Spectral Treatment of Distributed-Order Caputo Models: Applications to Nonlinear Fractional Sine and Klein-Gordon Differential Equations

In this study, we present a highly efficient spectral numerical approach for solving nonlinear Fractional Distributed-Order Sine-Gordon Differential Equations (FDO-SGDEs) and Fractional Distributed-Order Klein-Gordon Differential Equations (FDO-KGDEs) considering initial and Dirichlet boundary conditions. Our proposed method utilizes the fractional-order shifted Legendre-Gauss collocation method and shifted Chebyshev-Gauss collocation method, leveraging the Riemann-Liouville fractional derivative to transform the given problems into systems of algebraic equations. The effectiveness and accuracy of this technique are demonstrated through the successful resolution of four illustrative examples.