On a solution of the periodic boundary value problem for a hyperbolic equation with a fractional derivative

The article investigates a boundary value problem for a hyperbolic equation with the Riemann–Liouville fractional derivative, which is periodic in one variable. Such equations are widely used in modeling complex physical processes with memory effects, including viscoelasticity, anomalous diffusion, and thermoviscoelasticity phenomena, where classical integer-order models fail to adequately describe the hereditary properties of materials and transport processes. To solve this problem, an iterative algorithm is proposed based on domain decomposition and the reduction of the original problem to a system of integro-differential equations. A theorem on the existence and uniqueness of the solution is proved, and an estimate of the convergence rate of the method is obtained using matrix analysis and a strengthened Gronwall–Bellman inequality. It is established that the choice of the decomposition step plays a key role in ensuring the stability of the algorithm. The conducted analysis extends the class of problems for which efficient computational algorithms can be constructed and may serve as a foundation for studying more complex nonlinear cases and problems in irregular domains.