving the LFDE. Yang et al. ¹² presented the similarity solution method to solve the diffusion equation with LFDE. Cao et al. ¹³ proposed the functional method (FM) to deal with the LFDE. Su et al. ¹⁴ suggested the fractional complex transform (FCT) to discuss the local fractional wave equation (LFWE).

In this article, variational iteration algorithm II within local fractional derivative operator (local fractional variational iteration algorithm II (LFVIA-II)) (see Yang and Zhang ¹⁵ Liu et al., ¹⁶ Baleanu et al. ¹⁷ ) will be applied to deal with the non-differentiable problem in fractal heat transfer.

The layout of this article is as follows. In section “The non-homogeneous model for the non-differentiable heat flow,” we provide the basic theory of the non-differentiable heat flow. Section “Analysis of the methodology used” presents the LFVIA-II. Section “The non-homogeneous model for the non-differentiable heat flow” describes the solution for the non-homogeneous model associated with the non-differentiable heat flow. Finally, in section “Conclusion,” the conclusions are presented.

The non-homogeneous model for the non-differentiable heat flow

In this section, we will derive the non-homogeneous models for non-differentiable heat flow.

The first law of thermodynamics in fractal media takes the following form^1,2

$$\begin{array}{c}\multicolumn{1}{c}{\int \underset{S^{\left(\chi \right)}}{\int }{\kappa }^2{\nabla }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )•d{\mathit{S}}^{\left(\chi \right)}}\\ \multicolumn{1}{c}{+\underset{V^{\left(\nu \right)}}{\int \int \int }\varphi (\varsigma ,\xi ,\zeta ,\tau )d{V}^{\left(\nu \right)}}\\ \multicolumn{1}{c}{=\underset{V^{\left(\gamma \right)}}{\int \int \int }{\rho }_{\delta }{c}_{\delta }\frac{{\partial }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )}{\partial {\tau }^{\delta }}d{V}^{\left(\nu \right)}}\end{array}$$

(1)

which reduces to

$$\begin{array}{c}\multicolumn{1}{c}{\underset{V^{\left(\gamma \right)}}{\int \int \int }({\nabla }^{\delta }•\left({\kappa }^2{\nabla }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )\right)+\varphi (\varsigma ,\xi ,\zeta ,\tau )-{\rho }_{\delta }{c}_{\delta }\frac{{\partial }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )}{\partial {\tau }^{\delta }})}\\ \multicolumn{1}{c}{d{V}^{\left(\nu \right)}=0}\end{array}$$

(2)

By using the local fractional Gauss theorem of the fractal vector field given by (see Yang ¹ and Cattani et al. ² )

$$\underset{V^{\left(\nu \right)}}{\int \int \int }{\nabla }^{\delta }•\mathit{u}d{V}^{\left(\nu \right)}=\underset{S\left(\chi \right)}{∯}\mathit{u}•d{\mathit{S}}^{\left(\chi \right)}$$

(3)

where $\nu =(3/2)\chi =3\delta (1>\delta >0)$ , the local fractional surface integral of $u(r_P)$ is denoted by (see Yang ¹ and Cattani et al. ² )

$$\int \int u\left(r_P\right)d{\mathit{S}}^{\left(\chi \right)}=\underset{N\to \infty }{lim}\sum _{P=1}^{N}u\left(r_P\right){\mathit{n}}_P\Delta S_P^{\left(\chi \right)}$$

(4)

with $\Delta S_P^{(\chi )}\to 0$ as $N\to \infty$ , the local fractional volume integral of the function $\mathbf{u}(r_P)$ is denoted by

$$\int \int \int \mathbf{u}\left(r_P\right)d{V}^{\left(\nu \right)}=\underset{N\to \infty }{lim}\sum _{P=1}^N\mathbf{u}\left(r_P\right)\Delta V_P^{\left(\nu \right)}$$

(5)

with $\Delta V_P^{(\nu )}\to 0$ as $N\to \infty$ , and the parameters ${\rho }_{\delta }$ and $c_{\delta }$ are the density and the specific heat of the material, respectively.

The law of heat conduction in fractal media (the local fractional Fourier law) states as follows (see Yang ¹ and Cattani et al. ² )

$$\phi (\varsigma ,\xi ,\zeta ,\tau )=-\int \underset{S^{\left(\chi \right)}}{\int }{\kappa }^2{\nabla }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )•d{\mathit{S}}^{\left(\chi \right)}$$

(6)

where $\phi (\varsigma ,\xi ,\zeta ,\tau )$ is the amount of heat transferred per unit time and ${\kappa }^2$ is the thermal conductivity of the material in fractal media.

Let $\kappa$ , ${\rho }_{\alpha }$ , and $c_{\alpha }$ be constants. Then equation (2) can be written as follows (see Yang ¹ and Cattani et al. ² )

$${\rho }_{\delta }{c}_{\delta }\frac{{\partial }^{\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )}{\partial t^{\delta }}-{\kappa }^2{\nabla }^{2\delta }\vartheta (\varsigma ,\xi ,\zeta ,\tau )-\varphi (\varsigma ,\xi ,\zeta ,\tau )=0$$

(7)

where ${\nabla }^{2\delta }={\nabla }^{\delta }•{\nabla }^{\delta }$ .

This is the so-called non-homogeneous model for non-differentiable heat flow. The non-homogeneous model for non-differentiable heat flow in one-dimensional case is presented as follows (see Yang ¹ and Cattani et al. ² )

$$\frac{{\partial }^{\delta }\vartheta (\varsigma ,\tau )}{\partial {\tau }^{\delta }}-D_{\delta }\frac{{\partial }^{2\delta }\vartheta (\varsigma ,\tau )}{\partial {\varsigma }^{2\delta }}-\varphi (\varsigma ,\tau )=0$$

(8)

subject to initial-boundary conditions given by

$$-{\kappa }^2\frac{{\partial }^{\delta }\vartheta (0,\tau )}{\partial {\varsigma }^{\delta }}=\omega \left(\tau \right)$$

(9)

$$\vartheta (\varsigma ,0)=\theta \left(\varsigma \right)$$

(10)

$$\vartheta (0,\tau )=\upsilon \left(\tau \right)$$

(11)

Analysis of the methodology used

In this section, we will present the methodology of the LFVIA-II (see Yang and Zhang, ¹⁵ Liu et al., ¹⁶ Baleanu et al. ¹⁷ ).

The local fractional heat equation (8) for non-differentiable heat flow can be written in terms of the local fractional operator as follows

$$L_{\delta }\vartheta +R_{\delta }\vartheta =0$$

(12)

where $R_{\delta }=D_{\delta }(({\partial }^{2\delta }\vartheta (\varsigma ,\tau ))/\partial {\varsigma }^{2\delta })+\varphi (\varsigma ,\tau )$ and $L_{\delta }={\partial }^{\delta }\vartheta (\varsigma ,\tau )/\partial {\tau }^{\delta }$ , which is given by (see Yang and Zhang, ¹⁵ Liu et al., ¹⁶ Baleanu et al., ¹⁷ De Oliveira and Machado, ¹⁸ and Neamah ¹⁹ )

$$\frac{{\partial }^{\delta }\vartheta (\varsigma ,\xi )}{\partial {\varsigma }^{\delta }}|{}_{u=u_0}=\underset{\varsigma \to {\varsigma }_0}{lim}\frac{{\Delta }^{\delta }(\vartheta (\varsigma ,\xi )-\vartheta ({\varsigma }_0,\xi ))}{{(\varsigma -{\varsigma }_0)}^{\delta }}$$

(13)

with ${\Delta }^{\delta }(\vartheta (\varsigma ,\xi )-\vartheta ({\varsigma }_0,\xi ))\cong \Gamma (1+\delta )\Delta (\vartheta (\varsigma ,\xi )-\vartheta ({\varsigma }_0,\xi ))$ .

In this case, the correction local fractional functional is given by

$$\begin{array}{c}\multicolumn{1}{c}{{\vartheta }_{n+1}(\varsigma ,\tau )={\vartheta }_n(\varsigma ,\tau )+{}_0{I}_{\tau }^{\left(\delta \right)}}\\ \multicolumn{1}{c}{\left\{\mu [L_{\delta }{\vartheta }_n(\varsigma ,\tau )+R_{\delta }{\tilde{\vartheta }}_n(\varsigma ,\tau )]\right\}}\end{array}$$

(14)

where ${\tilde{\vartheta }}_n$ denotes a restricted local fractional variation, $\mu$ denotes a fractal Lagrange multiplier, and the local fractional integral operator is defined by (see Yang and Zhang, ¹⁵ Liu et al., ¹⁶ Baleanu et al., ¹⁷ De Oliveira and Machado, ¹⁸ Neamah ¹⁹ )

$$\begin{array}{c}\multicolumn{1}{c}{{}_l{I}_m^{\left(\delta \right)}\psi \left(s\right)=\frac{1}{\Gamma (1+\delta )}}\\ \multicolumn{1}{c}{\underset{l}{\overset{m}{\int }}\psi \left(s\right){\left(\mathrm{ds}\right)}^{\delta }=\frac{1}{\Gamma (1+\delta )}\underset{\Delta s\to 0}{lim}\sum _{j=0}^{j=N-1}\psi \left(s_j\right){\left(\Delta s_j\right)}^{\delta }}\end{array}$$

(15)

with $\Delta s=max\{\Delta s_1,\Delta s_2,\Delta s_j,\dots \}\mathrm{and}\Delta s_j=s_{j+1}-s_j(s_0=\sigma ,s_N=\upsilon ,j=0,\dots ,N-1)$ .

In view of the local fractional variational principle,^1,4,7,17,19 we set

$$\mu =-1$$

(16)

Therefore, from equations (12), (14), and (16), the LFVIA-II can be written as follows

$${\vartheta }_{n+1}(\varsigma ,\tau )={\vartheta }_0(\varsigma ,\tau )-{}_0{I}_{\tau }^{\left(\delta \right)}[D_{\delta }\frac{{\partial }^{2\delta }{\vartheta }_n(\varsigma ,\tau )}{\partial {\varsigma }^{2\delta }}+\varphi (\varsigma ,\tau )]$$

(17)

where

$${\vartheta }_0(\varsigma ,\tau )=\theta \left(\varsigma \right)$$

(18)

Finally, from equation (17), the non-differentiable solution of (8) takes the following form

$$\vartheta =\underset{n\to \infty }{lim}{\vartheta }_n$$

(19)

The analytical solution for thenon-homogeneous model for thenon-differentiable heat flow

The homogeneous model for the non-differentiable heat flow is presented as follows

$$\frac{{\partial }^{\delta }\vartheta (\varsigma ,\tau )}{\partial {\tau }^{\delta }}-5\frac{{\partial }^{2\delta }\vartheta (\varsigma ,\tau )}{\partial {\varsigma }^{2\delta }}-1=0$$

(20)

subject to initial-boundary conditions given by

$$\frac{{\partial }^{\delta }\vartheta (0,\tau )}{\partial {\varsigma }^{\delta }}=0$$

(21)

$$\vartheta (\varsigma ,0)={\sin }_{\delta }\left({\varsigma }^{\delta }\right)$$

(22)

$$\vartheta (0,\tau )=0$$

(23)

In view of equation (17), the LFVIA-II for equation (20) can be expressed in the following form

$${\vartheta }_{n+1}(\varsigma ,\tau )={\vartheta }_0(\varsigma ,\tau )-{}_0{I}_{\tau }^{\left(\delta \right)}[5\frac{{\partial }^{2\delta }{\vartheta }_n(\varsigma ,\tau )}{\partial {\varsigma }^{2\delta }}+1]$$

(24)

Assuming that ${\vartheta }_0(\varsigma ,\tau )={\sin }_{\delta }({\varsigma }^{\delta })$ , we have that

$$\begin{array}{c}\multicolumn{1}{c}{{\vartheta }_1(\varsigma ,\tau )={\vartheta }_0(\varsigma ,\tau )-{}_0{I}_{\tau }^{\left(\delta \right)}[5\frac{{\partial }^{2\delta }{\vartheta }_0(\varsigma ,\tau )}{\partial {\varsigma }^{2\delta }}+1]}\\ \multicolumn{1}{c}{={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_0{I}_{\tau }^{\left(\delta \right)}[5\frac{{\partial }^{2\delta }{\sin }_{\delta }\left({\varsigma }^{\delta }\right)}{\partial {\varsigma }^{2\delta }}+1]}\\ \multicolumn{1}{c}{={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_0{I}_{\tau }^{\left(\delta \right)}[-5{\sin }_{\delta }\left({\varsigma }^{\delta }\right)+1]}\\ \multicolumn{1}{c}{={\sin }_{\delta }\left({\varsigma }^{\delta }\right)(1+\frac{5{\tau }^{\delta }}{\Gamma (1+\delta )})-\frac{{\tau }^{\delta }}{\Gamma (1+\delta )}}\end{array}$$

(25)

$$\begin{array}{l}{\vartheta }_2\left(\varsigma ,\tau \right)={\vartheta }_0\left(\varsigma ,\tau \right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }{\vartheta }_1\left(\varsigma ,\tau \right)}{\partial {\varsigma }^{2\delta }}+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }}{\partial {\varsigma }^{2\delta }}\left({\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left(1+\frac{5{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}\right)+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^2\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)},\end{array}$$

(26)

$$\begin{array}{l}{\vartheta }_3\left(\varsigma ,\tau \right)={\vartheta }_0\left(\varsigma ,\tau \right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }{\vartheta }_2\left(\varsigma ,\tau \right)}{\partial {\varsigma }^{2\delta }}+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }}{\partial {\varsigma }^{2\delta }}\left({\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^2\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}\right)+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^3\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)},\end{array}$$

(27)

$$\begin{array}{l}{\vartheta }_4\left(\varsigma ,\tau \right)={\vartheta }_0\left(\varsigma ,\tau \right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }{\vartheta }_3\left(\varsigma ,\tau \right)}{\partial {\varsigma }^{2\delta }}+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }}{\partial {\varsigma }^{2\delta }}\left({\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^3\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}\right)+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^4\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)},\\ ⋮\end{array}$$

(28)

$$\begin{array}{l}{\vartheta }_n\left(\varsigma ,\tau \right)={\vartheta }_0\left(\varsigma ,\tau \right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }{\vartheta }_{n-1}\left(\varsigma ,\tau \right)}{\partial {\varsigma }^{2\delta }}+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)-{}_{0}I{{}_{\tau }}^{\left(\delta \right)}\left[5\frac{{\partial }^{2\delta }}{\partial {\varsigma }^{2\delta }}\left({\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^{n-1}\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}\right)+1\right]\\ ={\sin }_{\delta }\left({\varsigma }^{\delta }\right)\left({\displaystyle \sum _{i=0}^n\frac{5^i{\left(-1\right)}^i{\tau }^{i\delta }}{\Gamma \left(1+i\delta \right)}}\right)-\frac{{\tau }^{\delta }}{\Gamma \left(1+\delta \right)}.\end{array}$$

(29)

Therefore, the non-differentiable solution of equation (20) is found to be expressed as follows

$$\vartheta (\varsigma ,\tau )=\underset{n\to \infty }{lim}{\vartheta }_n(\varsigma ,\tau )={\sin }_{\delta }\left({\varsigma }^{\delta }\right)E_{\delta }(-5{\tau }^{\delta })-\frac{{\tau }^{\delta }}{\Gamma (1+\delta )}$$

(30)

where $E_{\delta }({\tau }^{\delta })=\sum _{n=0}^{\infty }{\tau }^{n\delta }/\Gamma (1+n\delta )$ , and its graph is shown in Figure 1 when $\delta =\ln 2/\ln 3$ .

Figure 1.

The non-differentiable solution of equation (20) in the closed form when $\delta =\ln 2/\ln 3$ .

Conclusion

Local fractional vector calculus is applied to describe the non-differentiable heat flow. In this work, a new application of the LFVIA-II to the non-homogeneous model for the non-differentiable heat flow is given. The non-differentiable solution of the heat flow on Cantor sets in the closed form with the graph is discussed. The presented method is easy, simple, efficient, and accurate to solve PDEs by employing the local fractional derivatives.