An Euler-type method for two-dimensional Volterra integral equations of the first kind

Two-dimensional first-kind Volterra integral equations (VIEs) are studied. The first-kind equations are reduced to second kind, and by obtaining an appropriate integral inequality, existence and uniqueness are demonstrated. The equivalent discrete integral inequality then permits convergence of discretization methods; and this is illustrated for the Euler method. Finally, a class of nonlinear telegraph equations is shown to be equivalent to (two-dimensional) Volterra integral equations, thereby providing existence and uniqueness results for this class of equations. Furthermore, the telegraph equation may be solved by the numerical method for two-dimensional VIEs, and a simple numerical example is given.