Research paperA generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations
Introduction
The distributed-order differential equation can be regarded as a natural generalization of the single-order and the multi-term fractional differential equation. In recent decades, distributed-order fractional differential equations (DOFDEs) have been used to accurately modelize more phenomena in the fields of optimal control [1], [2], [3], diffusions [4], [5], [6], [7], [8], [9], [10], viscoelastic [11], [12], signal processing [13], electrochemistry [14], biosciences [15], [16], finance [17], [18], and dielectrics [19].
Analytical methods to study the solutions for DOFDEs have been studied, for instance, in Bagley and Torvik [20], Diethelm and Ford [21], Gorenflo et al. [22], Luchko [23]. Usually it is difficult, sometimes impossible, to determine the exact solutions of DOFDEs. Therefore, constructing efficient numerical solution methods for solving DOFDEs is an important task. Available methods include a method using the trapezoidal quadrature rule [24], linearization technique [25], finite difference schemes [26], Gauss-Legendre quadrature method [27], hybrid functions of block-pulse functions (HFBPF) and Bernoulli polynomials [28], the Petrov-Galerkin spectral method [29], Chebyshev collocation method [30], finite volume method [31], Laguerre Petrov-Galerkin spectral method in Lischke et al. [32], Legendre spectral element method [33], HFBPF and Taylor polynomials, Jibenja et al. [34], and Legendre wavelet method [35]. Different from the DOFDEs of dimension one, only a few numerical techniques for solving distributed-order fractional partial differential equations (DOFPDEs) are reported. These include meshless method [36], finite element method [37], fractional pseudo-spectral method [38], and Legendre operational matrix methods [39], [40].
Recently wavelets, say played important tools for numerically solving problems in fractional calculus. The wavelets used include cosine-and-sine wavelets [41], Chebyshev wavelets [42], Legendre wavelets [43], [44] and Haar wavelets [45]. In these methods, the operational matrices, say were used in the approximationwhere is the Riemann-Liouville fractional integral operator (RLFIO). Following a different direction, instead of constructing the operational matrix, Mashayekhi and Razzaghi obtained an exact formula for computing when are the HFBPF and Bernoulli polynomials and then used the formula to produce a more accurate numerical method [28]. Using similar arguments, exact formulas for the integral of Taylor wavelets are given in Toan et al. [46], Vichitkunakorn et al. [47].
The solutions of fractional differential equations (FDEs) can contain some fractional power terms which cannot be approximated by using classical integer order bases [48]. To increase the efficiency of the numerical methods, fractional-order polynomials and wavelets were introduced by applying the transformation of variables to the integer-order polynomials and wavelets, such as: fractional-order Legendre polynomials [49], fractional-order Laguerre polynomials [50], fractional-order Bernoulli polynomials [51] and fractional-order Bernoulli wavelets [52], [53]. However, no exact formula for the RLFIO for any of the above fractional-order polynomials and wavelets are reported.
In the current work, we consider the following two-dimensional DOFPDE:with the initial - Dirichlet boundary conditionsHere, are continuous functions; and is a continuous non-negative weight function satisfying [54]The proofs for the existence, continuity, smoothness and uniqueness of the solution were given in Luchko [54]. In this article, we focus on constructing a new numerical method to solve Eqs. (1) and (2) effectively. In our method, we use the fractional-order Chebyshev wavelets (FOCW). It is noted that Chebyshev polynomials are related to Fourier cosine functions with the advantage that they are polynomials rather than the infinite series defining the cosine. Chebyshev polynomials are therefore applied to a wide variety of computing problems [55]. We provide an exact formula in terms of the regularized beta functions for the RLFIO of FOCW. The regularized beta functions can be computed effectively by using Mathematica. We use the FOCW and the exact formula to reduce the given DOFPDE to a system of algebraic equations. The exact formula allows us to achieve very accurate numerical solutions. We demonstrate the feasibility and effectiveness of the new method by including several examples. In Example 6.1, we will show that for the cases when the exact solution is a polynomial, we can get this solution. In addition, we will show in Examples 6.2 and 6.3, for the cases in which the exact solutions contain fractional-order power term, by applying the present method, we get the exact solutions. The exact solutions in Examples 6.1–6.3 were not obtained previously in the literature. To the best of our knowledge, this formula can be used for several fractional partial differential equations.
In Section 2, we give some basic definitions from fractional calculus. Section 3 introduces the FOCW and the exact formula for the RLFIO of the FOCW. The numerical method and the error bound are reported in Sections 4 and 5 respectively, and the numerical examples are included in Section 6.
Section snippets
Preliminaries
A detailed introduction to fractional calculus can be found in Toan et al. [46], Vichitkunakorn et al. [47] Definition 2.1 The RLFIO of order of a function is defined by Definition 2.2 The Caputo fractional derivative of order of a function is defined by
The and are linear operators. Furthermore, we have Definition 2.3 The regularized beta-function is defined as [56]
FOCW
The shifted Chebyshev polynomials of order are defined on [0,1] and given by [57]These polynomials are orthogonal in where i.e.,Let be non-negative integers. The Chebyshev wavelets for and on the interval [0,1] are defined bywhereThe Chebyshev wavelets are
Numerical method for solving DOFPDEs
For fixed non-negative integers and letIntegrating Eq. (17) with respect to and using the initial condition, we haveIntegrating Eq. (18) twice with respect to and using the Dirichlet boundary conditions, we getComputing the Caputo’s derivative of order in Eq. (19) with respect to , we
Error bound
In this section, we include the following theorem concerning the error bound of convergence: Theorem 5.1 Let be a smooth function on and suppose that is the best approximation of out ofwith respect to the norm in . Thenwhere are constants satisfying
Illustrative examples
We show the advantage of the present method via comparison with previously known methods, such as: finite difference schemes [26], Chebyshev collocation method [30], and Legendre operational matrix method [40]. Our method produces numerical solutions with more accuracy. The examples in this paper are solved by using Mathematica software version 11.3.0.0. Example 6.1 Consider the two-dimensional DOFPDEwithThe exact See [26, Example 4.1]
Conclusion
In the present work, we gave the fractional-order Chebyshev wavelets based on Chebyshev wavelets. The exact formula of the Riemann-Liouville fractional integral operator of the fractional-order Chebyshev wavelets is calculated by using the regularized beta function. The fractional-order Chebyshev wavelets and the exact formula were used to construct a numerical method to solve 2-dimensional distributed-order fractional partial differential equations effectively. The error bound was
CRediT authorship contribution statement
Quan H. Do: Conceptualization, Data curation, Investigation, Validation, Visualization, Writing - original draft, Writing - review & editing. Hoa T.B. Ngo: Formal analysis, Investigation, Validation, Visualization, Writing - original draft, Writing - review & editing. Mohsen Razzaghi: Conceptualization, Methodology, Formal analysis, Investigation, Validation, Visualization, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors wish to express their sincere thanks to the anonymous referees for valuable suggestions that improved the final version of the manuscript.
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