This paper focuses on obtaining the traveling wave solutions of the nonlinear Gilson-Pickering equation (GPE), which describes the prorogation of waves in crystal lattice theory and plasma physics. The solution of the GPE is approximated via the finite difference technique and the localized meshless radial basis function generated finite difference. The association of the technique results in a meshless approach that does not require linearizing the nonlinear terms. At the first step, the PDE is converted to a system of nonlinear ODEs with the help of the radial kernels. In the second step, a high-order ODE solver is adopted to discretize the nonlinear ODE system. The global collocation techniques pose a considerable computationl burden due to the calculation of the dense algebraic system. The proposed method approximates differential operators over the local support domain, leading to sparse differentiation matrices and decreasing the computational burden. Numerical results and comparisons are provided to confirm the efficiency and accuracy of the method.

This study looks at the mathematical model of internal atmospheric waves, often known as gravity waves, occurring inside a fluid rather than on the surface. Under the shallow-fluid assumption, internal atmospheric waves may be described by a nonlinear partial differential equation system. The shallow flow model's primary concept is that the waves are spread out across a large horizontal area before rising vertically. The Fractional Reduced Differential Transform Method(FRDTM) is applied to provide approximate solutions for any given model. This aids in the modelling of the global atmosphere, which has applications in weather and climate forecasting. For the integer-order value (α=1), the FRDTM solution is compared to the precise solution, EADM, and HAM to assess the correctness and efficacy of the proposed technique.

Highlights

● Under the shallow fluid assumption, the mathematical model of internal atmospheric waves is considered.

● In this model, waves are spread out across a large horizontal area before rising vertically.

● The fractional reduced differential transform method (FRDTM) is applied to find the approximate solution of time-fractional non-linear system of PDEs.

● The fractional solution provides a pragmatic look into the physical behaviour of internal waves, which can be helpful to refine climate and weather models.

● The obtained results have been compared with the numerical solution, EADM, and HAM for integer order to check the correctness and efficacy of the proposed technique.

The primary objective of this research problem is to analyze the Rayleigh wave propagation in homogeneous isotropic half space with mass diffusion in Three Phase Lag (TPL) thermoelasticity at two temperature. The governing equations of thermodiffusive elastic half space have been solved using the normal mode analysis in order to obtain the Rayleigh wave frequency equation at relevant boundary conditions. The variation of various parameters like non-dimensional speed, attenuation coefficient, penetration depth and specific loss corresponding to thermodiffusion parameter, relaxation time, wave number and frequency has been obtained. The effect of these parameters on Rayleigh wave propagation in thermoelastic half space are graphically demonstrated and variations of all these parameters have been compared within Lord-Shulman (L-S), Green-Nagdhi (GN-III) and Three Phase Lag (TPL) theory of thermoelasticity.

Highlights

● Rayleigh wave propagation in homogeneous isotropic half space with mass diffusion in Three Phase Lag thermoelasticity at two temperature.

● Normal mode analysis is used to obtain the Rayleigh wave frequency equation at relevant boundary conditions.

● Relevant cobalt material parameter has been considered to demonstrate the variation of various parameters of Rayleigh wave propagation.

● Variation of non-dimensional speed, attenuation coefficient, penetration depth and specific loss has been demonstrated graphically.

● Variation of parameters have been compared within Lord-Shulman (L-S), Green-Nagdhi (GN-III) and Three phase lag theory of thermoelasticity.

In this paper, the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques. The quadratic-case and cubic-case are investigated for the proposed model. Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation. Some graphical analysis is presented to support the findings of the paper. Finally, we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.

Highlights

● The conformable-time-fractional generalized dissipative Kawahara equation is presented and solved by using the tanh-coth-expansion and sine-cosine-function methods.

● The quadratic-case and cubic-case are investigated for the proposed model.

● The new solutions highlighted the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation.

● Some graphical analysis is presented to support the findings of the current work.

The Cahn-Hilliard system was proposed to the first time by Chan and Hilliard in 1958. This model (or system of equations) has intrinsic participation energy and materials sciences and depicts significant characteristics of two phase systems relating to the procedures of phase separation when the temperature is constant. For instance, it can be noticed when a binary alloy (“Aluminum + Zinc” or “Iron + Chromium”) is cooled down adequately. In this case, partially or totally nucleation (nucleation means the appearance of nuclides in the material) is observed: the homogeneous material in the initial state gradually turns into inhomogeneous, giving rise to a very accurate dispersive microstructure. Next, when the time scale is slower the microstructure becomes coarse. In this work, to the first time, the unified method is presented to investigate some physical interpretations for the solutions of the Cahn-Hilliard system when its coefficients varying with time, and to show how phase separation of one or two components and their concentrations occurs dynamically in the system. Finally, 2D and 3D plots are introduced to add more comprehensive study which help to understand the physical phenomena of this model. The technique applied in this analysis is powerful and efficient, as evidenced by the computational work and results. This technique can also solve a large number of higher-order evolution equations.

Highlights

● Different wave structures for abundant solutions to the Chan–Hilliard system are investigated.

● Performance was done using the strategy of the unified method.

● Physical explanations are discussed for the obtained solutions.

In this article, we suggest a new form of modified Kudryashov's method (NMK) to study the Dual-mode Sawada Kotera model. We know very well that the more the solutions depend on many constants, the easier it is to study the model better by observing the change in the constants and what their impact is on the solutions. From this point of view, we developed the modified Kudryashov method and put it in a general form that contains more than one controllable constant. We have studied the model in this way and presented figures showing the correctness of what we hoped to reach from the proposed method. In addition to the results we reached, they were not sufficient, so we presented an extensive numerical study of this model using the finite differences method. We also came up with the local truncation error for the difference scheme is h6k2(1+k2). In addition, the analytical solutions we reached were compared with the numerical solutions, and we presented many forms that show that the results we reached are a clear contribution to this field.

Highlights

● We suggest a new form of modified Kudryashov's method (NKM) to study the Dual-mode Sawada Kotera model.

● We know very well that the more the solutions depend on many constants, the easier it is to study the model better by observing the change in the constants and what their impact is on the solutions.

● From this point of view, we developed the modified Kudryashov method and put it in a general form that contains more than one controllable constant.

● We have studied the model in this way and presented figures showing the correctness of what we hoped to reach from the proposed method.

● In addition to the results we reached, they were not sufficient, so we presented an extensive numerical study of this model using the finite differences method.

● We also came up with the local truncation error for the difference scheme is $h^6{k}^2(1+k^2)$ .

● In addition, the analytical solutions we reached were compared with the numerical solutions, and we presented many forms that show that the results we reached are a clear contribution to this field.

This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived. We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system. Furthermore, some 2D and 3D graphical illustrations of the obtained solutions have been presented. The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted. The derived solutions may contribute to comprehending wave propagation in shallow water. So, the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.

Highlights

● Simplified modified Camassa-Holm equation with conformable and M- fractional derivative order is investigated.

● The novel solutions of considered equations are obtained analytically.

● The solutions of the conformable and $M-$ truncated model are graphically compared in the figures for different values of $\alpha$ and $\beta$ that are in order of the derivative operator.

● The considered method suggests trigonometric functions producing dark, singular, and trigonometric solitons etc.