article · Frontiers in Physics
The damped Burger equation describes the behaviour of one-dimensional nonlinear shock waves under damping effects, which are relevant in fluid dynamics and plasma physics. This research evaluates approximate solutions for the time-fractional version of this equation using the Caputo operator alongside two analytical techniques: the Aboodh residual power series method and the Aboodh transform iteration method. Both approaches provide systematic, accurate, and flexible means to solve complex nonlinear fractional evolution equations. Derivations of approximate solutions are verified through numerical error tables and two- and three-dimensional graphical analyses. Comparisons against exact solutions in the integer case confirm the precision of both computational schemes, while investigations into the fractional parameter illustrate its specific impact on the physical wave dynamics.
Nonlinear shock waves occur widely in natural and industrial fluid flows and plasmas. Solving the mathematical equations that describe these waves helps scientists model complex physical systems with greater precision. Demonstrating robust computational techniques allows researchers to analyse fractional differential equations more efficiently, improving simulations where damping and memory effects influence wave behaviour over time.
This work represents early-stage foundational research focused on computational mathematics. Potential users include academic researchers and simulation software developers working in fluid dynamics or plasma physics. Because the study focuses on theoretical method demonstration rather than an applied engineering tool, real-world commercialisation is distant and would require integration into commercial engineering simulation packages.
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Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.
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DOI: 10.3389/fphy.2024.1374481
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