In this work, the non-integrable nonplanar (cylindrical and spherical) damped Kawahara equation (ndKE) is solved and analyzed analytically. The ansatz method is implemented for analyzing the ndKE in order to derive some high-accurate and more stable analytical approximations. Based on this method, two-different and general formulas for the analytical approximations are derived. The obtained solutions are applied for studying the distinctive features for both cylindrical and spherical dissipative dressed solitons and cnoidal waves in a complex plasma having superthermal ions. Moreover, the accuracy of the obtained approximations is numerically examined by estimating the global maximum residual error. Also, a general formula for the nonplanar dissipative dressed solitons energy is derived in detail. This formula can recover the energy of the nonplanar dissipative dressed solitons, the planar dressed solitons, the planar damped dressed solitons, and the nonplanar dressed solitons. Both the suggested method and obtained approximations can help a large sector of authors interested in studying the nonlinear and complicated phenomena in various fields of science such as the propagating of nonlinear phenomena in physics of plasmas, nonlinear optics, communications, oceans, and seas.

This investigation employs the “Tantawy Technique,” a novel, highly accurate, and fast technique, to solve and analyze the fourth-order time-fractional Cahn–Hilliard (TFCH) models. We also implement the variational iteration transform method (VITM) and the homotopy perturbation transform method (HPTM) within the Yang transform framework to analyze the fourth-order TFCH models. These iterative procedures enable the acquisition of solutions in a convergent series. We focus on two nonlinear issues in the current investigation to demonstrate the validity and accuracy of the suggested approaches. We check the derived approximations against the exact solutions for the integer cases and then find their absolute error to ensure their accuracy and stability. The present approaches show how different fractional orders affect the profile of the derived approximations. The presented issues demonstrate the precision and effectiveness of the suggested techniques in tackling strong nonlinear fractional differential equations. Implementing the suggested methodologies indicates that as the value of the fractional parameter transitions from fractional to integer order, the result approaches the exact solution for the integer case. These methods also make it straightforward to use fractal calculus in real life. The numerical analysis results demonstrate that the proposed iterative techniques are viable and user-friendly tools with high computational efficiency when used in various physical models in science and engineering.

In this paper, the modulational instability (MI) of the high-frequency electron-acoustic waves (EAWs) is reported in a non-Maxwellian plasma composed of two distinct types of electrons and stationary ions. One type of electrons is treated as a cold inertial fluid, whereas the other type is considered as inertialess species following κ-deformed Kaniadakis distribution. The fluid equations to the current model are reduced via a reductive perturbation technique to a nonlinear Schrödinger equation, which is then used to compute the MI and the growth rate of the EAWs. It is instructive to note that the deformation parameter (which develops the Kaniadakis entropy) and the hot-to-cold electron density ratio (hot electron concentration) significantly affect the conditions for MI. The modulated envelope black (dark and gray) solitons are investigated. The current results are beneficial in analyzing the spectrum of the cosmic rays, which violates manifestly the Boltzmann–Gibbs statistics. Moreover, the obtained results can be used to understand the mystery of many observations in stars where the presence of non-Maxwellian particles dominates.

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues.

The in vitro fabrication of big osteoarticular implants integrating biomaterials and cells is of tremendous interest because these tissues have a limited ability to regenerate. However, the growth of such cells in vitro is highly problematic, especially later in the culture, when the extracellular matrix has almost filled the initial porous network. Thus, the fluid flow required to properly perfuse the sample cannot be obtained by the hydraulic driving force alone. Fluid pumping is a central concern of a microfluidic system and electro-osmotic pumps (EOPs) are commonly employed for this purpose. Using electro-kinetic equations as a basis, this study analyzed the variations of a two-fluid electro-osmotic flow of viscoelastic fluid flow through a channel. The behavior of the fluid was studied through the Ellis equation. This is how the electro-osmotic pump functions, as demonstrated in the literature that it electrically drags a conducting fluid across a non-conducting fluid through interfacial dragging force along the channel. A steady-state analytical solution for the system in a conducting fluid channel was studied by undertaking an interface planner for fluids exhibiting Newtonian rheological properties. The pumping characteristics were studied in detail by using the Ellis model’s parameters. The fluid rheology was studied, which showed the viability of this technique.