A sine-cosine wavelet method for the approximation solutions of the fractional Bagley-Torvik equatio

Abstract

Keywords: Sine-cosine wavelet; Bagley-Torvik equation; Caputo derivative; Block pulse function

Introduction

Many physical and engineering problems are modeled accurately via fractional order differential equations (FDEs). [1–11] Therefore, applied mathematicians and scientists focus on analyzing FDEs. The paper is focused on the numerical solution of fractional Bagley– Torvik (B-T) equation because it plays an important role in modelling dynamic physical phenomena. One such field is fluid mechanics, in particular, the analysis of the motion of a rigid plate soaked in a Newtonian fluid. Analytical, semi-analytical and numerical algorithms have been established for the solution of B-T equations.

To name a few, a fractional linear multistep method was considered in [12]. The Hybrid functions method was proposed in [13]. Legendre artificial neural network method [14] was studied for solving B-T equations. Bagley-Torvik equations was examined with Fermat Operational Tau Method [15]. Sumudu transformation method [16] was applied to solve B-T equations. Legendre collocation method was used for solving the B-T equations [17]. The operational formulation of collocation methods [18], wavelet methods [19–20] and fractional Taylor methods [21] are implemented for the solution of this fractional equation.

*Corresponding author. *E-mail address: artur@yildiz.edu.tr This paper was recommended for publication in revised form by Regional Editor Vildan Çetkin Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Sigma J Eng Nat Sci, Vol. 40, No. 1, pp. 150–154, March, 2022

The basis functions used to obtain sine-cosine wavelets have compact support and they are orthogonal. Therefore the resulting operational matrices for fractional integration using sine-cosine wavelets end up being sparser. Which in turn is important for less computational load and speed considerations which is the reason why we intend to explore the sine-cosine wavelet in this study. In this paper, the following form of Bagley-Torvik (B-T) equation is analyzed:

Sine-Cosine Wavelet Approximation

Sine-Cosine Wavelet Wavelets involve a mother wavelet and dilated and translated versions of the mother wavelet. If the dilation parameter a and the translation parameter b vary continuously, family of continuous wavelets is written as follows [22]:

1  t − b ψ a , b (t ) = a − 2 ψ  , a, b ∈ R, a ≠ 0  a 

When the dilation parameter and the translation parameter are chosen 2 and 1, the family of discrete wavelets are obtained: k

where m, c, k and g(t) represent the mass, damping, stiffness coefficients and external force, respectively. y(t) is the displacement function and Dα is Caputo fractional differential operator of order 3/2. We utilize a new operational matrix method based on the sine-cosine wavelet (SCW) to solve initial value problem of B-T equation. By means of operational matrix of fractional integration, we obtain a system of linear algebraic equations. The paper is organized as follows. The fundamental concepts and definitions are provided in section II. The definition and the properties of sine-cosine wavelets are covered in section III and also the operational matrices for fractional integration for sine-cosine wavelets are obtained. In section IV the method is applied to the fractional B-T equation. The numerical results are presented in section V.

Preliminiaries

In this section we provide the definition and the basic properties of the Caputo fractional derivative which are used in this paper as follows: The Caputo definition of fractional derivative operator is expressed in the form: t f (n) (τ ) 1 ( D f )(t ) = Γ (n − α ) ∫0 (t − τ )1−n+α dτ , 0 ≤ n − 1 < α ≤ n. (3)

where ψkn (t) forms an orthonormal basis [23]. Sine-cosine wavelets ψnm = ψ(k,n,m,t) consist of four arguments: k = 1,2,3,…, n = 1,2,3,…,2k–1, the values m are given in (9) and t represents normalized time. Sine-cosine wavelets defined as follows:

and 1  ,  2  f m (t ) =  cos (2mπ t ) , sin (2 (m − L )π t )  

where ∈[0,1), L ∈ Z+. It is obvious that the set of sine-cosine wavelet forms orthonormal set. Approximating a function A function f defined over [0,1) is given by the following expanding 2k −1

where C and ψ(t) are m' × 1 (m' = 2k (2L + 1)) vectors expressed as: T c0,0 , c0,1 ,…, c0,2 L , c1,0 ,…,  C=  c1,2 L ,…, c2k−1, ,0 ,…, c2k−1 ,2 L 

Ψ0,0 ,Ψ0,1 ,…,Ψ0,2 L ,Ψ1,0 ,…,  Ψ =  Ψ1,2 L ,…,Ψ2k−1, ,0 ,…,Ψ2k−1 ,2 L 

Sigma J Eng Nat Sci, Vol. 40, No. 1, pp. 150–154, March, 2022

φm′xm′ = Ψ (t1 ) Ψ (t 2 ) Ψ (t 3 )  Ψ (t m′ ) , where t i =

Description Of The Method

Operational matrix of the fractional integration We determine the operational matrix of fractional order integration by means of block pulse functions (BPFs). The set of BPFs is defined as 1, i − 1 , ≤ t < i ,  bi (t ) =  m m  0, otherwise

We solve the Bagley–Torvik equation (1) and (2) via sine-cosine wavelet method and we approximate D2y(t) and g(t) as follows:

where Bm'(t) = [b1(t), b2(t),..., bm'(t)]T. The operational matrix of the fractional integral Fα related with the BPFs vector can be represented as:

2 D 3 / 2 y (t ) = (I 0.5 D 2 y )t ≈ C T Pm1 /′xm ′ ψ (t ) ,

The sine–cosine wavelet may be expanded in terms of BPFs as:

where G = [g0, g1, … gm'–1]. By equations (4) and (20), we have

by substituting Eqs. (21), (22), (23) and (25) in Eq. (1) we obtain a system of algebraic equations as: T 2 2 mC T ψ (t ) + cC T Pm1 /′xm ′ψ (t ) + kPm ′ xm ′ψ (t ) = G ψ (t ) . (26)

The vector of unknown coefficients C is computed. By using Eq. (25), we can obtain y(t).

with ξk = (k + 1)α+1 – 2kα+1 + (k – 1)α+1. [24] By using Fα, sine-cosine wavelet operational matrix of the fractional integration Pmα′xm′ can be expressed Pmα′xm′ ≈ φm′xm′ F α φm−1′xm′ .

Figure 1. Sine-cosine wavelet and the exact solution for Example 1.

Sigma J Eng Nat Sci, Vol. 40, No. 1, pp. 150–154, March, 2022

Table 1. The absolute errors of sine-cosine wavelet method for different k values for Example 1

Table 2. The absolute errors of sine-cosine wavelet method for different k values for Example 2

suggest, when the values of k increase, the absolute error decreases and our solution converges to the exact solution which proves the validity of the method. Example 2: In this example, we consider the following B-T equation: 15 15 t+ πt + t2 t , 4 8 y (0) = y ′ (0) = 0, t ∈[0,1) ,

Figure 2. Sine-cosine wavelet and the exact solution for Example 2.

Illustrative Examples

In this section, we apply the method to two examples and we have carried out all of the numerical calculations using Matlab R2020a. Example 1: We first consider following B-T equation 8 , Γ ( o.5) t 0.5 t ∈[0,1) ,

D 3 / 2 y (t ) + D 2 y (t ) + y (t ) = t 3 + 6t + y (0) = y ′ (0) = 0,

with exact solution is y(t) = t2.5. By using the proposed technique presented in section IV, we obtain the approximate solution of B-T equation. Table 2 represents absolute errors for k = 6, 7, 8 and L=1. In addition, the comparison of the numerical solutions and the exact solutions for k = 6 and L=1 is given in Figure 2. As can be seen, numerical results show the efficiency of our solution. Numerical results also demonstrate that the method is fast and can be applied to realtime problems.

with exact solution is y(t) = t3. By applying the method described in section IV, the values of the unknown matrix CT are obtained. Figure 1 represents the sine -cosine wavelet solution for k = 6, L = 1 and the exact solution. From Figure 1, it is clear that sine-cosine wavelet solution provides good approximation with exact solution. The absolute errors for different k values of sine-cosine wavelet solution (L = 1) are given in Table 1. As results

In this study, we propose sine-cosine wavelet method with block pulse functions to solve fractional Bagley– Torvik differential equations. This method converts this fractional differential equation to system of linear algebraic equations. We obtain the numerical solutions of the Bagley-Torvik equations by solving this system. Since the basis functions of sine–cosine wavelets are orthogonal, the implementation of present approach is simple and easy. Numerical examples are included to illustrate the validity and the accuracy of the proposed numerical method.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Sigma J Eng Nat Sci, Vol. 40, No. 1, pp. 150–154, March, 2022

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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