Cubic orthogonal spline collocation method for the viscoelastic hyperbolic integro-differential equation with nonlinear-nonlocal damping

Lei Ouyang and Kexin Li

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 249, issue C, 157-178

Abstract: This work is devoted to the numerical analysis of a class of nonlinear hyperbolic integro-differential equations arising in viscoelasticity, featuring nonlinear nonlocal damping and memory kernels of variable-sign or exponential-logarithmic type. A high-order cubic orthogonal spline collocation (OSC) method is developed and rigorously analyzed. Under appropriate assumptions on the nonlinear coefficient, the global stability and uniqueness of the semi-discrete OSC approximation are established by employing a splitting technique, Gauss–Legendre quadrature, and structural properties of the kernels. Furthermore, a fully discrete OSC scheme is constructed using a three-point central difference for the temporal derivative and a second-order quadrature rule for the integral term. The optimal high-order error estimates are derived for both semi- and fully discrete schemes. Numerical experiments are provided to confirm the theoretical results and demonstrate the accuracy and stability of the proposed OSC method.

Keywords: Hyperbolic integro-differential equations; Nonlinear-nonlocal damping; Orthogonal spline collocation; Variable-sign and exponential-logarithmic kernels; Global stability and uniqueness; Error estimate (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:249:y:2026:i:c:p:157-178

DOI: 10.1016/j.matcom.2026.05.012

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