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A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form x β (z − x) −α g(y(x)), where α ∈ (0, 1), β > 0 and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the L ∞ and the L 2 norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

We analyze the existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations, including equations with weakly singular kernels. Of particular interest are those integral equations that can be transformed into cordial Volterra integral equations whose underlying integral operator may be non-compact. 2010 AMS Mathematics subject classification. Primary 45A05, 45D05, 45E99. Keywords and phrases. Volterra integral equation of the third kind, cordial Volterra integral equation, existence and regularity of solutions.