Abstract
The fractional population diffusion model is crucial for pest prevention. This paper presents an adaptive hierarchical collocation method for solving this model, enhancing the efficiency of algorithms based on Low-Complexity Shannon-Cosine wavelet derived from combinatorial identity theory. This function, an improvement over previous constructs, mitigates the need for iterative computation of parameters and boasts advantages like interpolation, symmetry, and compact support. The method’s extension to other time-fractional partial differential equations (PDEs) is also possible. The algorithm’s complexity analysis illustrates the concise function’s efficiency advantage over the original expression when solving time-fractional PDEs. Comparatively, the method exhibits superior numerical performance to alternative wavelet spectral methods like the Shannon–Gabor wavelet.
1. Introduction
Fisher, in his 1937 article, “The wave of advance of advantageous genes,” proposed the population diffusion model, which is utilized in population dynamics to describe the spatial spread of an advantageous allele, and explored its traveling wave solutions [1]. In recent years, the population diffusion model has found broad applications in numerous fields such as epidemics [2], population growth prediction, and propagation of invasive species [3], among others. While the population diffusion model falls under the category of reaction-diffusion equations, obtaining its analytical solution remains challenging despite it being one of the simplest nonlinear reaction-diffusion equations.
In the case of the population diffusion model, many different numerical methods have been proposed to approximate the solution, such as the homotopy perturbation method (HPM) [4] and the residual power series method (RPSM) [5]. Compared with the above methods, the spectral method [6] offers higher accuracy but lower efficiency. Wavelet analysis [7], which has emerged since the mid-80s, serves as a set of mathematical tools to solve a variety of problems in signal and image analysis [2, 8]. As an efficient and effective tool, wavelets [9] have been widely used in many areas, playing a crucial role, especially in signal analysis.
The Shannon wavelet [10] possesses many excellent numerical properties, such as orthogonality, interpolation, smoothness, vanishing moments, and symmetry, but lacks compact support. The property of compact support is useful in improving numerical precision and efficiency. To overcome the shortcoming of the Shannon wavelet function, Mei and Gao [11] constructed the Shannon-Cosine wavelet. It acts as a real wavelet function that retains most of the excellent properties of the Shannon wavelet. Moreover, the proposed wavelet function overcomes the shortcoming of the common windowed or truncated Sinc function, that is, the approximation error does not vanish as the sampling step tends to zero.
The Shannon-Cosine wavelet, widely used in various fields such as image denoising [8, 12], inpainting [13], geometry modeling [14], signal processing [15], option pricing [16], computational mathematics, and control engineering [14], includes a trigonometric series. Interval theory has excellent versatility in describing imprecise data. The theory has been continuously improved and developed, and it has now become an active branch in the field of computational mathematics. Combining interval theory with wavelet function optimization can effectively improve wavelet performance. Mei et al. [17] proposed an interval wavelet construction method with interpolation properties based on the generalized variational principle.
More specifically, the constructed wavelet contains the trigonometric series; however, the coefficients of the cosine function need to be calculated in advance, and there is indeed a possibility to avoid the calculation of coefficients. To overcome this shortcoming, Low-Complexity Shannon-Cosine wavelet is proposed, and it has been proved that the Low-Complexity Shannon-Cosine wavelet is equivalent to the Shannon-Cosine wavelet of trigonometric series by using the relevant properties of combinatorial numbers. As a result, the number of types of parameters in the Shannon-Cosine wavelet function is successfully reduced.
To make the distinction clear, our convention is to use italic fonts for scalar functions, roman fonts for vector functions, and bold roman fonts for matrix functions. We thus follow the notation described below:
The rest of the article is organized as follows. The Shannon-Cosine wavelet and Low-Complexity Shannon-Cosine wavelet are introduced in Section 2, and the properties of the Low-Complexity Shannon-Cosine wavelet and the proof process, demonstrating that it is essentially equivalent to the Shannon-Cosine wavelet in the trigonometric series, are given in Section 3. In Section 4, numerical results are provided. Finally, Section 5 focuses on the conclusions.
2. Definitions of the Shannon-CosineWavelet and Low-Complexity Shannon-Cosine Wavelet
Mei and Gao [11] successfully constructed the Shannon-Cosine wavelet by applying the windowed function to the Shannon wavelet. The Shannon-Cosine kernel function is a real wavelet function, which differs from the Shannon–Gabor wavelet that can only be regarded as a quasi-wavelet.
This kernel function is a compactly supported function that preserves the interpolation property and satisfies the wavelet normalization requirement. It overcomes the limitations of other functions like the Shannon–Gabor wavelet and truncated Sinc functions, specifically the nonvanishing reconstruction error as sampling approaches zero. Offering high precision and efficiency, the Shannon-Cosine function is particularly effective for solving various partial differential equations and can be generalized for multiscale cases.
The Shannon-Cosine scale function is composed of three parts: the Sinc function, the weighting function, and the Heaviside function, which is defined as follows:whereand , is a constant that satisfies , the value of is not unique, and is the Heaviside function, which is defined as follows:
The contrasts between functions and are illustrated in Figure 1. When compared to , demonstrates the compact support property and adheres to the partition of unity.
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The simplified Shannon-Cosine scale function is obtained by modifying the original Shannon-Cosine scale function, specifically by raising the cosine function to a positive even power number. This modification highlights the scale’s dependence on cosine similarity and results in a more streamlined function, i.e.,
To facilitate differentiation from the simplified form of the Shannon-Cosine wavelet below, the Shannon-Cosine wavelet [11] and the simplified Shannon-Cosine wavelet are denoted as and . The difference between the two forms of the Shannon-Cosine wavelet function is just the difference between and . is a power function with respect to . So, the time complexity of is . For the original Shannon-Cosine wavelet function, the corresponding expressed in equation (2) can be rewritten as follows. For convenience, let ; then,
To improve the numerical efficiency, can be calculated employing the recurrence formula as follows:
Therefore, the time complexity of the original Shannon-Cosine wavelet function is , that is, in big notation. It is straightforward to see that the time complexity of the new form of the Shannon-Cosine wavelet is much lower than that of the original format. This is the reason why we propose a new form of the Shannon-Cosine wavelet in this research.
To more clearly demonstrate the advantages of the constructed wavelet, the simplified Shannon-Cosine wavelet is renamed as the Low-Complexity Shannon-Cosine wavelet.
Indeed, it is easy to see that does hold for some not too large , which seems to imply
3. Equivalence Proof of Two Forms of Shannon-Cosine Wavelets
The differences between the two forms of Shannon-Cosine wavelet mainly appear in the window function, that is, and , the latter is a trigonometric series, and the former is a positive even power of the cosine function.
In this section, it will be proved that the two forms of the Shannon-Cosine wavelet are equivalent, and the Low-Complexity form possesses smaller time complexity than the original form.
The relevant properties of combinatorial numbers are introduced in reference [18], the difference method is an effective method for proving certain combinatorial constants. The difference operator of an arbitrary function is defined as , which is called the value of the first-order difference of at .
The multiorder difference operator deduced from the first-order difference operator is , .
The other two operators also need to be introduced in this paper, one for the constant operator and the other for the displacement operator .
It is easy to know the following formula, and , so .
In other words, , from which we obtain
Lemma 1. Let be any function; then,
Proof. By using two different representations , the following equation holds:Let be a series; if its -order difference series is not a zero series and its -order difference series is a zero series, then is said to be a -order equivariant series.
Lemma 2. If is a -order polynomial, then the series is an equivariant series of the order .
Proof. and .
Thus, the series is an equivariant series of order.
Lemma 3. If a is an equivariant series of order and , then
Proof. From Lemmas 1 and 2, it can be derived that .
Lemma 4. If is a polynomial of order and , then .
Proof. Assuming that is a polynomial of order , we can conclude from Lemma 2 that is an equivariant series of order , and the conclusion above holds based on Lemma 3.
Theorem 5. The original Shannon-Cosine scale function and the Low-Complexity scale function are equivalent, i.e., .
Proof. Let andThe column vector consisting of the coefficients of in the scale function satisfies the equation [11]When , .
When , some mathematical notations are defined in advance:Expand the determinant by the first row according to the rules for calculating determinants.
If is an arbitrary even number and , the value of the determinant of the matrix isIf is an arbitrary odd number and , the value of the determinant of the matrix isBecause , and the rank of the matrix is and the equation has a unique solution.
Regarding the expansion of the high power of the cosine function, there is a conclusion that holds , and let , soConsider the first row of the matrix:Consider the second row of the matrix:Consider the third to row of the matrix, The validity of Theorem 5 can be established due toandis aorder polynomial to the variable, which satisfies the conditions of Lemma 4 so , since, , , hence, the equation (16) has a unique solution and, , then, and the rest of the Original Shannon-Cosine wavelet scale function and the Low-Complexity Shannon-Cosine wavelet scale function are equal, so Theorem 5 holds.
Specific parameter values can be used to verify Theorem 5, and for the case when , the parameter values are provided in Table 1. By comparing the values in the literature [11], the two coefficients are found to be exactly equal.
4. Multiscale Shannon-Cosine Wavelet Interpolation Operator
In this section, a multiscale wavelet interpolation operator is constructed firstly based on the interpolation property of the Shannon-Cosine wavelet, which can be used to represent the smooth signal sparsely. Therefore, it can be used to solve time-fractal PDEs efficiently. Then, we analyze the time complexity of the interpolation operators constructed based on the two forms of the Shannon-Cosine wavelet, respectively.
4.1. Construction of the Multiscale Wavelet Interpolation Operation
The two forms of Shannon-Cosine wavelet and the Low-Complexity Shannon-Cosine wavelet function and have been introduced in Section 2, and the sequence of scale functions and the sequence of wavelet functions are, respectively, defined aswhere , .
Define the mapping as follows:
The inverse function of the function in this paper is denoted by , when , the definition of is complemented by , and then there is a unique (j, k) corresponding to any integer . We denote this mapping relationship by the following notation .
The definition of is
The data , along with its first-order derivative and second-order derivative , can be represented in matrix format as follows:where , and are defined as equation (34).
Let , which represents the space of measurable square-integrable functions. Then, the infinite series expansion of with Shannon-Cosine wavelet converges uniformly to , i.e.,
4.2. Fractional Population Diffusion Model
To compare and illustrate the advantages of the new Shannon-Cosine wavelet, we applied the multiscale Shannon-Cosine wavelet collocation method to obtain the numerical solutions of the time-fractional population diffusion model. The differences in key algorithms for implementing two Shannon-Cosine wavelets are first introduced and analyzed in terms of time complexity.
The Time-Fractional Population Diffusion Model (TFPDM) is a well-defined fractional partial differential equation (PDE) that highlights the spatial and temporal propagation of a virile gene in an infinite medium [19]. The TFPDM can be described as follows:
The initial and boundary conditions arewhere , denotes the Caputo [20] fractional derivative in time, and , , and are some known functions. The following specific TFPDM is considered forandwith the initial and boundary conditions, which has an exact solution [1]
The finite difference format [21] of the fractional order derivative part of equation (36) is given bywhere , Using the wavelet allocation method, the differential form of the right-hand side of equation (36) is obtained. Moreover, the nonlinear term is linearized by the method introduced by Rubin and Graves [22]. The resulting expressions are as follows:
Bringing the difference form to the equation (36), ignoring the infinitesimal terms gives the finite difference form of the equation.
The solution of the general iterative method of TFPDM in matrix form is given by
The boundary conditions generate other equations for and :
Let
The matrix form of the equation (36) based on the multiscale wavelet allocation method [23] isand where is defined by equation (34), both and within that formula refer either to the Shannon-Gabor wavelet function [24] or the Shannon-Cosine wavelet function.
The solution to equation (40) is given bywhere
The calculation of the relative numerical error is given by
We use a combination of the multiscale Shannon-Cosine wavelet spectral method and iteration technique for the numerical solution of equations (30) and (40) compared to Shannon–Gabor [24] wavelet.
The core idea of this algorithm is to iteratively solve the linear system by updating the matrix and vector at each step, using the wavelet-based method to approximate the solution. The algorithm successively refines the solution until convergence is achieved. The specific process is shown in Algorithm 1.
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Table 2 presents the average error of the TFPDM approximate solution at using the Shannon-Cosine wavelet spectral method, with parameters for varying values of and . This indicates that the Shannon-Cosine wavelet spectral method offers an approximation for TFPDM that is closely aligned with the exact solution. Furthermore, the numerical error is notably smaller for specific values that meet the normalization condition compared to those that round towards negative or positive infinity.
Table 3 displays a comparison of error among two wavelet spectral methods to approximate the solution of the TFPDM, which demonstrates that the multiscale Shannon-Cosine wavelet spectral method has better numerical stability compared to Shannon–Gabor wavelet. The figure shown in Figure2 displays the numerical solution with parameters , , , , and .
Figure 3 displays the numerical solutions and relative numerical errors when using the multiscale Shannon-Cosine wavelet spectral methods for the TFPDM, across different values. The parameters adopted in this analysis are , , , and . The findings reinforce the superior numerical stability provided by the multiscale Shannon-Cosine wavelet spectral method under these conditions.
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Figure 2 illustrates the numerical solutions using the multiscale Shannon-Cosine wavelet spectral methods to approximate the solution of TFPDM for different values of , when the time . The comparison shows that the multiscale Shannon-Cosine wavelet spectral method displays superior numerical stability.
According to the analysis of experimental results, an increase in the scale parameter is positively correlated with a decrease in experimental error. This indicates that increasing can improve the stability and consistency of the experiment. Similarly, an increase in the fractional-order exponent parameter also leads to a reduction in experimental error, potentially reducing error sources and enhancing experimental accuracy. However, the increase in the time variable is positively associated with an increase in experimental error, possibly due to system or condition variations. In summary, the scale parameter and the fractional-order exponent parameter have a positive impact on reducing experimental error, while the increase in the time variable is linked to error increase. These findings provide valuable insights into the reliability and accuracy of the experimental results.
5. Conclusion
In this paper, we present the Low-Complexity Shannon-Cosine wavelet by substituting the Original Shannon-Cosine with the positive and even power of the cosine function. It has been rigorously demonstrated that the Low-Complexity Shannon-Cosine wavelet is equivalent to the Original Shannon-Cosine wavelet via trigonometric series, utilizing the pertinent properties of combinatorial numbers.
The primary advantage of the Low-Complexity Shannon-Cosine wavelet lies in the elimination of certain parameter calculations present in the earlier version, thereby enhancing the efficiency of programming. Numerical experiments have established that the Shannon-Cosine wavelet serves as an invaluable technique for achieving high-precision numerical solutions to various fractional-order partial differential equation problems in engineering applications.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (grant no. 61871380), Beijing Natural Science Foundation (grant no. 4172034), Shandong Provincial Natural Science Foundation (grant no. ZR2020MF019), and Beijing Innovation Consortium of Agriculture Research System (BAIC11-2023).