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<p>In this paper we have focused on the Thomas–Fermi boundary value problems. Thomas–Fermi problems are widely used in the field of astrophysics to determine how the electrons are distributed within an atom. However, Thomas–Fermi problems are difficult to solve numerically due to their coefficient singularity and strong nonlinearity. In this proposed scheme we have implemented the use of higher derivative of the unknown function and approximated it with the help of Haar wavelet, thereby changing them into operational matrices of integration. Subsequently the governing nonlinear differential equation is modified into a system of nonlinear equations. This system is then solved using the Newton–Raphson and Newton–Krylov methods. In addition, comparisons between both the solvers are made. The regularity of the solution is ensured by properly addressing the singularity at origin. Finally comparisons with the exact solution and other known methods are made to demonstrate the effectiveness of this scheme.</p>

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A higher-order Haar wavelet collocation method with Jacobian-free Newton solver for the Thomas–Fermi model equation

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Naaz Siddiqui,
  • Sachin Sharma,
  • Bharti Sharma

摘要

In this paper we have focused on the Thomas–Fermi boundary value problems. Thomas–Fermi problems are widely used in the field of astrophysics to determine how the electrons are distributed within an atom. However, Thomas–Fermi problems are difficult to solve numerically due to their coefficient singularity and strong nonlinearity. In this proposed scheme we have implemented the use of higher derivative of the unknown function and approximated it with the help of Haar wavelet, thereby changing them into operational matrices of integration. Subsequently the governing nonlinear differential equation is modified into a system of nonlinear equations. This system is then solved using the Newton–Raphson and Newton–Krylov methods. In addition, comparisons between both the solvers are made. The regularity of the solution is ensured by properly addressing the singularity at origin. Finally comparisons with the exact solution and other known methods are made to demonstrate the effectiveness of this scheme.