Interpolated Solutions of Abel Integral Equations Using Barycentric ‎Lagrange Double Interpolation

We provide Interpolant solutions to the Apple integral equations that emerge in climate, ‎‎atmosphere, heat transfer, superfluid, astrophysics, solid mechanics, scattering theory, ‎‎spectroscopy, stereology, elasticity theory, and plasma physics, and other fields. We ‎developed ‎adequate formulas for the optimal distribution of kernel nodes to address the ‎kernel’s ‎singularity, ensuring that the kernel does not reach infinity when one of the two ‎variables ‎approaches the other. Four matrices represent the data function, whereas five ‎matrices ‎represent the kernel. We achieved two formulas for the matrix-vector single ‎interpolated ‎solution, the first based on interpolated the data function while the second based ‎on ‎interpolated the kernel only. The matrix-vector single interpolated solution has two ‎formulas: ‎the first is based on interpolating the data function and the kernel, while the second ‎is based ‎on interpolating only the kernel. The first formula simply involves the calculation of ‎two ‎matrices: the elements of the first matrix are correspond to the functional values of the ‎data ‎function, and the elements of the second matrix correspond to the functional values of ‎the ‎kernel at the two sets of nodes that are associated with the kernel’s variables. When ‎compared ‎to the solutions provided by other approaches, the lower-degree interpolated ‎solutions to three ‎cases were found to be convergent to the exact solutions with a minimum ‎CPU time and high ‎accuracy, demonstrating the novelty and simplicity of the proposed ‎method as well as the ‎accuracy of the results.‎