A hybrid Meshless approach for solving the distributed-order fractional Klein-Gordon equation based on generalized moving least squares and finite difference method - Habibirad, Ali; Ordokhani, Yadollah; Azin, Hadis; İnç, Mustafa (Taylor & Francis Ltd, 2026)
Recent studies have investigated various forms of the time-fractional and distributed-order nonlinear Klein-Gordon equations (KGEs). In this work, we focus on the nonlinear KGE involving distributed time derivatives and propose a novel numerical framework for its efficient and accurate solution. For cases where the nonlinear component in the KGE appears as sin ( $ \psi (v) = \sin (v) $ psi(v)=sin(v)), this particular form of the equation is commonly known as the Sine-Gordon equation, whose numerical solution holds significant importance. A Gaussian quadrature-based approach is employed to approximate the distributed derivative term, ensuring high precision in the numerical treatment. The proposed method combines the Generalized Moving Least Squares (GMLS) technique for spatial discretization with a first-order accurate finite difference approximation in the time domain. The experimental findings provide substantial evidence for the efficacy of our novel methodology in accurately capturing the complex dynamics of the equation, highlighting its potential for solving similar problems with distributed derivatives in diverse computational domains.
Keywords: Klein-Gordon equation; generalized moving least squares; distributed-order time-fractional derivative; finite difference; Gaussian quadrature-based method