article · Physics of Fluids
The time-fractional generalised Burger–Fisher equation provides a mathematical framework for modelling complex natural systems that feature memory and non-local effects. Such processes appear in fluid flow dynamics, gas dynamics, shock-wave formation, heat transfer, population growth, and diffusion transport. To better analyse these nonlinear phenomena, an iterative analytical approach has been established by merging the Laplace transform with the residual power series method. This combined technique allows researchers to approximate shock-wave solutions for both standard integer and fractional orders. Validation on two numerical test problems confirms the precision and performance of the method when compared against exact solutions for integer-order cases. Additionally, graphical and tabular assessments illustrate how changing the fractional order influences the behaviour of the resulting solutions.
Many natural and physical systems cannot be described accurately by classical calculus because past events and distant interactions influence their behaviour. Fractional differential equations capture these memory-dependent dynamics in fluid flow, heat distribution, and gas movements. By providing an efficient way to approximate shock waves in these complex equations, this computational approach aids in understanding critical physical transitions across engineering and applied sciences.
This study is early-stage theoretical and mathematical research focused on computational equation solving. It could eventually assist engineering simulation developers, aerodynamicists, and fluid dynamics researchers who model shock waves, diffusion, and heat transfer. However, because the work is demonstrated exclusively on two abstract mathematical test problems, it remains distant from commercial deployment and requires integration into applied engineering software tools before real-world practical adoption.
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The time-fractional generalized Burger–Fisher equation (TF-GBFE) has various applications across various scientific and engineering disciplines. It is used for investigating various phenomena, including the dynamics of fluid flow, gas dynamics, shock-wave formation, heat transfer, population dynamics, and diffusion transport, among other areas of research. By incorporating fractional calculus into these models, researchers can more effectively represent the non-local and memory-dependent effects frequently observed in natural phenomena. Due to the importance of the family of TF-GBFEs, this work introduces a changed iterative method for analyzing this family analytically to gain a deep understanding of many nonlinear phenomena described by this family (e.g., shock waves). The proposed approach combines two algorithms: the Laplace transform and the residual power series method. The suggested technique is thoroughly discussed. Two numerical problems are discussed to check the effectiveness and accuracy of the proposed method. The approximations for integer and fractional orders are compared with the exact solution for integer-order problems. Finally, to investigate how the fractional order affects these problems, the obtained results are discussed graphically and numerically in the tables.
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DOI: 10.1063/5.0187127
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