This study provides analytical solutions of the fractional Richards equation for modeling water subdiffusion in porous media. Our aim is to provide some nontrivial analytical solutions which can serve as benchmark references for testing numerical methods for solving the fractional Richards equation. The derivation of the analytical solutions of the fractional Richards equation is based on the use of the Gardner model for capillary pressure to allow the linearization of the problem. The solutions are derived for the Caputo and Atangana-Baleanu-Caputo fractional derivatives using the method of separation of variables and Fourier series expansion. In our case, we used a nonlinear form of the equation through another bounded auxiliary variable. Comparisons are conducted between the analytical solutions and the corresponding numerical ones which are obtained using a numerical scheme based on the localized radial basis function (LRBF) method across various porous media having different physical characteristics.
The Whitham–Broer–Kaup (WBK) model describes tsunami type wave pattern due to involvedness of dispersion, dissipation as well as weedy nonlinear effects. As the wave evolutions rise with time, we are going to investigate time varying wave solutions of the nonlinear model. To do that this article applied modified simple equation (MSE) and enhanced modified simple equation (EMSE) schemes on the WBK model to extract time varying wave pattern. As a result, we found the time varying solitary wave solutions as kink and anti-kink, kinky-periodic, parabolic-periodic, multi-period and periodic wave for some free parameters with their numerical values carried some massive soliton wave which can produces tsunami. Some plentiful 3D figures, density plot and counter plot are sketch from the obtained solutions to understand the topic and their description are deliberated properly. By analyzing the achieved results, it is vibrant that the EMSE and MSE scheme is informal, reliable and effective to arise the time varying wave evolutions of such model applicable in water-wave mechanism as tsunami type massive wave.
The behavior of solitons in multimode fibers is investigated using the Connected Higher-Order Nonlinear Schrödinger Equations (CHNSE). A sophisticated algebraic technique is employed to analyze the CHNSE, yielding a variety of exact solutions, including bright and dark solitons, Weierstrass elliptic function solutions, periodic singular solutions, and hyperbolic and Jacobi elliptic solutions. These solutions are graphically represented using density and three-dimensional graphics to illustrate their physical characteristics. The CHNSE model provides deeper understanding of nonlinear optical processes in multimode fibers, essential for high-speed communication systems. The findings contribute to the development of new optical communication technologies.
This study proposes an analytical-numerical model to investigate wave resonance attenuation in hybrid coastal protection systems consisting of mangroves and trapezoidal breakwaters. The proposed model is adopted from the modified Linear Shallow Water Equations, which include the effect of vegetation-induced drag and the breakwater geometry. An analytical solution is derived to determine the natural period of the basin, whereas a staggered finite volume method is used to simulate wave propagation and resonance. The results show that mangroves can suppress resonance by themselves when sufficient vegetation-induced friction is provided, while the combination of mangroves and a rough trapezoidal breakwater can achieve resonance mitigation with a lower roughness requirement. Further sensitivity analyses demonstrate that wider mangrove belts and a shorter distance between the mangroves and the breakwater are more effective in attenuating waves. These results provide practical guidelines for the design of sustainable hybrid systems for coastal defence capable of alleviating resonance-induced wave amplification.
In this paper, we present an effective approach for constructing Lyapunov functions for a specific class of partial differential equations (PDEs) involving the fractional p-Laplacian operator, both with and without delay. We systematically examine special cases of classical and fractional spatial diffusion equations falling under this class. Additionally, the proposed method is applied to a compartmental epidemic model with general incidence rate as well as to a nonlinear delayed system describing the spread of disease in a population under an anomalous diffusion.
