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In this research article, a nonlinear time–space fractional order (4+1)-dim Fokas wave equation that is crucial for examining the corporal marvels of waves on and inside the surface of water is examined. For this purpose, a well-known analytical method is utilized, namely, the Sardar sub-equation (SSE) method along with a truncated M-fractional derivative. As a result, many new families of solitary wave solutions, such as kink-type solitons, singular and periodic solitons, dark and bright solitons, are established. By using the SSE method, the outcomes are portrayed in 3-dim, 2-dim, and contour plots for distinct parametric values. The attained hyperbolic and trigonometric function-type results demonstrate the capability of recognizing the exact solutions of the other nonlinear evolution equations through the executed technique.

The nonlinear conformable time-fractional modified Camassa-Holm (MCH) equation plays an important role in physics. It is an interesting model to define change waves with weak nonlinearity. The aim of this study is to present the new exact solutions of conformable time-fractional MCH equation. For this purpose, an effective method which is the Improved Bernoulli Sub-Equation Function Method (IBSEFM) has been used. The 2D and 3D graphs and contour surfaces acquired from the values of the solutions are plotted by the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional partial differential equations arising in mathematical physics. [ABSTRACT FROM AUTHOR]

We consider the quadratic differential pencil l(y) = y" + (λ² - 2λp(x) - q(x))y with the periodic boundary conditions y(π) - y(0) = 0, y'(π) - y'(0) = 0 where p(x) ∈ C(2) [0, π], q(x) ∈ C(1)[0, π] are complex-valued functions. We compute new accurate asymptotic expressions of the fundamental solutions of the quadratic differential pencil. We obtain the asymptotic formulas for eigenvalues and eigenfunctions of the differential pencil problem. [ABSTRACT FROM AUTHOR]

In this study, we implement the Improved Bernoulli Sub-Equation Function Method (IBSEFM) to construct new travelling wave solutions of a nonlinear conformable time fractional partial differential equation arising in nonlinear optics. We successfully obtain the solutions and choosing some suitable values of parameters, we acquire the numerical simulation of all the obtained solutions. All the computations and graphics in this article are performed by the aid of mathematics software. [ABSTRACT FROM AUTHOR]

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry. [ABSTRACT FROM AUTHOR]