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<p>In this study, a novel meshless kernel-based approximation method is developed for solving the Sobolev equation. Radial kernels are employed as approximating basis functions within the least-squares framework, enabling spatial discretization on trial spaces spanned by translates of positive-definite radial kernels. The interior and boundary data are then decomposed, allowing the conversion of the underlying differential algebraic equations into a system of first-order ordinary differential equations (ODEs) for interior data only. Finally, an efficient and high-order ODE solver is invoked for stable time integration. The proposed meshless scheme is a low-cost computational scheme that circumvents the ill-conditioning and improves eigenvalue stability. The method performance is evaluated on a two-dimensional Sobolev equation over rectangular and complex-shaped domains with regular and scattered nodes. Some recommendations are made for resolving discontinuous and singular solutions.</p>

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Kernel-based meshless approximation method for a fluid flow model: a least-square approach with decomposition of interior and boundary data

  • 期刊论文

  • 发表日期：

    2025年1月30日

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  • Manzoor Hussain

摘要

In this study, a novel meshless kernel-based approximation method is developed for solving the Sobolev equation. Radial kernels are employed as approximating basis functions within the least-squares framework, enabling spatial discretization on trial spaces spanned by translates of positive-definite radial kernels. The interior and boundary data are then decomposed, allowing the conversion of the underlying differential algebraic equations into a system of first-order ordinary differential equations (ODEs) for interior data only. Finally, an efficient and high-order ODE solver is invoked for stable time integration. The proposed meshless scheme is a low-cost computational scheme that circumvents the ill-conditioning and improves eigenvalue stability. The method performance is evaluated on a two-dimensional Sobolev equation over rectangular and complex-shaped domains with regular and scattered nodes. Some recommendations are made for resolving discontinuous and singular solutions.