Research Article  Open Access
A Jacobi Collocation Method for Solving Nonlinear BurgersType Equations
Abstract
We solve three versions of nonlinear timedependent Burgerstype equations. The JacobiGaussLobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parameters a and ß. In addition, the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time. This system may be solved by any standard numerical techniques. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions produce highaccurate results. Numerical results show that the proposed method is of high accuracy and is efficient to solve the Burgerstype equation. Also the results demonstrate that the proposed method is a powerful algorithm to solve the nonlinear partial differential equations.
1. Introduction
Spectral methods (see, e.g., [1–3] and the references therein) are techniques used in applied mathematics and scientific computing to numerically solve linear and nonlinear differential equations. There are three wellknown versions of spectral methods, namely, Galerkin, tau, and collocation methods. Spectral collocation method is characterized by the fact of providing highly accurate solutions to nonlinear differential equations [3–6]; also it has become increasingly popular for solving fractional differential equations [7–9]. Bhrawy et al. [5] proposed a new Bernoulli matrix method for solving highorder Fredholm integrodifferential equations with piecewise intervals. Saadatmandi and Dehghan [10] developed the Sinccollocation approach for solving multipoint boundary value problems; in this approach the computation of solution of such problems is reduced to solve some algebraic equations. Bhrawy and Alofi [4] proposed the spectralshifted JacobiGauss collocation method to find an accurate solution of the LaneEmdentype equation. Moreover, Doha et al. [11] developed the shifted JacobiGauss collocation method to solve nonlinear highorder multipoint boundary value problems. To the best of our knowledge, there are no results on JacobiGaussLobatto collocation method for solving Burgerstype equations arising in mathematical physics. This partially motivated our interest in such method.
For timedependent partial differential equations, spectral methods have been studied in some articles for several decades. In [12], Ierley et al. investigated spectral methods to numerically solve timedependent class of parabolic partial differential equations subject to periodic boundary conditions. TalEzer [13, 14] introduced spectral methods using polynomial approximation of the evolution operator in the Chebyshev LeastSquares sense for timedependent parabolic and hyperbolic equations, respectively. Moreover, Coutsias et al. [15] developed spectral integration method to solve some timedependent partial differential equations. Zhang [16] applied the Fourier spectral scheme in spatial together with the Legendre spectral method to solve timedependent partial differential equations and gave error estimates of the method. Tang and Ma [17] introduced the Legendre spectral method together with the Fourier approximation in spatial for timedependent firstorder hyperbolic equations with periodic boundary conditions. Recently, the author of [18] proposed an accurate numerical algorithm to solve the generalized FitzhughNagumo equation with timedependent coefficients.
In [20], Bateman introduced the onedimensional quasilinear parabolic partial differential equation, while Burgers [21] developed it as mathematical modeling of turbulence, and it is referred as onedimensional Burgers’ equation. Many authors gave different solutions for Burgers’ equation by using various methods. Kadalbajoo and Awasthi [22] and Gülsu [23] used a finitedifference approach method to find solutions of onedimensional Burgers’ equation. CrankNicolson scheme for Burgers’ equation is developed by Kim, [24]. Nguyen and Reynen [25, 26], Gardner et al. [27, 28] and Kutluay et al. [29] used methods based on the PetrovGalerkin, LeastSquares finiteelements, and Bspline finite element methods to solve Burgers’ equation. A method based on collocation of modified cubic Bsplines over finite elements has been investigated by Mittal and Jain in [30].
In this work, we propose a JGLC method to numerically solve the following three nonlinear timedependent Burgers’type equations:(1) timedependent 1D Burgers’ equation: (2) timedependent 1D generalized BurgerFisher equation: (3) timedependent 1D generalized BurgersHuxley equation:
In order to obtain the solution in terms of the Jacobi parameters and , the use of the Jacobi polynomials for solving differential equations has gained increasing popularity in recent years (see, [31–35]). The main concern of this paper is to extend the application of JGLC method to solve the three nonlinear timedependent Burgerstype equations. It would be very useful to carry out a systematic study on JGLC method with general indexes . The nonlinear timedependent Burgers’type equation is collocated only for the space variable at points, and for suitable collocation points, we use the nodes of the JacobiGaussLobatto interpolation which depends upon the two general parameters ; these equations together with the twopoint boundary conditions constitute the system of ordinary differential equations (ODEs) in time. This system can be solved by one of the possible methods of numerical analysis such as the Euler method, Midpoint method, and the RungeKutta method. Finally, the accuracy of the proposed method is demonstrated by test problems.
The remainder of the paper is organized as follows. In the next section, we introduce some properties of the Jacobi polynomials. In Section 3, the way of constructing the GaussLobatto collocation technique for nonlinear timedependent Burgerstype equations is described using the Jacobi polynomials, and in Section 4 the proposed method is applied to three problems of nonlinear timedependent Burgerstype equations. Finally, some concluding remarks are given in Section 5.
2. Some Properties of Jacobi Polynomials
The standard Jacobi polynomials of degree (,??) with the parameters are satisfying the following relations: Let ; then we define the weighted space as usual, equipped with the following inner product and norm: The set the of Jacobi polynomials forms a complete orthogonal system, and Let be the set of polynomials of degree at most , and due to the property of the standard JacobiGauss quadrature, it follows that for any , where () and () are the nodes and the corresponding Christoffel numbers of the JacobiGaussquadrature formula on the interval , respectively. Now, we introduce the following discrete inner product and norm: For , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for , the Chebyshev of the first and second kinds and the Legendre polynomials, respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases (the Chebyshev polynomials of the third and fourth kinds) are also recovered.
3. Jacobi Spectral Collocation Method
Since the collocation method approximates the differential equations in physical space, it is very easy to implement and be adaptable to various problems, including variable coefficient and nonlinear differential equations (see, for instance [4, 6]). In this section, we develop the JGLC method to numerically solve the Burgerstype equations.
3.1. (1 + 1)Dimensional Burgers’ Equation
In 1939, Burgers has simplified the NavierStokes equation by dropping the pressure term to obtain his onedimensional Burgers’ equation. This equation has many applications in applied mathematics, such as modeling of gas dynamics [36, 37], modeling of fluid dynamics, turbulence, boundary layer behavior, shock wave formation, and traffic flow [38]. In this subsection, we derive a JGLC method to solve numerically the (1 + 1)dimensional Burgers’ model problem: where subject to the boundary conditions and the initial condition Now we assume that and if we make use of (6)–(8), then we find and accordingly, (14) takes the form or equivalently takes the form The spatial partial derivatives with respect to in (9) can be computed at the JGLC points to give where Making use of (17) and (18) enables one to rewrite (9) in the form: where Using Equation (19) and using the twopoint boundary conditions (11) generate a system of ODEs in time: where Then the problem (9)–(12) transforms to the SODEs: which may be written in the following matrix form: where The SODEs (24) in time may be solved using any standard technique, like the implicit RungeKutta method.
3.2. (1 + 1)Dimensional BurgerFisher Equation
The BurgerFisher equation is a combined form of Fisher and Burgers’ equations. The Fisher equation was firstly introduced by Fisher in [39] to describe the propagation of a mutant gene. This equation has a wide range of applications in a large number of the fields of chemical kinetics [40], logistic population growth [41], flame propagation [42], population in onedimensional habitual [43], neutron population in a nuclear reaction [44], neurophysiology [45], autocatalytic chemical reactions [19], branching the Brownian motion processes [40], and nuclear reactor theory [46]. Moreover, the BurgerFisher equation has a wide range of applications in various fields of financial mathematics, applied mathematics and physics applications, gas dynamic, and traffic flow. The BurgerFisher equation can be written in the following form: where subject to the boundary conditions and the initial condition The same procedure of Section 3.1 can be used to reduce (26)–(29) to the system of nonlinear differential equations in the unknown expansion coefficients of the soughtfor semianalytical solution. This system is solved by using the implicit RungeKutta method.
3.3. (1 + 1)Dimensional Generalized BurgersHuxley Equation
The Huxley equation is a nonlinear partial differential equation of second order of the form It is an evolution equation that describes the nerve propagation [47] in biology from which molecular CB properties can be calculated. It also gives a phenomenological description of the behavior of the myosin heads II. In addition to this nonlinear evolution equation, combined forms of this equation and Burgers’ equation will be investigated. It is interesting to point out that this equation includes the convection term and the dissipation term in addition to other terms. In this subsection, we derive JGLC method to solve numerically the ()dimensional generalized BurgersHuxley equation: where subject to the boundary conditions: and the initial condition: The same procedure of Sections 3.1 and 3.2 is used to solve numerically (30)–(34).
4. Numerical Results
To illustrate the effectiveness of the proposed method in the present paper, three test examples are carried out in this section. The comparison of the results obtained by various choices of the Jacobi parameters and reveals that the present method is very effective and convenient for all choices of and. We consider the following three examples.
Example 1. Consider the nonlinear timedependent onedimensional generalized BurgersHuxley equation: subject to the boundary conditions: and the initial condition: The exact solution of (35) is
The difference between the measured value of the approximate solution and its actual value (absolute error), given by where and , is the exact solution and the approximate solution at the point , respectively.
In the cases of , and , Table 1 lists the comparison of absolute errors of problem (35) subject to (36) and (37) using the JGLC method for different choices of and with references [19], in the interval . Moreover in Tables 2 and 3, the absolute errors of this problem with and various choices of for , in both intervals and , are given, respectively. In Table 4, maximum absolute errors with various choices of for both values of are given where , in both intervals and . Moreover, the absolute errors of problem (35) are shown in Figures 1, 2, and 3 for , and with values of parameters listed in their captions, respectively, while in Figure 4, we plotted the approximate solution of this problem where , , , and for . These figures demonstrate the good accuracy of this algorithm for all choices of , , and and moreover in any interval.




Example 2. Consider the nonlinear timedependent one dimensional Burgerstype equation: subject to the boundary conditions and the initial condition