Subdomain finite element method with quartic B-splines for the modified equal width wave equation

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

“…The present error values are nearly 100 times smaller than earlier applications. The present invariant values do not change during the simulation. Application For the comparison with the other applications [8–17] the same parameters and time increment Δ t = 0.05 are used. Present results are compared with 11 different methods and given in Table 3.…”

“…For the last application of the single solitary wave, the same parameters with earlier works [11,[13][14][15] and small time increment Δt = 0.01 have been used. Numerical results are given with comparison of earlier works in Table 12.…”

“…Because of the importance of the MEW equation in nonlinear media, many researchers have tried to obtain the numerical solutions of the equation. For example, Evans and Raslan used collocation method [8], Esen used lumped Galerkin method [9], Çelikkaya used operator splitting method [10], Esen and Kutluay used finite difference method [11], Essa used multigrid method [12], Karakoç et al used collocation method based on different linearization techniques [13], Geyikli and Karakoç used subdomain method [14], Karakoç and Geyikli used sextic B-spline based subdomain method [15], Geyikli and Karakoç used Petrov-Galerkin method [16], Karakoç and Geyikli used lumped Galerkin method [17], Raslan et al used finite difference method [18], Dereli used collocation method via radial basis functions [19], Zaki used Petrov-Galerkin method [20], Saka used collocation method based on quintic B-splines [21].…”

“…Application 1.6. Single solitary wave: C = 0.5, = 1, 0 ≤ x ≤ 80, Δt = 0.05 For the comparison with other applications[11,[13][14][15] the same parameters and time increment Δt = 0.01 are used. Comparison of the present results with other applications are given inTable 7.…”

A numerical scheme for solutions of a class of nonlinear differential equations

“…Most of these equations do not have analytical solutions, but numerical techniques may be used to obtain approximate solutions. For example, various nonlinear differential equations have been solved using the Taylor matrix method [9], the closed-form method [10], Chebyshev polynomial approximations [11], the variational iteration method [12], the subdomain finite element method [13], the differential quadrature method [14,15], the variational iteration method [16,17], He's variational iteration method [18], the cubic B-spline scaling functions and Chebyshev cardinal functions [19], the homotopy perturbation method [20,21], the variation of parameters method [22], the Adomian decomposition method [23,24], the quintic B-spline differential quadrature method [25], the variational iteration method, the power series method [26], the Adomian decomposition method [27], the Pade series method [28], the Legendre polynomial function approximation [29], the Taylor method [30], the Chebyshev series method [31] and the modified variational iteration method [32]. Recently, the Taylor, Chebyshev, Legendre, Bernstein and Bessel matrixcollocation methods have been used [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] to solve some types of differential, integ...…”