An application of the method of simplest equation for real solutions for advection-diffusion interaction
R. Nikolov, I. Jordanov
, E. Nikolova
, V. Boutchaktchiev
Abstract: Objectivite: The objective of this research is to derive a novel solution for a model system described by the advection-diffusion equation using a specific instance of the Simple Equations Method (SEsM), particularly the Modified Method of Simplest Equation and its extended versions. The study aims to investigate the propagation of nonlinear waves in tissue biomechanics and analyze their features through numerical simulations. Materials and Methods: This study employs the Modified Method of Simplest Equation, a recent advancement in the Simple Equations Method (SEsM), to derive solutions for the advection-diffusion equation. Numerical simulations are conducted to illustrate and analyze the characteristics of traveling wave solutions. Mathematical modeling focuses on advection (transport driven by bulk movement) and diffusion (transport due to concentration gradients) within the context of tissue biorheology. Results: The study derives a new traveling wave solution for the model system, showcasing the propagation of nonlinear waves within the tissue environment. Numerical simulations highlight the spatial and temporal behavior of these waves, demonstrating the advection and diffusion processes in the model. Discussion: The findings provide insights into the dynamics of nonlinear wave propagation in biological tissues, emphasizing the role of the advection-diffusion equation in modeling transport phenomena. This approach enhances understanding of how substances such as nutrients, oxygen, and signaling molecules navigate through the extracellular matrix, facilitating cellular communication. Conclusion: The study analyzed a new traveling wave solution for the advection-diffusion equation using the Simple Equations Method (SEsM). The results underline the importance of nonlinear PDEs in modeling tissue biomechanics and offer a valuable framework for studying the transport of substances in biological systems.
Series on Biomechanics, Vol.38, No.4(2024), 151-156
DOI: 10.7546/SB.20.04.2024
Keywords: advection-diffusion model; analytical solution; Nonlinear dynamics; simple equations method (SEsM)
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| Date published: 2024-12-11
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