Simulation and Decay Prediction of Time-Fractional Dynamic Waves Evolution Based on Meshless Methods.

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Bibliographic Details
Title:	Simulation and Decay Prediction of Time-Fractional Dynamic Waves Evolution Based on Meshless Methods.
Authors:	Zhao, RuI¹ 107552300623@stu.xju.edu.cn, Yang, Huanzhu¹ 596352335@qq.com, Sabir, Amina² amina_sabir1@126.com, Imin, Rahmatjan³ rahmatjanim@xju.edu.cn
Source:	IAENG International Journal of Applied Mathematics. Jul2026, Vol. 56 Issue 7, p2405-2418. 14p.
Subjects:	Meshfree methods, Fractional calculus, Energy dissipation, Burgers' equation, Computer simulation
Abstract:	This paper presents, for the first time, a meshless scheme coupling the Symmetric Kernel Derivative Free Smoothed Particle Hydrodynamics (SKDF-SPH) method and L1 discretization for solving the time-fractional Burgers equation (TFBE), aiming to simulate dynamics wave evolution and predict decay. Specifically, the L1 method is employed to discretize the temporal Caputo derivative to capture the fractional "memory effect", while the SKDF-SPH handles spatial terms. This approach avoids computing kernel derivatives and can solve first-order and second-order derivatives, thus improving accuracy and efficiency in complex scenarios. Building on existing theoretical frameworks, this paper presents a proof of the decay theorem for time-fractional dynamic waves, clarifies the quantitative relationship between the norm decay characteristics of solutions and fractional orders, and validates them through three types of numerical examples. The examples with an analytical solutions demonstrates that the proposed method yields small errors in both 2 L and L norms, achieving a second-order convergence rate; the complex-boundary example without an analytical solution successfully captures the spatiotemporal coupled evolution law of dynamic waves; the long-term simulation of the homogeneous equation reveals the regulatory effect of the fractional order on decay -- a larger leads to more significant energy dissipation and diffusion. It is also determined that 4 serves as the optimal parameter for the decay rate, verifying the suboptimal decay property. The results demonstrates that this meshless scheme provides a reliable tool for simulating time-fractional dynamic waves and can be extended to related engineering fields. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

Description
Abstract:	This paper presents, for the first time, a meshless scheme coupling the Symmetric Kernel Derivative Free Smoothed Particle Hydrodynamics (SKDF-SPH) method and L1 discretization for solving the time-fractional Burgers equation (TFBE), aiming to simulate dynamics wave evolution and predict decay. Specifically, the L1 method is employed to discretize the temporal Caputo derivative to capture the fractional "memory effect", while the SKDF-SPH handles spatial terms. This approach avoids computing kernel derivatives and can solve first-order and second-order derivatives, thus improving accuracy and efficiency in complex scenarios. Building on existing theoretical frameworks, this paper presents a proof of the decay theorem for time-fractional dynamic waves, clarifies the quantitative relationship between the norm decay characteristics of solutions and fractional orders, and validates them through three types of numerical examples. The examples with an analytical solutions demonstrates that the proposed method yields small errors in both 2 L and L norms, achieving a second-order convergence rate; the complex-boundary example without an analytical solution successfully captures the spatiotemporal coupled evolution law of dynamic waves; the long-term simulation of the homogeneous equation reveals the regulatory effect of the fractional order on decay -- a larger leads to more significant energy dissipation and diffusion. It is also determined that 4 serves as the optimal parameter for the decay rate, verifying the suboptimal decay property. The results demonstrates that this meshless scheme provides a reliable tool for simulating time-fractional dynamic waves and can be extended to related engineering fields. [ABSTRACT FROM AUTHOR]
ISSN:	19929978