Elsevier

Chaos, Solitons & Fractals

Volume 162, September 2022, 112495
Chaos, Solitons & Fractals

Chebyshev cardinal polynomials for delay distributed-order fractional fourth-order sub-diffusion equation

https://doi.org/10.1016/j.chaos.2022.112495Get rights and content

Highlights

  • The distributed-order time fractional fourth-order sub-diffusion equations is defined.

  • The Chebyshev cardinal polynomials are employed to solve these problems.

  • A distributed-order fractional derivative matrix is extracted.

  • The accuracy of the proposed method is investigated in some numerical examples.

Abstract

In this work, a category of delay distributed-order time fractional fourth-order sub-diffusion equations is investigated. The Chebyshev cardinal polynomials (as a proper class of basis functions) are employed to make an appropriate methodology for these problems. To this end, some matrix relationships regarding the distributed-order fractional differentiation (in the Caputo kind) of these polynomials are extracted and applied in generating the desired approach. The provided method converts solving these problems into obtaining the solution of systems of algebraic equations. The reliability of the technique is evaluated by solving three examples.

Introduction

The subject of fractional derivatives (which is a natural extension of the classical derivatives) is of interest to researchers because of its wide range of applications [1]. A very important point about fractional derivatives that is highly regarded is their non-local feature [2]. This property makes the future state of a dynamical system modeled by such derivatives dependent on its past and present states. In recent years, many fundamental problems in engineering and practical science have been modeled with good accuracy and finesse via fractional derivatives. For instance, they have been used in modeling the behavior of coronavirus [3], teletraffic problems [4], [5], dynamics of tuberculosis and HIV diseases [6], the vibration of a plate excited with supports movement [7], etc.

A serious point regarding fractional differential equations (i.e., equations that possess at least one term with a fraction derivative) is to find their analytic solution, which is often very arduous and in most cases, infeasible. In the face of such problems, the use of numerical methods is usually fruitful. Some well-known numerical approaches that have recently been applied for this category of problem include finite difference methods, meshless methods and spectral methods. For instance, see [8], [9], [10], [11], [12], [13], [14].

Various definitions of fractional derivatives have been provided in different texts. The most popular classical fractional derivatives are the Caputo and Riemann–Liouville derivatives [1]. As an extension of the traditional fractional derivatives, the distributed-order fractional derivatives are defined by integrating the classical cases over the order of the differentiation within a known domain [15], [16]. These derivatives play a middle role between the ordinary and fractional derivatives [17]. The differential equations produced by this kind of fractional derivatives can be propounded as generalizations of single and multiple fractional differential equations [16], [18] In recent years, fractional differential equations with distributed-order derivatives have been successfully applied to more accurately model many problems in the fields of signal processing [19], viscoelastic [20], electrochemistry [21], control [22], diffusion [23], etc.

Fractional diffusion equations of fourth-order model many important problems in engineering, such as beam vibration and groove formation on a metal surface [24]. There are several real models that are constructed via the diffusion-wave equations with distributed-order fractional derivatives. In [25], the authors used distributed-order fractional form of diffusion-wave equations to study waves in a viscoelastic rod. Some novel fractional differential equations based upon the Fokker–Planck equation with distributed-order fractional derivative are introduced in [26] for infiltration and absorption. In [27], the authors used the theory of distributed-order fractional derivatives regarding diffusion-wave equations in studying the radial groundwater flow and also investigated its application to pumping and slug tests.

Due to the importance of the fractional fourth-order sub-diffusion equations, we are interested in proposing a suitable technique to solve a delay version of such equations. So, we concentrate on the distributed-order fractional problem 01ρ(α)Dtα0Cw(z,t)dα+λwzzzz(z,t)=h(z)w(z,t)+gz,t,w(z,t),wz,tκ1,wz,tκ2,,wz,tκq, with (z,t)0,L×0,T, under the conditions w(z,t)=ψ(z,t),(z,t)0,L×[κ,0],and w0,t=β0(t),wL,t=β1(t),wzz0,t=β2(t),wzzL,t=β3(t),in which λ is a non-zero real number, h, g, ψ and βi for i=0,1,2,3 are given smooth functions, κi>0 and κ=max1iqκi are given constants. Also, Dtα0Cw(z,t) denotes the Caputo’s fractional differentiation of order α relative to t of w. Moreover, the distribution function ρ:[0,1]R+{0} satisfies the following conditions [28]: α[0,1],ρ(α)0,and01ρ(α)dα=c0>0.Note that the linearized second-order numerical method is developed in [24] to solve the above problem. The implicit compact difference method is developed in [29] for solving the above fractional problem.

Since the distributed-order fractional differentiation is based upon the integration of a classical fractional differentiation, the polynomial basis functions can be useful in solving fractional problems with this type of derivative. Among basis functions of polynomial type, cardinal polynomials can be a good choice for solving these problems because their expansion coefficients are easily calculated. In recent years, cardinal polynomials have been extensively applied to solve diverse problems. Some of these uses can be found in [30], [31], [32], [33].

In this study, we utilize the Chebyshev cardinal polynomials (CCPs) [34] to generate a suitable and accurate numerical method for the above introduced delay distributed fractional problem. A derivative matrix is derived to compute the distributed-order fractional differentiation of these polynomials, and employed in making the provided method. The expressed methodology turns the problem solving into solving a simple algebraic problem by approximating the unknown function of the problem by these polynomials and exploiting the tau technique. Note that the interpolation property makes it easy to calculate nonlinear term.

The overview of this study is as follows: Required preparations are given in Section 2. A review of the CCPs with their properties is provided in Section 3. A matrix relationship is provided in Section 4 for distributed-order fractional differentiation of the CCPs. The generated methodology is explained in detail in Section 5. Several examples are investigated in Section 6. Eventually, the conclusion of this study is summarized in Section 7.

Section snippets

Definitions and preliminaries

Here, we provide some information that is required in this study.

Definition 2.1

[1]

Let g is a differential function over c1,c2 and 0<α1 is a real number. The Caputo fractional differentiation of order α of g is given by Dtαc1Cg(t)=1Γ(1α)c1t(ts)αg(s)ds,0<α<1,g(t),α=1.Note that for α=0, we have Dt0c1Cg(t)=g(t).

Property 2.2

[1]

For 0<α<1 and kN{0}, we have Dtαc1Ctc1k=0,k=0,k!tc1kαΓ(kα+1),k=1,2,.

Definition 2.3

An Nˆ-point Gauss–Legendre quadrature formula can be defined over [0,1] as follows [35]: 01f(t)dt12i=1Nˆw̄ift̄i,where w̄i=2

Chebyshev cardinal polynomials

Here is a brief overview of CCPs along with some of their features.

Definition 3.1

For a given MZ+, the CCPs of order M are generated on [0,L] as [34] φL,i(z)=1θi(L)k=0Mbik(L)zMk,where θi(L)=l=0liM(zizl),bik(L)=1k=0,1kl=0kail(L)bikl(L),k0,ail(L)=r=0riMzrl, and zl=L21cos(2l+1)π2(M+1) for l=0,1,,M.

Property 3.2

A continuous function f on [0,L] can be expressed by these polynomials as f(z)i=0MfziφL,i(z)FL,MΦL,M(z),where FL,M=fz0fz1fzM,and ΦL,M(z)=φL,0(z)φL,1(z)φL,M(z).

Theorem 3.3

The first derivative of ΦL,M(z) can

Distributed-order fractional derivative matrix

In the sequel, we introduce a matrix relationship for the distributed-order fractional derivative of the CCPs.

Lemma 4.1

Let φT,i(t)’s are the functions defined in (3.8). Then, for 0α1, we have Dtα0CφT,i(t)φT,i(α)(t)=1θi(T)k=0Nbik(T)tNk,α=0,1θi(T)k=0N1bik(T)(Nk)!Γ(Nkα+1)tNkα,0<α<1,1θi(T)k=0N1bik(T)(Nk)tNk1,α=1.

Proof

Using the formula expressed in (3.1) and Property 2.2, the proof can be easily performed. 

Theorem 4.2

The Caputo fractional differentiation of order 0α1 of ΦT,N(t) in (3.8), can be computed as

Description of the method

Here, we establish a numerical method for the problem expressed in (1.1)–(1.3) by approximating the unknown w(z,t) in terms of the CCPs. To this end, we let w(z,t)i=0Mj=0NwijφL,i(z)φT,j(t)ΦL,M(z)WMNΦT,N(t),where WMN is an unknown matrix as WMN=w00w01w0Nw10w11w1NwM0wM1wMN.From Corollary 3.4 and the above approximation, we get wzzzz(z,t)i=0Mj=0Nwijd4φL,i(z)dz4φT,j(t)ΦL,M(z)DM(4)WMNΦT,N(t). Using Theorem 4.3, the distributed-order fractional differentiation of (5.1), can be

Numerical examples

We here investigate some examples to show the reliability of our method. The following relations will be used to measure the values of error generated by the provided approach: E=max(z,t)[0,L]×[0,T]|w(z,t)w̃(z,t)|,E2=0L0Tw(z,t)w̃(z,t)2dtdz1/2, where w is the analytic solution and w̃ is the solution derived using the proposed scheme.

Example 1

[24], [29]

Consider the equation 01Γ92αDtα0Cw(z,t)dαwzzzz(z,t)=z2zw(z,t)+gz,t,w(z,t),wz,t0.1,(z,t)[0,1]×[0,1], where gz,t,w(z,t),wz,tκ=Γ92t52(t1)ln(t)ezz2z+w2z,t

Conclusion

In this paper, the Chebyshev cardinal polynomials were employed to make a suitable scheme for delay distributed-order time fractional fourth-order sub-diffusion equation. Because of this, some matrix relations regarding the distributed-order fractional derivative of these polynomials were derived and applied in the desired approach. The adopted technique was a matrix method to convert the main problem into a problem of algebraic equations. Some examples were considered for numerical evaluation

CRediT authorship contribution statement

M.H. Heydari: Conceptualization, Methodology, Software, Validation, Writing – original draft, Visualization, Supervision. M. Razzaghi: Conceptualization, Methodology, Software, Validation, Review and editing. J. Rouzegar: Conceptualization, Methodology, Software, Validation, Review and 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.

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