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Ultrasound imaging stands as a cornerstone of modern medical diagnostics, revolutionising clinical practice with its non-invasive, real-time visualisation of internal structures. Central to this technique is the propagation of ultrasound waves and their intricate interplay with biological tissues, culminating in the generation of intricate and detailed images. This study delves into the symbiotic relationship between solitons and ultrasound imaging within the framework of the Westervelt equation, a fundamental model governing ultrasound propagation. Employing the generalised Riccati equation mapping method and the generalised exponential rational function method, a diverse array of soliton solutions is elucidated, encompassing dark, kink, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Visualisation of these solutions through 3D plots, contour plots, and 2D plots at varying time intervals offers a captivating insight into their dynamic nature. We provide a comparison of these solutions through 2D plots at different parameter values, highlighting their varying impacts. Central to this study is the exploration of how these soliton solutions can be harnessed to enhance the quality and accuracy of ultrasound images in medical imaging. Through meticulous analysis of their characteristics, this research seeks to illuminate their potential applications, paving the way for a new era of precision diagnostics in healthcare. By conducting thorough mathematical analyses and numerical simulations, we seek to elucidate the complex relationship between soliton theory and ultrasound imaging, connecting the theoretical aspects of nonlinear wave phenomena with their practical applications in medical diagnostics. An intensive literature review underscores the novelty of our work.

This study investigates the chaotic behavior and stability of a nonlinear fractional-order financial system utilizing the Caputo fractional derivative. Initially, we formulate the nonlinear fractional-order financial model and establish the problem framework. Next, we prove the existence and uniqueness of solutions by applying the Banach and Schauder fixed-point theorems to the proposed system. Additionally, we analyze the Ulam–Hyers stability and discuss other significant findings related to the system’s stability. To simulate the proposed model, we develop numerical schemes based on fractional calculus, employing Lagrange polynomial interpolation. Finally, we present the numerical simulations to validate the theoretical results, highlighting the significant impact of fractional-order derivatives on the system’s behavior. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for a nonlocal singular elliptic coupled system involving N-Laplace operator. The source term of the system is expressed as the sum of two components: one with subcritical, critical, or supercritical exponential growth controlled by the Trudinger–Moser inequality, and the other being singular at the origin. We also investigate the asymptotic behavior of the solutions with respect to the parameters. Furthermore, we prove that no solutions exist for our system in dimension N = 2 . [ABSTRACT FROM AUTHOR]

This work investigates positive solutions for equations involving the anisotropic variable exponent operator, a mathematical framework critical in modeling nonstandard growth phenomena. By deriving new existence results, the study advances the understanding of nonlinear partial differential equations in anisotropic and variable exponent settings. These findings have significant implications for real-world applications, such as fluid dynamics, image processing, and material science, where variable growth conditions arise naturally. More specifically, the study employs methods from functional analysis to investigate the existence, nonexistence, and minimal solutions of certain anisotropic equations within the framework of generalized Sobolev spaces. [ABSTRACT FROM AUTHOR]

The main aim of this study is to examine the Hyers–Ulam stability of fractional derivatives in Volterra–Fredholm integro-differential equations using Caputo fractional derivatives. We explore the existence and uniqueness of solutions for the proposed integro-differential equation using Banach and Krasnoselskii's fixed-point techniques. Furthermore, we examine the Hyers–Ulam stability of the equation under the Caputo fractional derivative by deriving suitable sufficient conditions. We analyze the graphical behavior of the obtained results to demonstrate the efficiency of the analytical method, highlighting its ability to deliver accurate and precise approximate numerical solutions for fractional differential equations. Finally, numerical applications are presented to validate the stability of the proposed integro-differential equation. [ABSTRACT FROM AUTHOR]

This study explores the motivational factors behind alcohol consumption and its societal impacts. It identifies key reasons for drinking, such as improving sociability, boosting confidence, coping with challenges, participating in rituals, and seeking detoxification. The study highlights the substantial societal harm caused by the use of alcohol and drugs, with a particular emphasis on the increase in sexual violence, both within and outside families, as a result of impaired behavior. In this work, we present a newly developed mathematical model within the Caputo fractional framework, designed to reflect the lifestyle and behaviors of individuals in the alcohol model. We analyze the existence and uniqueness of solutions using the Lipschitz growth condition and fixed point theory. In addition, we explore equilibrium points, compute basic reproduction numbers, and explore the Hyers–Ulam stability of the alcohol model. Finally, we develop a numerical scheme for computational analysis of the proposed fractional model. [ABSTRACT FROM AUTHOR]

In this work, we will investigate a contact problem that arises in the interaction of two thermo-electro-viscoelastic bodies with adhesion and damage. The bilateral contact is modeled through Tresca's friction law, and the materials undergo damage as a result of elastic deformations. The existence and uniqueness of the weak solution is obtained through the derivation of a variational formulation. The proof mainly relies on the results of variable evolutionary inequalities, ordinary differential equations and fixed-point theory. [ABSTRACT FROM AUTHOR]

In this article, we consider the dynamics of a viscoelastic plate equation with internal fractional damping, a nonlinear logarithmic source, and infinite memory effects. The existence of a local weak solution is shown effectively through the framework of semigroup theory. Furthermore, we show that the blow-up in finite time of the local solution may occur under specific conditions and is demonstrated within the development of a suitable Lyapunov functional. Our result offers an insight into the challenges presented by this class of equations and their relevance to physical systems. [ABSTRACT FROM AUTHOR]

This work addresses a value problem concerning a system of high-order nonlinear equations with viscoelastic terms acting in both equations and homogeneous Dirichlet conditions. Initially, we demonstrate that the system has a weak local solution by combining fixed point theory and Galerkin's method. Under certain conditions on the variable exponents p (.) , as well as conditions related to the kernel functions η (t) ≤ μ (t) , for all t ≥ 0 , we establish that the solution with positive initial energy exhibits exponential growth. In addition, we study some numerical examples to illustrate our theoretical results and demonstrate the effectiveness of our new approach. [ABSTRACT FROM AUTHOR]