Abstract
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 operational matrix 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.
1. 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 [11–17], Jacobi, Legendre, and Chebyshev polynomials methods [18–21], 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 one-dimensional fractional convection diffusion equations in [30]. In this paper, we apply the second kind Chebyshev wavelet method to obtain the numerical solutions of one-dimensional 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.
2. 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 , 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 , are continuous functions with first-order derivative.
3. 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 of a function is defined as follows [4]:Some properties of the operator are as follows:
Definition 2. The Caputo fractional derivative of a function is defined as follows [4]:Some properties of the Caputo fractional derivative are as follows:
4. The Second Kind Chebyshev Wavelet and Its Operational Matrix of Fractional Integration
4.1. The Second Kind Chebyshev Wavelet and Its Properties
The second kind Chebyshev wavelet has four arguments, . They are defined on the interval as follows [19]:withHere are the second kind Chebyshev polynomials which are orthogonal with respect to the weight function and satisfy the following recursive formula:
A function defined over may be expanded in terms of the second kind Chebyshev wavelet as follows:whereand the weight function Moreover, and are column vectors given byTake the collocation points as follows:We define the second kind Chebyshev wavelet matrix as
An arbitrary function of two variables defined over may be expanded into Chebyshev wavelets basis as follows:where and .
The following theorem discusses the convergence and accuracy estimation of the proposed method.
Theorem 3. Let be a second-order derivative square-integrable function defined over with bounded second-order 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) where .
4.2. Operational Matrix of Fractional Integration
On the interval , we defined a – set of block-pulse functions (BPFs) asThe functions are disjoint and orthogonal:Similarly, the second kind Chebyshev wavelet may be expanded into an -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. Letwhere is called the second kind Chebyshev wavelet operational matrix of fractional integration and it can be given by
For More details, see [29].
5. Numerical Implementation
In this section, we use the second kind Chebyshev wavelets method for numerically solving the nonsteady fractional-order heat transfer model with initial-boundary conditions. In order to solve this problem, we assumewhere is an unknown matrix which should be determined, and is the vector defined in (15). By integrating (28) from to , we obtainMaking use of the initial condition (3) enables one to put (29) in the following form: Then we haveBy integrating (30) two times from to , we obtainand, by putting in (32), we getwhereBy one time differentiation of (33) with respect to , we obtainwhereNow by substituting (31) and (35) into (2) and combining (4) and taking collocation points , , , we obtain the following linear system of algebraic equations:By solving this system to determine , we can get the numerical solution of this problem by substituting into (33).
6. 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:where the parameters , , , and with initial-boundary condition The analytical solution of this problem is The graph of the analytical solution is shown in Figure 2. The graphs of the numerical solutions when , , are shown in Figures 3–5. From Examples 4, 6, and 7, it can be concluded that the numerical solutions approximate to the analytical solution for a given value , as increases, or, for a given value , as increases.
Example 5. Consider the following fractional-order nonsteady heat transfer equation:with initial-boundary condition , The analytical solution of this problem is . When , , , the numerical solutions obtained by our method and those obtained by CAS wavelet method at some values of , are listed in Table 1.
Example 6. We consider the following second-order nonsteady heat transfer model:in such a way that , , . The analytical solution of the system is The absolute errors between the numerical and analytical solutions obtained by our method and CAS wavelet method at some values of , when , are shown in Table 2. Table 2 shows that our method has a better approximation than CAS wavelet method.
Example 7. Consider the following second-order nonsteady heat transfer model:where the parameters , , , and , in such a way that , The analytical solution of this problem is The graphs of the analytical and numerical solutions, when , , are shown in Figures 6–9.
Example 8. Consider (41), with ; the numerical solutions when at are shown in Figure 10. This example is introduced to verify the robustness of the proposed method; when the fractional order gradually approaches to , the numerical solutions are in agreement with the analytical solution.
7. 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment and the Natural Science Foundation of Shanxi Province (201701D221135), Dr. Startup Funds of Taiyuan University of Science and Technology (20122054), and Postdoctoral Funds of Taiyuan University of Science and Technology (20152034).