Wavelet approximation scheme for distributed order fractional differential equations

doi:10.1016/j.camwa.2020.08.016

Computers & Mathematics with Applications

Volume 80, Issue 8, 15 October 2020, Pages 1985-2017

https://doi.org/10.1016/j.camwa.2020.08.016 Get rights and content

Abstract

This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre quadrature formula and standard Tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations. For the better understanding of the method, numerical algorithms are also provided for the considered problems. In order to verify the desired accuracy of the proposed method, five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method. Also, the convergence analysis, error bounds, error estimate and numerical stability of presented method via Legendre and Chebyshev wavelets are investigated. Moreover, comparison of the numerical results obtained by the proposed approach is provided with the results of existing method.

Introduction

In mathematical science, the topic of fractional differential equations (FDEs) is ancient. FDEs are generalization of ordinary differential equations (ODEs) and partial differential equations (PDEs) to an arbitrary non-integer order. Fractional derivative is a great tool, as it is used to explain the consciousness and biological characteristics of various techniques and materials. This is the main reason why FDEs are used to model electrical and mechanical properties of actual materials and archaeological properties of rocks and several other fields. For the development history of fractional differential operator (FDO) one can see [1]. The FDEs are widely used in different fields of science and engineering such as viscoelastic damped structure model [2], hysteretic damping model [3], transient analysis of viscoelastic structure [4], electrochemistry [5] and ultraslow kinetics [6]. To the best of our fundamental understanding, two separate approaches based on spectral methods were carried out for FODEs and FPDEs. The first approach contains the classical orthogonal functions as a trail function to approximate the corresponding FDEs. In the second approach, Jacobi polyfractonomials were used as a trail function to find the approximate solutions of FODEs and FPDEs (see for example [7] and references therein). The discussion of fractional-order control and system has attracted many researchers to work on as an important application field of fractional calculus.

In sciences and engineering there are number of existing problems which are governed by the distributed order FDEs [8], [9]. If the fractional calculus operators operate on $f (η)$ , and we integrate $D_{η}^{γ} f (η)$ with respect to the order $γ$ , then distributed-order differential/integral equations can be obtained. In 1969, Caputo was the first who presented the concept of distributed-order equation and solved it later in 1995. The distributed order equation is designed to model the input–output relationship of a linear time-invariant system based on observation of the frequency domain response. In other words, the distributed-order equation is the time-domain representation of the input–output relationship perceived and constructed in the frequency domain. The distributed-order differential equations and their applications in engineering fields have recently gained much attention. For instance, Bagley and Torvik [10] discussed the general solution of linear distributed-order differential equation and also examined the existence of solution of DOFDEs. In [11], distributed-order equations were introduced in the constitutive dielectric media. In 2016, the fractional kinetics of distributed-order equations was discussed by Sokolov et al. [12]. In 2006, Umarov and Steinberg [13] proposed multi-dimensional random walk models governed by distributed fractional order differential equations. In particular, the distributed-order operator is a more reliable method for understanding and defining certain actual physical processes such as the complexities of nonlinear systems [14], [15], networked structures [16] and nonhomogeneous phenomena [9].

In modeling many fields of mathematical physics, chemistry, biology and electronics, the study of distributed order fractional differential equations (DFDEs) plays major role; such as in the distributed order oscillator [17], complex system [18], viscoelastic model [19], PI-control systems [20], distributed order system identification [21], frequency control system [22], and many more (see [23] and references therein). It has already been shown that both integer and fractional order differential equations are the special cases of DFDEs. One can see [10], [24] regarding the uniqueness and existence of the solution of DFDEs. Since most of the DFDEs do not have precise analytical solutions therefore an efficient numerical methods are needed to solve them. For this purpose, computational methods such as Petrov–Galerkin and spectral collocation methods [25], Legendre collocation method [26], pseudo-spectral method [27], improved composite collocation method [28], spectral method [29], an improved meshless method [30], finite element method [31], Legendre spectral element method [32], implicit difference scheme [33], Chebyshev collocation method [34], Legendre–Gauss collocation method [35], block-pulse wavelet [36] are widely used.

The operational matrix method has been proved simple and efficient computational technique for solving fractional differential equations. In 2010, Saadatmandi and Dehghan [37] proposed and derived the operating matrix of fractional derivative (OMFD) for the shifted Legendre polynomials (SLPs). Li et al. [38] developed the haar wavelet OMFD in 2010. Keshavarz et al. [39] also launched the Bernoulli wavelets OMFD in 2014. The OMFD for B-spline functions was also constructed in 2012 by Lakestani et al. [40]. In 2013, Bhrawy et al. [41] obtained the OMFD for the generalize Laguerre polynomials. In 2014 Kayedi-Bardeh et al. [42] developed the OMFD using the fractional jacobi function. The interested readers can also see [43], [44] for more details of operational matrices.

Recently, M. Pourbabaee and Saadatmandi [45] provided a useful technique based on the operational matrices of Legendre polynomials for finding the approximate solution of DFDE along with initial conditions. Based on the facts and to the best of our knowledge, there is no numerical technique available based on wavelets operational matrix for solving DFDEs. Motivated by the work of M. Pourbabaee and Saadatmandi [45] and usefulness of operational matrices, the aim of this paper is to create the distributed order fractional derivative operational matrix for Legendre and Chebyshev wavelets. The purpose of using Legendre and Chebyshev wavelets in this paper are the properties of their basis functions which are constructed by orthogonal polynomials. Thus, Legendre and Chebyshev wavelets basis have advantages of both infinitely differentiable functions and small compact supports. Furthermore, the operational matrices of Legendre and Chebyshev wavelets have the sparsity and thus contain many zeros, which make computation simple and fast. These operational matrices are utilized along with Gauss–Legendre quadrature rule and standard Tau approach to solve linear DFDEs and distributed order time fractional diffusion equations (DTFDEs). The motivation of this method is the provide good experimental results with less computational complexity.

The remainder of the paper is presented as follows. Section 2 briefly presents some basic definitions which include fractional derivative, the distributed differential operator, Legendre wavelet, Chebyshev wavelet and their approximation properties. Also, we define Gauss Legendre quadrature formula in this section. Section 3 is devoted to the operational matrices. In this section, operational matrices of derivative of Legendre and Chebyshev wavelets for integer order and distributed order are derived. The implementation of the method to one and two-variables distributed order equations is described in Section 4. Convergence analysis and error bounds with the help of Legendre as well as Chebyshev wavelets for the mentioned scheme are given in Section 5. Error estimation of the presented method is provided in Section 6. Finally, numerical experiments are performed alongwith discussion of numerical stability in Section 7.

Section snippets

Definition of fractional derivative in the Caputo sense

The Caputo fractional derivative with order $γ > 0$ is defined as [1] $D_{η}^{γ} g (η) = \{\begin{matrix} \frac{1}{Γ (n - γ)} \int_{0}^{η} \frac{g^{(n)} (τ)}{{(η - τ)}^{γ + 1 - n}} d τ, & n - 1 < γ < n \\ \frac{d^{n}}{d η^{n}} g (η), & γ = n \in N . \end{matrix}$

Here $Γ (.)$ is the Gamma function. This Caputo operator has some following properties:

•
$D_{η}^{γ} C = 0$ , where, C is any arbitrary constant.
•
The Caputo derivative of $g (η) = η^{i}, i \in N$ is given as: $D_{η}^{γ} η^{i} = \{\begin{matrix} 0, & i < ⌈ γ ⌉, \\ \frac{Γ (i + 1)}{Γ (i + 1 - γ)} η^{i - γ}, & i \geq ⌈ γ ⌉, \end{matrix}$ where, $⌈ \cdot ⌉$ is the ceiling function.
•
$D_{η}^{γ}$ satisfies the linearity property, i.e. $D_{η}^{γ} (\sum_{l = 1}^{k} b_{l} g_{l} (η)) = \sum_{l = 1}^{k} b_{l} D_{η}^{γ} g_{l} (η),$ where, $b_{l}$ are arbitrary constants for $l = 1, 2, \dots, k$

Legendre wavelets integer order derivative matrix

Theorem 3.1

Let $Ψ^{L} (η)$ be the Legendre wavelets vector defined in Section 2.3. The derivative of $Ψ^{L} (η)$ can be expressed as [48] $\frac{d Ψ^{L} (η)}{d η} = D_{L} Ψ^{L} (η),$ where $D_{L}$ is the $2^{K} (m + 1)$ operational matrix defined as follows: $D_{L} = [\begin{bmatrix} F & 0 & 0 & \dots & 0 \\ 0 & F & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & F \end{bmatrix}],$ in which F is $(m + 1) \times (m + 1)$ matrix and its $(r, s)$ th element is defined as follows: $F_{r, s} = \{\begin{matrix} 2^{K + 1} \sqrt{(2 r - 1) (2 s - 1)} r = 2, 3, \dots, (m + 1), s = 1, 2, \dots, r - 1 a n d (r + s) o d d, \\ 0, otherwise. \end{matrix}$

The above matrix can be generalized for higher orders as $\frac{d^{n_{1}} Ψ^{L} (η)}{d η^{n_{1}}} = {(D_{L})}^{n_{1}} Ψ^{L} (η), n_{1} = 1, 2, 3, \dots$

Chebyshev wavelets integer order derivative matrix

Theorem 3.2

Let $Ψ^{C} (η)$ be the Chebyshev wavelets

Numerical method: Application to the distributed order linear equations

In this section we discuss numerical method for solving distributed order linear fractional differential equations in one and two variables. We provide two schemes namely scheme (I) and scheme (II) for dealing with distributed order fractional differential equations in one and two-variable, respectively.

Convergence analysis and error bound by using Legendre wavelets

Theorem 5.1

Let ${(f (ξ, η))}_{N u m}, {(v (ξ, η))}_{N u m}, {(w (ξ, η))}_{N u m}$ be the approximation of continuous functions $f (ξ, η), v (ξ, η), w (ξ, η)$ , respectively defined on $Ω (= [0, 1] \times [0, 1])$ with bounded mixed second derivative, say $| \frac{\partial^{4} f (ξ, η)}{\partial ξ^{2} \partial η^{2}} | \leq B, | \frac{\partial^{4} v (ξ, η)}{\partial ξ^{2} \partial η^{2}} | \leq B_{1}, | \frac{\partial^{4} w (ξ, η)}{\partial ξ^{2} \partial η^{2}} | \leq B_{2}$ for some positive constant $B, B_{1}, B_{2}$ , where, $v = \frac{\partial u (ξ, η)}{\partial η}$ and $w = \frac{\partial^{2} u (ξ, η)}{\partial ξ^{2}}$ . Then

$(a)$ $H (ξ, η)$ can be expanded as an infinite sum of Legendre wavelets and the series converges uniformly to the function $H (ξ, η)$ , that is $H (ξ, η) = \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} \sum_{n^{'} = 1}^{\infty} \sum_{m^{'} = 0}^{\infty} C_{n m n^{'} m^{'}}^{L} Ψ_{n}$

Error estimation for one variable case

In this part, we discuss an error estimation for one variable distributed order fractional differential equation. We have one-variable distributed fractional differential equation as $\int_{α_{1}}^{α_{2}} ρ (γ) D_{η}^{γ} u (η) = f (η), η \in [0, 1],$ under initial conditions $u^{(n)} (0) = u_{n}, n = 0, 1, \dots, ⌈ α_{2} ⌉ - 1 .$ First, we assume $E_{N u m} (η) = u (η) - u_{N u m} (η)$ be the error function, where, $u (η)$ is exact and $u_{N u m} (η)$ is approximate solutions of Eq. (6.1). Hence $u_{N u m}$ satisfies the following equations $\int_{α_{1}}^{α_{2}} ρ (γ) D_{η}^{γ} u_{N u m} (η) = f (η) + R_{N u m} (η),$ under initial

Numerical examples

In this part, we consider five test functions in order to verify the usefulness and accuracy of the method. The numerical results obtained via the presented method are provided in the form of Tables and Figures for different values of $M, N$ and $K$ . Numerical results of $L_{2}$ -errors, $L_{\infty}$ -errors and pointwise errors along with used CPU time are provided in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11 and 18, 19. Comparison of numerical results produced by

Conclusion

In this paper, a new approach based on operational matrices of wavelets and standard Tau method is developed for solving distributed order fractional differential equations. Operational matrices of Legendre and Chebyshev wavelets together with standard Tau method and Legendre Gauss quadrature formula convert the original problem into a system of linear algebraic equations. The testing of method is done on several test problems. To verify the efficiency and accuracy of the method $L_{2}$ -errors, $L_{\infty}$

CRediT authorship contribution statement

Yashveer Kumar: Conceptualization, Software, Writing. Somveer Singh: Methodology, Editing. Nikhil Srivastava: Visualization, Investigation. Aman Singh: Writing - review. Vineet Kumar Singh: Supervision.

Acknowledgments

The first & fourth authors acknowledge the financial support from Council of Scientific & Industrial Research (CSIR), India, under Junior Research Fellow (JRF) scheme. The second author acknowledges the financial support from National Board for Higher Mathematics, Department of Atomic Energy, India, with sanction order no. 0204/17/2019/R & D-II/9814. The third author acknowledges the financial support from Ministry of Human Resource and Development (MHRD) , under JRF scheme.

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Wavelet approximation scheme for distributed order fractional differential equations

Abstract

Introduction

Section snippets

Definition of fractional derivative in the Caputo sense

Legendre wavelets integer order derivative matrix

Chebyshev wavelets integer order derivative matrix

Numerical method: Application to the distributed order linear equations

Convergence analysis and error bound by using Legendre wavelets

Error estimation for one variable case

Numerical examples

Conclusion

CRediT authorship contribution statement

Acknowledgments

Adv. Eng. Softw.

J. Comput. Phys.

J. Math. Anal. Appl.

Nonlinear Anal. TMA

ISA Trans.

Signal processing

J. Comput. Appl. Math.

Appl. Numer. Math.

Appl. Numer. Math.

Appl. Numer. Math.

Comput. Math. Appl.

Appl. Math. Comput.

Appl. Math. Model.

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

J. Franklin Inst. B

J. Quant. Spectrosc. Radiat. Transfer

Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

Fractional calculus-a different approach to the analysis of viscoelastically damped structures

AIAA J.

Damping description using fractional derivatives

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