Abstract

The approximation accuracy of the wavelet spectral method for the fractional PDEs is sensitive to the order of the fractional derivative and the boundary condition of the PDEs. In order to overcome the shortcoming, an interval Shannon-Cosine wavelet based on the point-symmetric extension is constructed, and the corresponding spectral method on the fractional PDEs is proposed. In the research, a power function of cosine function is introduced to modulate Shannon function, which takes full advantage of the waveform of the Shannon function to ensure that many excellent properties can be satisfied such as the partition of unity, smoothness, and compact support. And the interpolative property of Shannon wavelet is held at the same time. Then, based on the point-symmetric extension and the general variational theory, an interval Shannon-Cosine wavelet is constructed. It is proved that the first derivative of the approximated function with this interval wavelet function is continuous. At last, the wavelet spectral method for the fractional PDEs is given by means of the interval Shannon-Cosine wavelet. By means of it, the condition number of the discrete matrix can be suppressed effectively. Compared with Shannon and Shannon-Gabor wavelet quasi-spectral methods, the novel scheme has stronger applicability to the shockwave appeared in the solution besides the higher numerical accuracy and efficiency.

1. Introduction

In recent years, fractional calculus has been attracting more and more researchers in different fields of science and engineering and has been theoretically developed quickly over the last two decades [1–4]. It has been proved that the fractional-order differential equation models are more consistent with the biological phenomena [5] and hydrodynamics [6–8] than those of integer-orders. The Caputo and Riemann–Liouville fractional derivatives are the classical definition, and both of them have a kernel with singularity. To solve the problem of singular kernel, Caputo and Fabrizio proposed a derivative with fractional order based on the exponential function; their derivative in fact does not have singular kernel. In order to overcome the shortcoming of the nonlocal property of the exponential function, Ravichandran et al. [9–12] proposed two generalized fractional derivatives in Caputo and Riemann–Liouville sense [13, 14].

Despite a few special fraction PDEs having analytical solution [15], most of them should be solved by the numerical method. The solution of the fractional PDEs is sensitive to the iterative step, and so it is disabled to be solved by the traditional numerical method directly. Al-Mdallal has made outstanding contributions in this field, who has proposed many effective algorithms for solving the fractional PDEs, such as the fractional-order Legendre-collocation method [16] and fractional-Legendre spectral Galerkin method [17]. This inspired this work to try to construct a numerical method by means of the Shannon wavelet theory.

Sinc is a famous sampling function, but Shannon’s reconstruction formula is rarely used in practice because of the slow decay of the Shannon function [18]. Taking the window functions to modulate the sinc function is the common method to improve its decay rate. Many windows have been proposed such as rectangular window, Bartlett window, Hanning window, Hamming window, and Blackman window.

In recent years, wavelet analysis theory has been developed to be a powerful tool to solve the fractional partial differential equations in recent years [19–21]. Shannon wavelets have been constructed based on the sinc function. A complex Shannon wavelet is defined bywhere is the bandwidth and is the wavelet center frequency. Hoffman et al. [22] have presented the Shannon-Gabor wavelet as follows:where is the width parameter (or called window size).

Both of the two Shannon-type wavelets are obtained by taking the Gaussian window to modulate the sinc function. The presence of the Gaussian window destroys the normative property possessed by the Shannon wavelet, that is,where is the error function. Obviously, is always less than unity except at the limit of . This is the reason why the windowed Shannon wavelet is not recommended. They fail to satisfy the partition of unity; this has the disturbing consequence that the reconstruction error will not vanish as the sampling step tends to zero.

Similar to Haar wavelet [23–25], B-Spline wavelet [26, 27], and Legendre wavelet [28, 29], Shannon wavelet [30, 31] possesses almost all the excellent numerical properties such as interpolative, relative sparse, and orthogonal properties besides the compact support property. Therefore, it is necessary to construct a novel window for sinc function, which can satisfy the partition of unity, so that it can be utilized to solve fractional PDEs efficiently [32]. In order to overcome the disadvantages in the Shannon-Gabor wavelet, a Shannon-Cosine wavelet is constructed by Mei et al. [32, 33], in which the waveform is used to meet the requirement of the partition of unity. Unfortunately, there are too many parameters in the Shannon-Cosine wavelet function to be identified, and this makes the expression to be very complex. In this paper, a simplified Shannon-Cosine wavelet function is proposed and the corresponding interval wavelet is constructed based on the point-symmetric extension [34, 35]. And then, the interval Shannon-Cosine wavelet is employed to construct a wavelet spectral method for the fractional PDEs.

2. Simplified Shannon-Cosine Scaling Function

As a basis function defined in Hilbert space, it could form a basic approximation space V as

This means that any function is characterized by a sequence of coefficients . The coefficients are the samples of the signal, and that is a kind of weight function. Therefore, it should meet the requirements as follows:①The coefficients sequence should be square-summable: .②The family of functions should form a Riesz basis of . This ensures that the representation is stable and unambiguously defined. The Riesz basis requirement has an equivalent expression in the Fourier domain: where is the Fourier transform of .③ should satisfy the partition of unity condition .

Sinc is a famous sampling function [20] based on approximating the Dirac delta function as a band-limited function and is given by

In order to overcome the shortcomings of the Shannon-Gabor function, we introduce the parametric cosine function instead of the exponential function to modulate Shannon function as follows:where N is a constant related to the support domain. is the Heaviside function defined as follows:

This ensures that new modulate Shannon function (7) is a real compact support function, and the support domain is [−N/2, N/2]. The function is named as Shannon-Cosine scaling function.

The comparison between the sinc and the Shannon-Cosine scaling function is shown in Figure 1.

(a)

(b)

(a)
(b)

Figure 1 

Comparison between Shannon and Shannon-Cosine scaling functions. (a) Shannon function. (b) Shannon-Cosine scaling function.

It is easy to prove that has the interpolative property; the coefficients in equation (4) are the samples of the signal. Therefore, obtained from any signal satisfies the first requirement.

Based on the Parseval equation, it is easy to understand that any basis function with meets the second requirement. This means that the third requirement puts the strongest constraint of the selection on an admissible generating function . It is well known that the sinc function satisfies the partition of unity condition, and this results in the Gaussian-windowed sinc function are not meeting this requirement. In the next section, we are going to prove that it can be satisfied by choosing the support domain parameter N.

3. Normalization and Choice of the Support Domain Parameter N

Shannon function has the waveform shown in Figure 2; its support domain is and it meets the normalization condition . It is easy to notice that the integration value of Shannon function in domain I₁ is larger than 1 and smaller than 1 in domain I₂. This reminds us that the reasonable choice of the support domain can ensure that the Shannon-Cosine scaling function satisfied the partition of unity condition. Theorem 1 reveals the relation between the support domain parameter N and the partition of unity condition.

Figure 2 

Shannon function.

Theorem 1. The Shannon-Cosine scaling function (m = 3) satisfies the normalization condition as follows:

Proof. The Fourier transform of is given byLet m = 3. Substituting and equation (6) into equation (10), we obtainIn fact, can be viewed as a continuous function with respect to the real number “N,” which is a parameter related to the length of the support domain. LetBased on the definition of Si(x), we know that is an oscillator function around the constant 1, that is,Obviously,That is, , is in correspondence with the extreme point of the function , which is a monotone function on the intervals (2n, 2n + 1) and (2n + 1, 2n + 2). Based on the mean value theorem, there must exist unique points and so as to P(N) = 1, i.e.,This completes the proof.

By means of the interval bisection method, it is easy to obtain the value of N which is in correspondence with P(N) = 1 as in Table 1.

It is easy to prove that the simplified Shannon-Cosine wavelet is equivalent to the Shannon-Cosine wavelet proposed in [36] using the parameter , but only the parameter N should be chosen in applications. So, it is convenient for solving the fractional derivative PDEs.

4. Point-Symmetric Interval Wavelet

In order to eliminate the boundary effect introduced by the wavelet transform, the point-symmetric extension and the general variational theory are employed to construct the interval wavelet. Compared to other extension methods such as the zero extension, symmetric extension, and periodic extension, the point-symmetric extension can ensure that the function is smooth at the endpoint.

Theorem 2. If the continuous function is symmetric on point , then the first-order derivative of the function is continuous at this point.

Proof. As the continuous function is symmetric on point , let ; we haveTherefore, we haveThis illustrates that the first derivative of the function continues to reach the point . This completes the proof.
For convenience, the values of at , ,…, , , …, are denoted as , ,…, , , …, . Based on the general variational theory, we can deduce the format of the interval wavelet function as follows:①,   …  … ②As ,   …  … And so, the function can be expressed asLet ; we havewhereSo, we haveTherefore, the interval interpolative wavelet function can be expressed asAccording to theory 2, it is easy to understand that equation (22) is able to reduce the boundary effect efficiently. In theory, the proposed interval wavelet can prevent the steep shock wave appearing near the boundary, which can introduce the large condition number of the discrete matrix and result in the decrease of the numerical precision.

5. Application in Solving Fractional PDEs

5.1. Interval Wavelet Spectral Method for Fractional Fokker–Planck Equation

The fractional Fokker–Planck equation is a typical fractional PDE, which is often used to describe a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In the presence of an externalforce field , the evolution of a test particle is usually described in terms of the Fokker–Planck equation (FPE):which defines the probability of finding the particle at a certain position x at a given time t. m denotes the mass of the diffusing particle, denotes the generalized diffusion coefficient with dimension , and is the generalized friction coefficient with dimension . The corresponding initial condition isand the boundary conditions are

Equation (2) uses the Riemann–Liouville fractional derivative of order , defined bywhere is the gamma function.

According to the properties of the Riemann–Liouville fractional derivative, it is easy to know that, if , equation (2) can be rewritten as follows:

According to the wavelet spectral method, the fractional Fokker–Planck equation can be approximately represented aswhere i = 0, 1, 2, …, 2^j. Let