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Article · Axioms · May 30, 2026

A Fast Chebyshev Spectral Collocation Method for a Coupled System of Nonlinear Klein–Gordon Equations with Caputo Fractional Memory

Yertay Kazez, Zh. A. Abdiramanov, Nauryzbay Adil, Abdumauvlen Berdyshev

Zhanars Abdiramanov
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Abstract

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (βNM=1/4) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders α≠β, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory.

Fractional Differential Equations SolutionsNumerical methods for differential equationsDifferential Equations and Boundary ProblemsChebyshev spectral collocationnonlinear Klein-Gordon equationcoupled systems

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