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International Journal of Differential Equations
Volume 2012 (2012), Article ID 587208, 15 pages
Numerical Solution of the Modified Equal Width Wave Equation
Department of Mathematics, Faculty of Education, İnönü University, 44280 Malatya, Turkey
Received 18 May 2011; Accepted 30 September 2011
Academic Editor: Sabri Arik
Copyright © 2012 Seydi Battal Gazi Karakoç and Turabi Geyikli. 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.
Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws and error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.
The modified equal width wave equation (MEW) based upon the equal width wave (EW) equation [1, 2] which was suggested by Morrison et al.  is used as a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. This equation is related with the modified regularized long wave (MRLW) equation  and modified Korteweg-de Vries (MKdV) equation . All the modified equations are nonlinear wave equations with cubic nonlinearities and all of them have solitary wave solutions, which are wave packets or pulses. These waves propagate in non-linear media by keeping wave forms and velocity even after interaction occurs. Few analytical solutions of the MEW equation are known. Thus numerical solutions of the MEW equation can be important and comparison between analytic solution can be made. Geyikli and Battal Gazi Karakoç [6, 7] solved the MEW equation by a collocation method using septic B-spline finite elements and using a Petrov-Galerkin finite element method with weight functions quadratic and element shape functions which are cubic B-splines. Esen applied a lumped Galerkin method based on quadratic B-spline finite elements which have been used for solving the EW and MEW equation [8, 9]. Saka proposed algorithms for the numerical solution of the MEW equation using quintic B-spline collocation method . Zaki considered the solitary wave interactions for the MEW equation by collocation method using quintic B-spline finite elements  and obtained the numerical solution of the EW equation by using least-squares method . Wazwaz investigated the MEW equation and two of its variants by the tanh and the sine-cosine methods . A solution based on a collocation method incorporated cubic B-splines is investigated by and Saka and Dağ . Variational iteration method is introduced to solve the MEW equation by Lu . Evans and Raslan  studied the generalized EW equation by using collocation method based on quadratic B-splines to obtain the numerical solutions of a single solitary waves and the birth of solitons. Hamdi et al.  derived exact solitary wave solutions of the generalized EW equation using Maple software. Esen and Kutluay studied a linearized implicit finite difference method in solving the MEW equation . In the present work we solve the MEW equation numerically by a lumped Galerkin method using cubic B-spline finite elements. The accuracy of the proposed method is demonstrated by two test problems: the motion of a single solitary wave and the interaction of two solitary waves. A linear stability analysis based on a Fourier method shows that the numerical scheme is unconditionally stable.
2. Cubic B-Spline Lumped Galerkin Method
The modified equal width wave (MEW) equation considered here has the normalized form  with the physical boundary conditions as , where is time, is the space coordinate, is a positive parameter, and is wave amplitude. In this study, boundary conditions are chosen from and the initial condition where is a localized disturbance inside interval . The interval is partitioned into uniformly sized finite elements of length by the knots such that and . The cubic B-splines , , at the knots are defined over the interval by  The set of functions forms a basis for functions defined over . The approximate solution to the exact solution is given by where are time-dependent parameters to be determined from the boundary and weighted residual conditions. Each cubic B-spline covers 4 elements so that each element is covered by 4 splines. In each element, using the following local coordinate transformation cubic B-spline shape functions in terms of over the element can be defined as All splines apart from , and are zero over the element . Variation of the function over element is approximated by where act as element parameters and B-splines as element shape functions. Using trial function (2.5) and cubic splines (2.4), the values of at the knots are determined in terms of the element parameters by where the symbols and denote first and second differentiation with respect to , respectively. The splines and its two principle derivatives vanish outside the interval . Use Galerkin’s method with weight function to obtain the weak form of (2.1) which is For a single element using transformation (2.6) into the (2.10) we obtain where is taken to be a constant over an element to simplify the integral. Integrating (2.11) by parts leads to where and . Taking the weight function with cubic B-spline shape functions given by (2.7) and substituting approximation (2.8) in integral equation (2.12) with some manipulation, we obtain the element contributions in the form In matrix notation this equation becomes where are the element parameters and the dot denotes differentiation with respect to . The element matrices , and are given by the following integrals: where the suffices take only the values for the typical element . A lumped value for is found from as By assembling all contributions from all elements, (2.14) leads to the following matrix equation: where is a global element parameter. The matrices , and are septadiagonal and row of each has the following form: where Replacing the time derivative of the parameter by usual forward finite difference approximation and parameter by the Crank-Nicolson formulation into equation (2.17), gives the septadiagonal matrix system where is time step. Applying the boundary conditions (2.2) to the system (2.21) we obtain an septadiagonal matrix system. This system is efficiently solved with a variant of the Thomas algorithm, but an inner iteration is also needed at each time step to cope with the non-linear term. A typical member of the matrix system (2.21) may be written in terms of the nodal parameters and as where which all depend on . The initial vector of parameter must be determined to iterate system (2.21). To do this, the approximation is rewritten over the interval at time as follows: where the parameters will be determined. are required to satisfy the following relations at the mesh points : The above conditions lead to a tridiagonal matrix system of the form which can be solved using a variant of the Thomas algorithm.
2.1. Stability Analysis
The stability analysis is based on the Von Neumann theory in which the growth factor of the error in a typical mode of amplitude , where is the mode number and the element size, is determined from a linearization of the numerical scheme. To apply the stability analysis, the MEW equation needs to be linearized by assuming that the quantity in the non-linear term is locally constant. Substituting the Fourier mode (2.27) into (2.22) gives the growth factor of the form where The modulus of is 1, therefore the linearized scheme is unconditionally stable.
3. Numerical Examples and Results
In this part, we consider the following two test problems: the motion of a single solitary wave and interaction of two solitary waves. All computations are executed on a Pentium 4 PC in the Fortran code using double precision arithmetic. For the MEW equation, it is important to discuss the following three invariant conditions given in , which, respectively, correspond to conversation of mass, momentum, and energy. The accuracy of the method is measured by both the error norm and the error norm to show how well the scheme predicts the position and amplitude of the solution as the simulation proceeds. The variable and its first derivative appearing in (3.1) can be computed from (2.9), respectively.
3.1. The Motion of Single Solitary Wave
For this problem we consider (2.1) with the boundary condition as and the initial condition An exact solution of this problem is given by  which represents the motion of a single solitary wave with amplitude , where the wave velocity and . For this problem the analytical values of the invariants are  For the numerical simulation of the motion of a single solitary wave, we choose the parameters , , , , through the interval . The analytical values for the invariants are , , . The invariants and change from their initial values by less than and respectively, during the time of running, whereas the changes of invariant approach to zero throughout. The computations are done until time , and we find , error norms and numerical invariants , , at various times. Results are documented in Table 1. One may also compare our results with those in the other studies [9, 16, 18]. According to both , error norms, agreement between numerical values and exact solution appears very satisfactorily through illustrations of three invariants and norms. Figure 1 shows that the proposed method performs the motion of propagation of a solitary wave satisfactorily, which moved to the right at a constant speed and preserved its amplitude and shape with increasing time as expected. Amplitude is 0.25 at which is located at , while it is 0.249900 at which is located at . The absolute difference in amplitudes at times and is so that there is a little change between amplitudes. The convergence rates for the numerical method in space sizes and time steps can be calculated by following formula , respectively, Table 2 displays the convergence rates for different values of space size and a fixed value of the time step . We have clearly seen that the convergence rates when is fixed are not good these for size. In addition, the time rate of the convergence for the numerical method is computed with various time step and fixed space step in Table 3. It can clearly be seen that the present method provides remarkable reductions in convergence rates for the smaller time.
3.2. Interaction of Two Solitary Waves
In this section, we consider (2.1) with boundary conditions as and the initial condition where .
Firstly we studied the interaction of two positive solitary waves with the parameters , , , , , , through the interval which used the earlier papers [9–11]. The analytic invariants are  , , and changes in , and are less than , , and percent, respectively, as can be seen in Table 4. The experiment was run from to to allow the interaction to take place. This condition is propagated to the right with velocities dependent upon their magnitudes and a stage is reached where the larger wave has passed through the smaller solitary wave and has emerged with their original shapes. Figure 2 shows the interactions of two positive solitary waves. Interaction started at about time , overlapping processes occurred between times and and, waves started to resume their original shapes after time . It can be seen that, at , the wave with larger amplitude is on the left of the second wave with smaller amplitude. The larger wave catches up with the smaller one as time increases. At , the amplitude of larger waves is 0.999581 at the point whereas the amplitude of the smaller one is 0.510464 at the point . It is found that the absolute difference in amplitude is for the smaller wave and for the larger wave for this algorithm. Finally, we have studied the interaction of two solitary waves with the following parameters, , , , , together with time step and space step in the range . The experiment was run from to to allow the interaction to take place. Figure 3 shows the development of the solitary wave interaction. As is seen from Figure 3, at , a wave with the negative amplitude is on the left of another wave with the positive amplitude. The larger wave with the negative amplitude catches up with the smaller one with the positive amplitude as the time increases. At , the amplitude of the smaller wave is 0.972910 at the point , whereas the amplitude of the larger one is −2.016990 at the point . It is found that the absolute difference in amplitudes is for the smaller wave and for the larger wave. The analytical invariants can be found as and changes in , and are less than , , and , percent, respectively. Table 4 lists the values of the invariants of the two solitary waves with amplitudes , and . It can be seen that the values obtained for the invariants are satisfactorily constant during the computer run. We have also compared the computed values of the invariants of the two solitary waves with results from  in Table 5.
In this paper, the cubic B-spline lumped Galerkin method has been successfully applied to obtain the numerical solution of the modified equal width wave equation. The efficiency of the method was tested on two test problems of wave propagation: the motion of a single solitary wave and the interaction of two solitary waves, and its accuracy was shown by calculating error norms and . It is clear that the error norms are adequately small and the invariants are satisfactorily constant in all computer run. The method can be also efficiently applied for solving a large number of physically important non-linear problems.
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