On the Differential Properties of Solutions to Fredholm Equations with Weakly Singular Kernels

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Abstract
The differential properties of a class of weakly singular integrals and of solutions to related Fredholm equations of the second kind are studied. Formulae for differentiation are first established for compound integrals, the kernels of which have surface discontinuities within a bounded region of Rn. Applying these formulae, the singularities of solutions to the Fredholm equations are resolved in terms of surface integrals. As an example of the analysis in the one-dimensional case, the integral transport equation is considered.