Higher-order modulation theory for resonant flow over topography

2017 Author(s). The flow of a fluid over isolated topography in the long wavelength, weakly nonlinear limit is considered. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included so that the flow is governed by a forced extended Korteweg-de Vries equation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores

Higher-order modulation theory for resonant flow

The flow of a fluid over topography in the long wavelength, weakly nonlinear limit is considered, for both isolated obstacles and steps or jumps. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow, so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included, so that the flow is governed by a forced extended Korteweg-de Vries equation. For the isolated obstacle, modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores. They are also compared with numerical solutions of the forced extended Benjamin-Bona-Mahony equation, which is asymptotically equivalent to the forced extended Korteweg-de Vries equation, but is numerically stable for higher amplitude waves. The usefulness of uniform soliton theory is also considered, for waves generated by an obstacle. It is based on the conservation laws of the extended Korteweg-de Vries equations for mass and energy and assumes that the upstream wavetrains is composed of solitary waves. We compare the solutions with theoretical and numerical solutions of the forced extended Korteweg-de Vries equation and the forced extended Benjamin-Bona- Mahony equation, to fully assess this approximation method for upstream solitary wave amplitude and wave speed. The flow of a fluid over a step or jump is also examined, and is a variation on the problem of flow over an isolated obstacle. Higher-order modulation theory solutions, based on the extended Korteweg-de Vries equation, for the undular bores generated upstream and downstream of the forcing are found. It is shown that an upstream propagating undular bore is generated by a positive step and formed by an elevation upstream of the step, and a downstream propagating undular bore is generated by a negative step and formed by a depression downstream of the step. An excellent comparison is obtained between the analytical and numerical solutions

Nonlinear Piecewise Caputo Fractional Pantograph System with Respect to Another Function

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel psi-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings

The Analytical Stochastic Solutions for the Stochastic Potential Yu–Toda–Sasa–Fukuyama Equation with Conformable Derivative Using Different Methods

We consider in this study the (3+1)-dimensional stochastic potential Yu–Toda–Sasa–Fukuyama with conformable derivative (SPYTSFE-CD) forced by white noise. For different kind of solutions of SPYTSFE-CD, including hyperbolic, rational, trigonometric and function, we use He’s semi-inverse and improved (G′/G)-expansion methods. Because it investigates solitons and nonlinear waves in dispersive media, plasma physics and fluid dynamics, the potential Yu–Toda–Sasa–Fukuyama theory may explain many intriguing scientific phenomena. We provide numerous 2D and 3D figures to address how the white noise destroys the pattern formation of the solutions and stabilizes the solutions of SPYTSFE-CD

Rotation impact on the radial vibrations of frequency equation of waves in a magnetized poroelastic medium

This research delves into the propagation of radial free harmonic waves in a poroelastic cylinder, conceptualized as a magnetically and rotationaly influenced hollow structure. The primary aim is to elucidate the magnetic field's and rotation impact on the vibrational behavior of such systems. The investigative method encompasses the resolution of motion equations, formulated as partial differential equations, through the application of Lame's potential theory. This analytical process is augmented by the implementation of fitting boundary conditions, culminating in the derivation of a comprehensive expression for the complex dispersion equation, predicated on the premise that the wavenumber embodies a complex entity. The precision of the model is corroborated through a comparative analysis with established literature, underpinned by an exploration of diverse scenarios. The research employed MATLAB for both numerical and graphical assessments, focusing on the dispersion and displacement attributes. Dispersion relations within the poroelastic medium were computed, considering varied magnitudes of magnetic field intensity, rotation and angular velocities. The outcomes are articulated through complex-valued dispersion relations, transcendental formulations, and numerical resolutions employing MATLAB's bisection technique. These insights hold substantial significance for the theoretical advancement in orthopedic research, particularly concerning cylindrical poroelastic media. This study deduces that the radial vibrational patterns and the corresponding frequency equation within a poroelastic medium are profoundly modified by the magnetic field's interference and rotation. This study formulate a novel governing equation for a poroelastic medium, highlighting the significance of radial vibrations and investigating the impact of magnetic field and rotation

Statistical and computational analysis for corruption and poverty model using Caputo-type fractional differential equations

Since there is a clear correlation between poverty and corruption, mathematicians have been actively researching the concept of poverty and corruption in order to develop the optimal strategy of corruption control. This work aims to develop a mathematical model for the dynamics of poverty and corruption. First, we study and analyze the indicators of corruption and poverty rates by applying the linear model along with the Eviews program during the study period. Then, we present a prediction of poverty rates for 2023 and 2024 using the results of the standard problem-free model. Next, we formulate the model in the frame of Caputo fractional derivatives. Fundamental properties, including equilibrium points, basic reproduction number, and positive solutions of the considered model are obtained using nonlinear analysis. Sufficient conditions for the existence and uniqueness of solutions are studied via using fixed point theory. Numerical analysis is performed by using modified Euler method. Moreover, results about Ulam-Hyers stability are also presented. The aforementioned results are presented graphically. In addition, a comparison with real data and simulated results is also given. Finally, we conclude the work by providing a brief conclusion