Exact and High-Accuracy Solutions to Fractional Burgers-Type and Reaction-Diffusion Equations Using the Laplace-Sba Method

This paper applies the hybrid Laplace-SBA (Somé-Blaise-Abbo) method to solve three classes of nonlinear fractional partial differential equations with respect to time in the Caputo sense: a fourth-order viscoelastic wave model, a reaction-advection-diffusion system, and a Burgers-type equation with time-variable coefficients. By coupling the Laplace transform with successive approximations based on the fixed-point principle, the scheme avoids linearization, spatial discretization, and the computation of Adomian polynomials. The exact solutions for the first two models are obtained in a single iteration. For the fractional Burgers equation, the series approximations converge continuously to the classical solution as \(α → 2\), achieving absolute errors as low as \(10⁻¹⁸\). These results demonstrate the efficiency and reliability of the Laplace-SBA method for nonlinear systems with memory effects.