Numerical approximation of modified Kawahara equation using Kernel smoothing method

A Regularization-Free Scheme for Recovering Large External Forces of Higher-Order Nonlinear Evolution Equations

“…For the engineering and mathematical problems of the Kawahara equation (KE), modified Kawahara equation (mKE), and sixth-order KdV equation, many direct approaches for acquiring the explicit travelling solitary wave solutions to the above-mentioned nonlinear evolution equations have been addressed, such as the algebraic method [20,21], the tanh method and the extended tanh method [22], the Adomian decomposition method [23], the Sine-Cosine method [24], the variational iteration method and the homotopy perturbation method [25,26], the dual-Petrov-Galerkin algorithm [27], the Crank-Nicolson differential quadrature algorithms [28], the variational iteration method and the Adomian decomposition method [29], the travelling wave ansatz [30], the multiplier techniques and compactness arguments [31], the general well-posedness principle [32], the optimal homotopy asymptotic method [33], the lattice Boltzmann model [34], the septic B-spline collocation method [35], the local radial basis functions method [36], the Crank-Nicolson discretization algorithm and the fifth-order quintic B-spline based differential quadrature method [37], the Kernel smoothing method and the Crank-Nicolson method [38], the complex method [39], and the generalized tanh-coth method [40]. The above-mentioned references did not mention the numerical solutions with noisy effect and did not demonstrate their real applications.…”