Numerical solutions of the MRLW equation using moving least square collocation method

Abstract

Abstract. In this paper, the Modi…ed Regularized Long Wave (MRLW) equation is solved by using moving least square collocation (MLSC) method. To show the accuracy of the used method several numerical test examples are given. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution is known L 2 , L1 error norms are calculated. Also mass, energy and momentum invariants are calculated for every test problem. Obtained numerical results are compared with some earlier works. According to the obtained results, the method is very e¢ cient and reliable.

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

“…To carry out the spatial integration of Eq (6), we split the spatial domain into uniformly finite elements at the knots To construct the trigonometric quintic B-spline functions, we need 10 more knots outside the spatial domain as and . Then, trigonometric quintic B-splines at these knots are defined as other ise (7) where and…”

“…From Figure , it is observed that the solitary wave propagates to the right keeping its original shape. A comparison of the results obtained by the proposed algorithm with the some existing techniques given in [2,3,7,11,12,13,14,15] is provided in Table . Comparison verifies that the suggested algorithm gives much better results than the other techniques given in Table . The conservation invariants, the temporal rate of convergence and the error norm are given in Table 2. It can be noticed from Table that for the fixed space step, when the temporal step size is decreased from to , the temporal order of convergence is almost three and the calculated invariants are in almost good agreement with their theoretical values.…”

“…It can be noticed from Table that for the fixed space step, when the temporal step size is decreased from to , the temporal order of convergence is almost three and the calculated invariants are in almost good agreement with their theoretical values. The plot of absolute error with , is exhibited in 4.84 [3] 1.65 [7] 0.80 [11] 1.08 [12] 0.85 [13] 9.06 [14] 5.44 [15] 1.25 Exact…”

“…Therefore, several computational techniques have been proposed to get the approximate solutions of the MRLW equation. Some of those methods are finite difference technique [1,2,3], the homotopy perturbation approach [4], the second-order Fourier pseudospectral method [5], the new approach based on the homotopy analysis [6], the moving least square collocation approach [7], the meshless method [8], Petrov-Galerkin method [9], subdomain technique with quartic B-spline functions [10], Galerkin method based on various B-spline functions [11,12], collocation approach with various Bsplines [13,14,15,16,17,18]. The main objective of the present work is to develop a numerical algorithm to obtain highly accurate numerical solutions of the MRLW equation.…”