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

Author

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  • Ouyang, Lei
  • Li, Kexin

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.

Suggested Citation

  • Ouyang, Lei & Li, Kexin, 2026. "Cubic orthogonal spline collocation method for the viscoelastic hyperbolic integro-differential equation with nonlinear-nonlocal damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 249(C), pages 157-178.
    • Handle: RePEc:eee:matcom:v:249:y:2026:i:c:p:157-178
    DOI: 10.1016/j.matcom.2026.05.012
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