Research Article  Open Access
Numerical Solution of Nonlinear SineGordon Equation by Modified Cubic BSpline Collocation Method
Abstract
Modified cubic Bspline collocation method is discussed for the numerical solution of onedimensional nonlinear sineGordon equation. The method is based on collocation of modified cubic Bsplines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic Bspline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSPRK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.
1. Introduction
In this paper we consider the onedimensional sineGordon equation with initial conditions The Dirichlet boundary conditions are given by The nonlinear sineGordon equation arises in many different applications such as propagation of fluxion in Josephson junctions [1], differential geometry, stability of fluid motion, nonlinear physics, and applied sciences [2]. The sineGordon equation (1) is a particular case of KleinGordon equation, which plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [3], given by where is a nonlinear force and is a constant.
In the literature several schemes have been developed for the numerical solution of sineGordon equation. BenYu et al. [4] proposed two difference schemes; Bratsos and Twizell [5] used method of lines to transform the initial/boundary value problem associated with (1) into a first order nonlinear initial value problem. Mohebbi and Dehghan [6] presented a combination of a compact finite difference approximation of fourth order and a fourthorder Astable DIRKN method. Kuang and Lu [7] proposed two classes of finite difference method for generalized sineGordon equation; Bratsos and Twizell [8] presented a family of finite difference method, in which time and space derivatives are replaced by finitedifference approximations and then the equation is converted into a linear algebraic system. Wei [9] used the discrete singular convolution algorithm for the integration of (1). A variational iteration method to obtain approximate analytical solution of the sineGordon equation without any discretization has been developed by Batiha et al. [10]. Zheng [11] presented a numerical solution of sineGordon equation defined on the whole real axis. Bratsos [12] used a fourthorder rational approximation to the matrix exponential term in a threetime level recurrence relation for the numerical solution of (1). Dehghan and Shokri [13] solved the equation using collocation points and approximate the solution using radial basis functions; Dehghan and Mirzaei [14] used a boundary integral equation method; Rashidinia and Mohammadi [15] developed two implicit finite difference schemes, by using spline function approximations. LiMin and ZongMin [16] presented a meshless scheme by using a multiquadric quasiinterpolation without solving a largescale linear system of equations, but a polynomial was needed to improve the accuracy of the scheme, while Jiang and Wang [17] proposed meshless approach by directly using high accuracy MQ quasiinterpolation without using any polynomial. A modified decomposition method for explicit and numerical solutions of the sineGordon equation in the form of convergent power series has been proposed by Kaya [18]. Uddin et al. [19] proposed a meshfree approach based on radial basis function for numerical solution of (1).
The Bspline possesses several properties such as minimal compact support and smoothness, which makes them suitable for the numerical solution of linear and nonlinear partial differential equations. Bspline with collocation provides a very simple solution procedure of differential equations. They also produce a spline function which is useful to obtain the solution at any point of the domain, while in finite difference methods [4, 6–8, 15], we can find the solution only at the selected knots. In the present method, approximate solutions of sineGordon equation are obtained using a modified cubic Bspline collocation method (MCBCM) in space and strong stability preserving RungeKutta (SSPRK54) scheme [20] in time. The equation is converted into a system of partial differential equations and then, using MCBCM, it reduces into a system of ordinary differential equations. Finally we use SSPRK54 scheme to solve the obtained system of ODEs. Numerical solution of nonlinear sineGordon equation has been obtained without using any transformation or without linearizing the nonlinear term.
The paper is organized as follows. In Section 2, cubic Bspline collocation method is explained. In Section 3, modified cubic Bspline basis functions are introduced and how to find the solution of (1)–(3) using these basis functions is explained. Initial vectors have been computed in Section 4. Numerical experiments are conducted in Section 5, to demonstrate the viability and the efficiency of the proposed method computationally, and results are compared with some previous results. Finally, brief conclusions drawn from the present study are presented in Section 6.
2. Description of Method
The solution domain is partitioned into a mesh of uniform length , where , such that .
In the cubic Bspline collocation method the approximate solution can be written as the linear combination of cubic Bspline basis functions for the approximation space under consideration. Our numerical treatment for solving (1) using the collocation method with cubic BSpline is to find an approximate solution to the exact solution in the form where are the time dependent quantities to be determined from boundary conditions and collocation from the differential equation.
The cubic Bspline at the knots is given by where the set of functions forms a basis for the function defined over the region with the obvious adjustment of the boundary base functions to avoid undefined knots. Each cubic Bspline covers four elements so that an element is covered by four cubic Bsplines. The values of and its derivatives are tabulated in Table 1.

Then, using approximate function (5) and Table 1, the approximate values of and its two derivatives at the knots are determined in terms of the time parameters as follows:
3. Numerical Scheme
We have used the following modified form of cubic Bspline basis functions [21] in the combination with collocation, to solve the sineGordon equation. Modified cubic Bspline basis functions have been used for handling the Dirichlet boundary conditions and finally we obtain a diagonally dominant system of differential equations. The procedure for modifying the basis functions is given as follows:
To find the numerical solution of sineGordon equation (1), first it is rewritten as a pair of coupled equations using the following transformation: Then (1) transforms into a coupled system of equations as
Now for solving the couple of (10), using collocation method with modified cubic Bspline basis functions, first we assume our solution as the linear combination of modified cubic Bspline basis functions:
Using the approximate solution (11), the approximate value of can be written as where is the derivative of with respect to time .
Using modified basis function (8) and Table 1 in (12), the value of at different knots can be written as
Using (11) in coupled system (10) and imposing the boundary conditions (3) at the boundary points, we have where and are and , respectively.
Now using (13) in (14) and (8) and Table 1 in (15), we get the following system of equations:
The systems (16) represent a system of first order differential equations and can be written as where
Once the vector has been determined at a specific time level, using (7), we can compute the approximate solution at the required knots. So first we solve system (17) for vector , by using Thomas algorithm only once at each time level . Then the obtained system with the system (18) will give first order ordinary differential equations and finally first order ordinary differential equations have been solved by SSPRK54 [20] scheme and consequently the approximate solution is computed.
4. Computation of Initial Vector
To find the solution at specific time level , we need the initial vectors and .
Using initial conditions (2), we have the following.
4.1. Initial Vector
Consider the following: System (20) is a tridiagonal system of equations, which can be written as where is tridiagonal matrix. Using Thomas algorithm the solution of (20) can be easily found.
4.2. Initial Vector
Using the initial condition we have which gives the initial vector .
5. Numerical Experiments
In this section, we consider four numerical examples to validate the proposed scheme. The accuracy of the scheme is verified by calculating , , and root mean square errors and results are also compared with some published work.
, , and RMS error norms are given by the following formulae:
Example 1. In this example the numerical solutions of (1) are obtained in the computational domain with the initial conditions The exact solution [6, 15, 16] is given as The boundary conditions (3) are obtained from the exact solution.
Case I (when ). First we solve the above example in the computational domain with and for space step sizes . In Table 2, we report the and errors and compare them with those given in Dehghan and Shokri [13]. We see that our results are in good agreement with [13], when we take . For our results are better than the result of [13] in terms of error. In Table 3, absolute errors, are reported at , with . A graph comparing the exact and numerical solutions at with and is depicted in Figure 1.


Case II. In the domain , the solutions of Example 1 are obtained with and . In Table 4, we report the and RMS errors at different time levels and results are compared with those of LiMin and ZongMin [16]. We noticed that our results are in good agreement with [16] in terms of error and in terms of RMS error our results are better than [16]. Figure 2 depicts the comparison of exact and numerical solution at .

Example 2. In this example we consider (1) in the computational domain with the initial conditions
and the exact solution [6, 15] is given by
where is the velocity of solitary wave and .
The boundary conditions (3) can be obtained from the exact solution.
In Table 5, we report and errors, for different values of with and . We also compare our results with those of Dehghan and Shokri [13]. From Table 5, it is clear that our scheme and that of [13] have approximately similar errors, for . For our results are better than [13] in terms of errors and approximately similar in terms of error norm. We also compute error norm for different values of with at and calculate order of convergence, which is shown in Table 7. The absolute errors, for , are also reported in Table 6. Figure 3 depicts the graph between exact and numerical solutions at .


