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Keywords = asymmetric functional-order diffusion–wave equations

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20 pages, 547 KB  
Article
An Efficient Spectral Method for a Class of Asymmetric Functional-Order Diffusion–Wave Equations Using Generalized Chelyshkov Wavelets
by Quan H. Do and Hoa T. B. Ngo
Symmetry 2025, 17(8), 1230; https://doi.org/10.3390/sym17081230 - 4 Aug 2025
Viewed by 403
Abstract
Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such [...] Read more.
Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications. Full article
(This article belongs to the Topic Numerical Methods for Partial Differential Equations)
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25 pages, 11561 KB  
Article
Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model
by Mahdiyar Khanpour, Abdolmajid Mohammadian, Hamidreza Shirkhani and Reza Kianoush
Water 2024, 16(10), 1394; https://doi.org/10.3390/w16101394 - 14 May 2024
Cited by 1 | Viewed by 1632
Abstract
This research aims to develop an efficient and accurate model for simulating tuned liquid dampers (TLDs) with sloped beds. The model, based on nonlinear shallow water equations, is enhanced by introducing new terms tailored to each specific case. It employs the central upwind [...] Read more.
This research aims to develop an efficient and accurate model for simulating tuned liquid dampers (TLDs) with sloped beds. The model, based on nonlinear shallow water equations, is enhanced by introducing new terms tailored to each specific case. It employs the central upwind method and Minmod limiter functions for flux and interface variable assessment, ensuring both high accuracy and reasonable computational cost. While acceleration, slope, and dissipation are treated as explicit sources, an implicit scheme is utilized for dispersion discretization to enhance the model’s stability, resulting in matrix equations. Time discretization uses the fourth-order Runge–Kutta scheme for precision. The performance of the model has been evaluated using several test cases including dam-breaks on flat and inclined beds and run-up and run-down simulations over parabolic beds, which are relevant to sloshing in tanks with sloped beds. It accurately predicts phenomena such as asymmetric sloshing waves, especially in sloped beds, where pronounced waves occur. Dispersion and dissipation terms are crucial for capturing these effects and maintaining stable wave patterns. An initial perturbation method assesses the tank’s natural period and numerical diffusion. Furthermore, the model integrates with a single-degree-of-freedom (SDOF) system to create a TLD model, demonstrating enhanced damping effects with sloped beds. Full article
(This article belongs to the Special Issue Advances in Hydraulic and Water Resources Research (2nd Edition))
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