Search Results (4)

In this study, the inverse engineering problems of the Ostrovsky equation (OE), Kawahara equation (KE), modified Kawahara equation (mKE), and sixth-order Korteweg-de Vries (KdV) equation will be investigated numerically. An effective numerical approach to tackle these inverse Coriolis dispersion problems and the above-mentioned inverse problems are still not available. To use different boundary shape functions, we must deal with the boundary data, initial conditions, and terminal time conditions of the OE, KE, mKE, and sixth-order KdV equations. The unknown Coriolis dispersion of OE and unknown large external forces of those three equations can be retrieved through back-substitution of the solution into the OE, KE, mKE, and sixth-order KdV equations while we obtain the solution with the symmetry property by employing the boundary shape function scheme (BSFS). Five numerical experiments with noisy data are carefully validated and discussed. Full article

The manuscript solves a modified Kawahara equation (mKE) within two cases with and without a damping term by applying the Laplace homotopy perturbation method (LHPM). Since the damped mKE is non-integrable (i.e., it does not have analytic integrals) and does not have exact initial conditions, this challenge makes many numerical methods fail to solve non-integrable equations. In this article, we suggested a new modification at LHPM by setting a perturbation parameter and an embedding parameter as the damping parameter and using the initial condition for mKE as the initial condition for non-damped mKE. The results proved that this mathematical approach is an effective method for solving damped mKE. Thus, we believe that the presented method will be helpful for solving many non-integrable equations that describe phenomena in sciences, such as nonlinear symmetrical wave propagation in plasma. Full article

In this investigation, the nonplanar (spherical and cylindrical) modified fifth-order Korteweg–de Vries (nmKdV5) equation, otherwise known as the nonplanar modified Kawahara equation (nmKE), is solved using the ansatz approach. Two general formulas for the semi-analytical symmetric approximations are derived using the recommended methodology. Using the obtained approximations, the nonplanar modified Kawahara (mK) symmetric solitary waves (SWs) and cnoidal waves (CWs) are obtained. The fluid equations for the electronegative plasmas are reduced to the nmKE as a practical application for the obtained solutions. Using the obtained solutions, the characteristic features of both the cylindrical and spherical mK-SWs and -CWs are studied. All obtained solutions are compared with each other, and the maximum residual errors for these approximations are estimated. Numerous researchers that are interested in studying the complicated nonlinear phenomena in plasma physics can use the obtained approximations to interpret their experimental and observational findings. Full article

In this work, a damped modified Kawahara equation (mKE) with cubic nonlinearity and two dispersion terms including the third- and fifth-order derivatives is analyzed. We employ an effective semi-analytical method to achieve the goal set for this study. For this purpose, the ansatz method is implemented to find some approximate solutions to the damped mKE. Based on the proposed method, two different formulas for the analytical symmetric approximations are formally obtained. The derived formulas could be utilized for studying all traveling waves described by the damped mKE, such as symmetric solitary waves (SWs), shock waves, cnoidal waves, etc. Moreover, the energy of the damped dressed solitons is derived. Furthermore, the obtained approximations are used for studying the dynamics of the dissipative dressed (modified Kawahara (mK)) dust-ion acoustic (DIA) solitons in an unmagnetized collisional superthermal plasma consisting of inertia-less superthermal electrons and inertial cold ions as well as immobile negative dust grains. Numerically, the impact of the collisional parameter that arises as a result of taking the ion-neutral collisions into account and the electron spectral index on the profile of the dissipative structures are examined. Finally, the analytical and numerical approximations using the finite difference method (FDM) are compared in order to confirm the high accuracy of the obtained approximations. The achieved results contribute to explaining the mystery of several nonlinear phenomena that arise in different plasma physics, nonlinear optics, shallow water waves, oceans, and seas, and so on. Full article