Numerical Simulation of One-Dimensional Fractional Nonsteady Heat Transfer Model Based on the Second Kind Chebyshev Wavelet

In the current study, a numerical technique for solving one-dimensional fractional nonsteady heat transfer model is presented. We construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional-order integration. The operationalmatrix of fractional-order integration is utilized to reduce the original problem to a system of linear algebraic equations, and then the numerical solutions obtained by our method are compared with those obtained by CAS wavelet method. Lastly, illustrated examples are included to demonstrate the validity and applicability of the technique.


Introduction
Fractional calculus is a branch of mathematics that deals with generalization of the well-known operations of differentiations to arbitrary orders.Many papers on fractional calculus have been published for the real-world applications in science and engineering such as viscoelasticity [1], bioengineering [2], biology [3], and more can be found in [4,5].Moreover fractional partial differential equations also are widely used in the areas of signal processing [6], mechanics [7], econometrics [8], fluid dynamics [9], and electromagnetics [10].As the analytical solutions of fractional partial differential equations are not easy to derive, the scholars are committed to obtain their numerical solutions of these equations.
In recent years, various numerical methods have been proposed for solving fractional diffusion equations, these methods include wavelets methods [11][12][13][14][15][16][17], Jacobi, Legendre, and Chebyshev polynomials methods [18][19][20][21], spectral methods [22,23], finite element method [24], wavelet Galerkin method [25], and finite difference methods [26,27].In [28], a new matrix method is proposed to solve two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions.In [29], the authors utilize the second kind Chebyshev wavelets to obtain the numerical solutions of the convection diffusion equations.Xie et al. use the Chebyshev operational matrix method to numerically solve onedimensional fractional convection diffusion equations in [30].In this paper, we apply the second kind Chebyshev wavelet method to obtain the numerical solutions of onedimensional fractional nonsteady heat transfer model.The obtained numerical solutions by our method have been compared with those obtained by CAS wavelet method.
The current paper is organized as follows: Section 2 introduces the basic definitions of fractional calculus.In Section 3, the mathematical model of nonsteady heat transfer problem is proposed.Section 4 illustrates the second kind Chebyshev wavelets and their properties.In Section 5, we apply the second kind Chebyshev wavelet for solving fractional nonsteady heat transfer model.Numerical examples are presented to test the proposed method in Section 6.Finally, a conclusion is drawn in Section 7.

One-Dimensional Nonsteady Heat Transfer Model
For one infinite plate sample, as shown in Figure 1, the height is , the upper surface and the edge are adiabatic, and the lower surface is contacted with the fluid, which its temperature is   .The heat conductivity coefficient of the sample is , the density is , and the specific heat capacity is   .The initial temperature is  0 , taking the origin of coordinates on the sample adiabatic surfaces, and the nonsteady heat transfer model with the initial-boundary condition can be defined as follows [31]: Obviously, when the sample density , heat conductivity coefficient , specific heat capacity   , and thickness  are known, we can obtain the temperature distribution at any position  and any time , which is the nonsteady heat conduction model with constant temperature boundary condition.Based on the above-mentioned model, we give the fractional-order nonsteady heat transfer model of the following form: with the initial condition: and the boundary conditions: where (, ) denotes source term, () is a given function, and  0 (),  1 () are continuous functions with first-order derivative.

Preliminaries of the Fractional Calculus
In this section, we give some necessary definitions and mathematical preliminaries on fractional calculus which will be used further in this paper.
Definition 1.The Riemann-Liouville fractional integral operator   ( > 0) of a function () is defined as follows [4]: Some properties of the operator   are as follows: Definition 2. The Caputo fractional derivative 0    of a function () is defined as follows [4]: Some properties of the Caputo fractional derivative are as follows:
The following theorem discusses the convergence and accuracy estimation of the proposed method.Theorem 3. Let () be a second-order derivative squareintegrable function defined over [0, 1) with bounded secondorder derivative, satisfying |  ()| ≤  for some constants ; then (1) () can be expanded as an infinite sum of the second kind Chebyshev wavelets and the series converge to () uniformly, that is, where   = ⟨(),   ()⟩  2  [0,1) . (2) where  ,, = ( ∫ 4.2.Operational Matrix of Fractional Integration.On the interval [0, 1), we defined a m -set of block-pulse functions (BPFs) as The functions {  ()} are disjoint and orthogonal: Similarly, the second kind Chebyshev wavelet may be expanded into an m-term block-pulse functions as Kilicman has given the block-pulse functions operational matrix of fractional integration   of following form: where Next, we derive the second kind Chebyshev wavelet operational matrix of fractional integration.Let where   m× m is called the second kind Chebyshev wavelet operational matrix of fractional integration and it can be given by For More details, see [29].
(37) By solving this system to determine , we can get the numerical solution of this problem by substituting  into (33).

Numerical Simulations
In this section, we use the proposed method to solve the initial-boundary problem of nonsteady heat transfer equations.The following numerical examples are given to show the effectiveness and practicability of the proposed method and the results have been compared with the analytical solution.
Example 4. Consider the following fractional-order nonsteady heat transfer model:  a given value , as  increases, or, for a given value , as  increases.
Example 8. Consider (41), with  = 2, 1.9, 1.8, 1.7; the numerical solutions when  =  = 4 at  = 0.3, 0.6, 0.95 are shown in Figure 10.This example is introduced to verify   the robustness of the proposed method; when the fractional order gradually approaches to 2, the numerical solutions are in agreement with the analytical solution.

Conclusions
This paper presents a numerical technique for approximating solutions of one-dimensional fractional nonsteady heat transfer model by combining the second kind Chebyshev wavelet with its operational matrix of fractional-order integration.In the proposed method, a small number of grid points guarantee the necessary accuracy.The main advantage of wavelet method for solving the kinds of equations is that, after dispersing the coefficients, matrix of algebraic equations is sparse.The solution is convenient, even though the size of increment may be large.Several examples are given to demonstrate the powerfulness of the proposed method.

Figure 1 :
Figure 1: Nonsteady heat transfer model with constant temperature boundary condition.