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.
