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.

It has been increasingly obvious in recent decades that fractional calculus (FC) plays a key role in many disciplines of applied sciences. Fractional partial differential equations (FPDEs) accurately model various natural physical phenomena and many engineering problems. For this reason, the analytical and numerical solutions to these issues are seriously considered, and different approaches and techniques have been presented to address them. In this work, the FC is applied to solve and analyze the time-fractional heat transfer equation as well as the nonlinear fractional porous media equation with cubic nonlinearity. The idea of solving these equations is based on the combination of the Yang transformation (YT), the homotopy perturbation method (HPM), and the Adomian decomposition method (ADM). These combinations give rise to two novel methodologies, known as the homotopy perturbation transform method (HPTM) and the Yang tranform decomposition method (YTDM). The obtained results show the significance of the accuracy of the suggested approaches. Solutions in various fractional orders are found and discussed. It is noted that solutions at various fractional orders lead to an integer-order solution. The application of the current methodologies to other nonlinear fractional issues in other branches of applied science is supported by their straightforward and efficient process. In addition, the proposed solution methods can help many plasma physics researchers in interpreting the theoretical and practical results.

Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with delayed neutrons in cylindrical reactors of fractional order. The method of solving used integrates the technique of residual power series (RPS) with the Laplace transform (LT) method. Anomalous neutron behavior is explained by examining the non-Gaussian scenario with various fractional parameters α. The LRPSM Laplace transform and residual power series method employed in this approach eliminates the complex difficulties. This simplicity makes the method particularly coherent with different fractional calculus applications. To validate the proposed method, numerical simulations are conducted with two different initial conditions representing distinct scenarios. The obtained results are presented in suitable tables and figures. It should be emphasized that this system is solved for the first time utilizing fractional calculus techniques. The outcomes are consistent with those achieved using the Adomian decomposition method.