Younas Usman, Muhammad Jan, Ali Qasim, Sediqmal Mirwais, Kedzia Krzysztof, Jan Ahmed Z
Department of Mathematics, Shanghai University, No. 99 Shangda Road, Shanghai, 200444, China.
Newtouch Center for Mathematics of Shanghai University, Shanghai, 200444, China.
Sci Rep. 2024 Oct 30;14(1):26080. doi: 10.1038/s41598-024-75494-y.
This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear -fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of -fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.
本研究工作聚焦于研究超声波的传播，超声波在材料内部传播分子或粒子的机械振动。超声成像在医学领域得到广泛应用且根基深厚。构成该领域许多不同用途基础的关键技术包括换能器、造影剂、脉冲压缩、波束整形、组织谐波成像、测量血流和组织运动的技术以及三维成像。三阶非线性分数阶韦斯特维尔特模型已被用作成像过程中的控制模型，以确保不同的波结构。提取了不同类型的精确解，包括混合孤子、暗孤子、奇异孤子、亮 - 暗孤子、亮孤子、复孤子和组合孤子。这些解是通过使用两种新引入的技术得到的，即修正的广义里卡蒂方程映射方法和修正的广义指数有理函数方法。此外，借助对数变换和不同的测试函数提取了呼吸子、团块波和其他波。所使用的方法极其有效，具有强大的计算能力，能够在这些系统中以高精度有效地求解不同的解。所使用的技术以有效、简单且足够灵活而闻名，能够将多个孤子系统整合到一个统一的框架中。此外，我们提供了二维和三维图形，解释了在分数阶导数影响下，解在各种参数值下的行为。本研究提供的结果可能会增进对特定系统非线性动态行为的理解，并证实所采用方法的有效性。我们期望我们的方法将对广泛的非线性模型及相关领域的其他问题有益。
