A collocation method based on ramp basis functions is proposed for the numerical solution ofnonlinear functional Volterra integral equations of Urysohn type. Exploiting the local supportof the ramp functions, the resulting collocation system has a lower-triangular structure, whichallows the approximate solution to be computed sequentially and avoids the solution oflarge nonlinear algebraic systems. A convergence analysis is presented, and error estimatesare derived under mild regularity assumptions using a nonlinear Gronwall inequality, whichguarantee first-order convergence of the method.Several numerical examples are provided to illustrate the accuracy and computationalefficiency of the proposed approach. In particular, a benchmark test problem is used to comparethe ramp collocation method with the Euler discretization method and the Heaviside–sigmoidalcollocation method. The numerical results confirm the theoretical findings and show that,for sufficiently smooth problems, higher convergence rates may be observed in practice. Thereported CPU times are consistent with the predicted (𝑛2) computational complexity anddemonstrate a favorable accuracy–efficiency balance of the proposed method.
