IMA Journal of Numerical Analysis (2000) 20, 423–440 An Euler-type method for two-dimensional Volterra integral equations of the first kind SEAN MCKEE † Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK TAO TANG Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada AND TERESA DIOGO Departamento de Matem ´ atica, Instituto Superior T´ ecnico, 1049-001 Lisbon, Portugal [Received 2 August 1998 and in revised form 17 June 1999] 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. 1. Introduction Consider the first-kind Volterra integral equation  t 0  s 0 k (t , s , u ,v)x (u ,v) du dv = f (t , s ), (t , s ) ∈ Ω := [0, T ] × [0, S]. (1.1) To be consistent, we require f (t , 0) ≡ 0, f (0, s ) ≡ 0, for (t , s ) ∈ Ω. (1.2) Assume that k and f are smooth and that k (t , s , t , s )  = 0 for all (t , s ) ∈ Ω. (1.3) By obtaining an appropriate Gronwall inequality we shall study the existence and uniqueness of (1.1). † Email: s.mckee@strath.ac.uk c  Oxford University Press 2000