A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

Taghipour, M.; Aminikhah, H.

doi:https://doi.org/10.1155/2021/8837846

Mathematical Problems in Engineering

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Advances in Numerical Optimisation: Theory, Models, and Applications

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Research Article | Open Access

Volume 2021 | Article ID 8837846 | https://doi.org/10.1155/2021/8837846

A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

M. Taghipour¹and H. Aminikhah^1,2

Academic Editor: Li-Tao Zhang

Received08 Sept 2020

Revised29 Dec 2020

Accepted08 Jan 2021

Published23 Jan 2021

Abstract

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

1. Introduction

The coupled Burgers equations are coupled partial differential equations which are capable of describing realitic polydispersive supensions. The coupled Burgers equation perdicts an interesting phenomenon, which is called phase shifts [1]. This equation is one of the fundamental models in fluid mechanics and arises in gas dynamics, chromatography, and flood waves in rivers [2]. The coupled viscous Burges equation is given bywith the initial conditionsand the boundary conditionswhere is a real constant and and are arbitrary constants depending on the system parameters such as Peclet number, Stokes velocity of particles due to gravity, and Brownian diffusivity [3].

Spline is a special function defined piecewise by polynomials. The spline approximation first appeared in a paper by Schoenberg [4]. Spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Applications of spline function in fractional partial differential equations can be found in [5–15].

In this article, we consider the following coupled Burgers equation with time fractional derivative:with initialand the boundary conditionswhere is order of time fractional derivative. Also, , , and are those ones we said before.

denotes the Caputo–Fabrizio derivative of the function defined as [16]where is a normalization function such that and .

Recently, the Caputo–Fabrizio derivative has received more attention from researchers due to their description of some physical phenomenon [17–26].

For the description of memory and some physical properties of various materials and processes, modeling with fractional derivatives is very appropriate. This is the main benefit of fractional derivatives in comparison with classical integer order models, in which such effects are missed. In recent years, the coupled system of Burgers equations with fractional derivatives has been the focus of attention. For example, in [27], the Adomian decomposition method is directly extended to study the coupled Burgers equations with time and space fractional derivatives. Khan et al. [28] proposed the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for time fractional Burgers and coupled Burgers equations. The fractional variational iteration method (FVIM) to solve a time and space fractional coupled burgers equations is given by Prakash et al. [29]. In [30], a q-homotopy analysis transform method (q-HATM) for time and space fractional coupled Burgers equations is introduced. Aminikhah and Malekzadeh [31] introduced a new homotopy perturbation method for system of variable coefficient coupled Burgers equations with time fractional derivative. In [32], the Laplace-Adomian decomposition method (LADM), the Laplace-variational iteration method (LVIM), and the reduced differential transform method (RDTM) are proposed to solve the one- and two-dimensional fractional coupled Burgers equations. Albuohimad and Adibi derived a hybrid spectral exponential Chebyshev method (HSECM) for time fractional coupled Burgers equations [33]. In [34], authors investigate the fractional coupled viscous Burgers equation involving Mittag–Leffler kernel. In [35], the generalized two-dimensional differential transform method (DTM) was applied to solve the coupled Burgers equations with space and time fractional derivatives. Ozdemir et al. used the Gegenbauer wavelets-based computational methods to find the approximate solutions of the coupled system of Burgers equations with time fractional derivative [36].

Our aim is to propose a Crank–Nicolson finite difference scheme using cubic B-spline quasi-interpolation to solve time fractional coupled viscous Burgers equations. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. This approximations have not been used for the fractional coupled Burgers equations before.

The paper is organized as follows. In Section 2, we present some basic definitions and concepts of quasi-interpolants. In Section 3, using the quasi-interpolant and Crank–Nicolson finite difference method, we obtain a numerical scheme. The stability of this method is studied in Section 4. In Section 5, some numerical examples are proposed. Finally, conclusions are given in Section 6.

2. Univariate Spline Quasi-Interpolants

In this section, we introduce the basic concepts about B-spline and univariate B-spline quasi-interpolants that we will use in Section 3.

According to [37], letand be an interval that has been partitioned into subintervals via a set of points with

We define the space of univariate polynomial splines of smoothness and degree with knots aswhere are given integers. We have

For a formal proof of this fact, see Theorem 4.4 of [38].

Given and , the associated extended partition is defined to be , where is the dimension of given in (11):

Given an extended partition , letfor , and letfor and . Let

We call these the normalized B-splines of order (or degree ) associated with the extended partition .

In [37], univariate B-spline quasi-interpolants can be defined as a formula of the formwhere are the B-splines forming a basis of .

Quasi-interpolants have been heavily studied in the literature. Some basic ideas and sources for further information can be found in [38]. For a good approximations, we need to make sure it reproduces polynomials, i.e., for all . For each , we assume that the coefficient is a linear functional defined on that can be computed from samples of at some set of points in .

According to [39], the error of a quasi-interpolation satisfieswhere , is the union of the supports of all B-splines , and denote the maximum norm of on and that is used to indicate proportionality. If the local mesh ratio is bounded, i.e., if the quotients of the lengths of adjacent knot intervals are , then the error of the derivatives on the knot intervals can be estimated byfor .

Suppose are equally spaced points in the interval . Let

Then, (17) defines a linear operator mapping into with for all cubic polynomials . For approximate derivatives of by derivatives of up to the order , we can evaluate the value of and at by and . We set , , and , where , , and , . By solution of the linear systems,we obtainwhere and is obtained as follows:

3. Numerical Scheme

We consider a grid , with . The step length in time is denoted by and . So, the domain is divided into a uniform grid of mesh points . The values of the function at the grid points are denoted and is the approximate solution at the point .

A discrete approximation to the at can be obtained by the following approximation [40]:where

Theorem 1. Suppose . LetThen,

Proof. Using the Taylor series expansion with integral remainder, we haveSince,and for and ; hence, the result will be achieved.
Now, using Theorem 1, we obtainWe introduce some lemmas which will be used in numerical scheme and stability analysis.

Lemma 1. (see [41]). Suppose ; then, we havewhere

Lemma 2. (see [42]). For the definition , we have and .

Lemma 3. (see [42]). Suppose ; it holds that and

Now, we present the numerical scheme for solving (4)–(7) based on the Crank–Nicolson method and cubic B-spline quasi-interpolant. We approximate equations (4) and (5) at the point by the Crank–Nicolson finite difference approximation:

The nonlinear terms in equations (33) and (34) are linearized using the following quasi-linearization [43]:

Now, using (22), (24), Lemma 1, and relations (36)–(38), we have the following difference scheme which is accurate of the order :

Therefore, in each time step we solve the following bi-pentadiagonal linear system of dimension :where .

4. Stability Analysis

To study the stability analysis of the proposed scheme, we use the Fourier method. In applying the Fourier stability method, the nonlinear terms are temporarily frozen, since the stability analysis is strictly only applicable to linear equations. Thus, we have linearized the nonlinear terms and in equation (4) by freezing and as a local constants and , respectively. We have

Substituting approximations (22) and (32) yield the following difference equation:

Let and be the approximate solutions of (43), and define , with corresponding vectors:

So, we havewhere .

Now, we define the grid functions as follows:

We expand the and into the following Fourier series expansions:where

Applying the Parseval equality,we have

Now, we suppose thatwhere . Substituting the above relations into (45) leads to

Set

We haveso that

Theorem 2. If is the solution of equation (55), then there are positive constants such that

Proof. We use the mathematical induction for proof. For , we haveUsing the convergence of the series on the right-hand side of equation (49), we know that there exists a positive constant such thatSo,Now, suppose thatBy equation (56), we haveNow, assume thatSo,

Remark 1. From (5), with similar way, there are positive constants such that

Theorem 3. The finite difference schemes (39) and (40) are unconditionally stable for .

Proof. According to Theorem 2 and Remark 1, using (51), we obtainso thatwhich shows that schemes (39) and (40) are unconditionally stable.

5. Numerical Results

In this section, we provide two examples to illustrate efficiency of schemes (39) and (40). All experiments are performed on a Windows 10 (64 bit) Intel(R) Core(TM) i7-7500U CPU 2.70 GHz, 8.0 GB of RAM using MATLAB R2017b. In all examples, we use the error normwhere and . We evaluate the convergence order with the following formula:

Example 1. Consider the coupled Burgers equation with time fractional derivative with exact solutions :We derive the functions and with the help from exact solutions. Numerical solution and exact solution have been demonstrated in Figure 1. Tables 1–4 give the approximation errors and CPU times for the difference schemes. We choose different space step sizes to obtain the numerical results and order of convergence. Figure 2 shows the comparison of numerical solution and exact solution for at and contour plot of numerical solution. Also, Figure 3 shows the pointwise errors for .

(a)

(b)

(a)

(b)

(a)

(b)

Example 2. Consider the following coupled Burgers equation with time fractional derivative with exact solutions :We solve this problem with the method developed in this article with several values of and . Graphs of numerical solution and exact solution at different times have been demonstrated in Figure 4. Figure 4 shows that the proposed method is efficient. Table 5 gives the approximation errors for with different and CPU times. We choose for space step size to obtain the numerical results. Figure 5 shows the comparison of numerical solution and exact solution for at . Also, Figure 6 shows the pointwise errors and contour plot of numerical solution for .

(a)

(b)

(a)

(b)

6. Conclusion

In this article, we constructed a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation to solve time fractional coupled Burgers equations. By the Fourier series method, we proved that this scheme is unconditionally stable. Numerical examples have been carried out to show the convergence orders and applicability of the scheme and error norms are calculated with respect to different space step sizes. From the error tables and graphs of exact and numerical solution, we can say that our method has a good accuracy. For the numerical computations, we have used Matlab.

Data Availability

All results have been obtained by conducting the numerical procedure and the ideas can be shared for the researchers.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2021 M. Taghipour and H. Aminikhah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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