Research paper
A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations

https://doi.org/10.1016/j.cnsns.2020.105597Get rights and content

Highlights

  • Two-dimensional distributed-order fractional differential equations (DOFDE) are used in different applications.

  • Recently, the Riemann-Liouville fractional integral operator (RLFIO) of wavelets was not calculated exactly.

  • We introduce fractional-order Chebyshev wavelets (FOCW) and use them to solve two-dimensional DOFDE.

  • By applying regularized beta functions we obtain an exact formula for the RLFIO for FOCW.

  • By using FOCW and the exact formula, we obtain a very effcient and accurate numerical solutions for two-dimensional DOFDE.

Abstract

We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.

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 Ψ(t), 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 Pβ, were used in the approximationIβΨ(t)PβΨ(t),where Iβ 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 IβΨ(t) when Ψ(t) 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 xtα(α>0) 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:01ρ(p)pu(x,t)tpdp=2u(x,t)x2+F(x,t),0x,t1,with the initial - Dirichlet boundary conditionsu(x,0)=f(x),u(0,t)=g0(t),u(1,t)=g1(t).Here, F,f,g0,g1 are continuous functions; and ρ(p) is a continuous non-negative weight function satisfying [54]ρ(p)0,and01ρ(p)dp=κ>0.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 β of a function u(x) is defined byIβu(x)=1Γ(β)0xu(t)(xt)β1dt,x0.

Definition 2.2

The Caputo fractional derivative of order β of a function u(x) is defined byDβu(x)=Iηβ(dηdxηu(x))=1Γ(ηβ)0xu(η)(t)(xt)ηβ1dt,η1<βη,ηN.

The Iβ and Dβ are linear operators. Furthermore, we haveIβxα=Γ(α+1)Γ(α+1+β)xα+β,α>1.Dβxα=Γ(α+1)Γ(α+1β)xαβ,α>η1.

Definition 2.3

The regularized beta-function is defined as [56]I(

FOCW

The shifted Chebyshev polynomials ρm(x) of order m are defined on [0,1] and given by [57]ρ0(x)=1,ρm(x)=mi=0m(1)mi22i(m+i1)!(mi)!(2i)!xi,m=1,2,These polynomials ρm(x) are orthogonal in Lω2[0,1] where ω(x)=11(2x1)2, i.e.,01ρn(x)ρm(x)ω(x)dx=δn,m.Let k,M be non-negative integers. The Chebyshev wavelets ψn,m(x) for n=0,1,,2k1 and m=0,1,,M on the interval [0,1] are defined byψn,m(x)={λm2k2ρm(2kxn),ifx[n2k,n+12k],0,otherwise,whereλm={2π,ifm=0,2π,ifm1,The Chebyshev wavelets ψn,m(x) are

Numerical method for solving DOFPDEs

For fixed non-negative integers k1,k2,M1 and M2, let3x2tu(x,t)Ψk1,M1α1(x)TΛΨk2,M2α2(t).Integrating Eq. (17) with respect to t and using the initial condition, we have2x2u(x,t)Ψk1,M1α1(x)TΛI1Ψk2,M2α2(t)+d2dx2f(x).Integrating Eq. (18) twice with respect to x and using the Dirichlet boundary conditions, we getu(x,t)I2Ψk1,M1α1(x)TΛI1Ψk2,M2α2(t)x.I2Ψk1,M1α1(1)TΛI1Ψk2,M2α2(t)+(1x)(g0(t)g0(0))+x(g1(t)g1(0))+f(x).Computing the Caputo’s derivative of order β in Eq. (19) with respect to t, we

Error bound

In this section, we include the following theorem concerning the error bound of convergence:

Theorem 5.1

Let u(x,t) be a smooth function on Ω=[0,1]×[0,1], and suppose that u¯(x,t) is the best approximation of u(x,t) out ofΠ=Span{ψiα1(x)ψjα2(t),i=1,2,,m^1,j=1,2,,m^2}.with respect to the norm in Lw*2(Ω). Thenu(x,t)u¯(x,t)w*2C1α1M1·2(k1+2)M11·M1!+C2α2M2·2(k2+2)M21·M2!+C3α1M1·α2M2·M1!M2!·2(k1+2)M1+(k2+2)M22,where C1,C2,andC2 are constants satisfyingmax(x,t)Ω|M1u(x,t)xM1|C1,max(x,t)Ω|M2u(x,t)tM2|

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

See [26, Example 4.1]

Consider the two-dimensional DOFPDE01Γ(3p)pu(x,t)tpdp=2u(x,t)x2+2t2+2tx(t1)(2x)lnt,withu(x,0)=0,u(0,t)=0,u(2,t)=0,0x2,0t1.The exact

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.

References (61)

  • J.T. Katsikadelis

    Numerical solution of distributed order fractional differential equations

    J Comput Phys

    (2014)
  • S. Nandal et al.

    Numerical treatment of non-linear fourth-order distributed fractional sub-diffusion equation with time-delay

    Commun Nonlinear Sci Numer Simul

    (2020)
  • X.Y. Li et al.

    A numerical method for solving distributed order diffusion equations

    Appl Math Lett

    (2016)
  • S. Mashayekhi et al.

    Numerical solution of distributed order fractional differential equations by hybrid functions

    J Comput Phys

    (2016)
  • M.L. Morgado et al.

    Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

    Appl Numer Math

    (2017)
  • J. Li et al.

    A novel finite volume method for the rRiesz space distributed-order diffusion equation

    Comput Math Appl

    (2017)
  • B. Yuttanan et al.

    Legendre wavelets approach for numerical solutions of distributed order fractional differential equations

    Appl Math Model

    (2019)
  • W. Fan et al.

    A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain

    Appl Math Lett

    (2018)
  • B.M. Moghaddam et al.

    Numerical approach for a class of distributed order time fractional partial differential equations

    Appl Numer Math

    (2019)
  • M. Pourbabaee et al.

    A novel Legendre operational matrix for distributed order fractional differential equations

    Appl Math Comput

    (2019)
  • H. Saeedi et al.

    A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order

    Commun Nonlinear Sci Numer Simul

    (2011)
  • L. Zhu et al.

    Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

    Commun Nonlinear Sci Numer Simul

    (2012)
  • M.H. Heydari et al.

    Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions

    Appl Math Comput

    (2014)
  • Y. Li et al.

    Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

    Appl Math Comput

    (2010)
  • F. Mohammadi et al.

    Fractional-order Legendre wavelet tau method for solving fractional differential equations

    J Comput Appl Math

    (2018)
  • S. Kazem et al.

    Fractional-order Legendre functions for solving fractional-order differential equations

    Appl Math Model

    (2013)
  • P. Rahimkhani et al.

    Fractional-order bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations

    Appl Numer Math

    (2017)
  • P. Rahimkhani et al.

    Fractional-order Bernoulli wavelets and their applications

    Appl Math Model

    (2016)
  • P. Rahimkhani et al.

    Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet

    J Comput Appl Math

    (2017)
  • X. Huang et al.

    Vertical electron density profiles from the digisonde network

    Adv Space Res

    (1996)
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