Numerical investigation of fractional-order advection–diffusion–reaction problems using an efficient B-spline scheme

Fractional partial differential equations (FPDEs) have proven to be powerful modeling tools for describing a wide range of complex phenomena across physics, chemistry, biology, geology, engineering, and finance. Applications include electrical networks, signal processing, optics, viscoelastic materials, fluid mechanics, acoustics, and predator–prey dynamics. Among these, the fractional-order diffusion–advection–reaction (FDAR) model plays a key role in capturing anomalous transport and nonlinear interactions. In this article, we present an efficient and accurate numerical scheme for solving the FDAR equation with initial and boundary conditions using the cubic exponential B-spline (CEBS) collocation method. The time derivative is treated in the Caputo sense and discretized using the L1 formula, while spatial derivatives are approximated by CEBS functions. The nonlinear terms are handled through quasi-linearization, and the unconditional stability of the method is established via von Neumann stability analysis. In addition, a convergence analysis is provided, showing that the proposed scheme achieves second-order accuracy. To evaluate the efficiency and accuracy of the proposed scheme, three test problems are examined. The obtained results are compared with those reported in earlier studies, demonstrating that the present technique achieves highly accurate results. Furthermore, the method is computationally efficient, requires less computational, and proves to be well suited for solving nonlinear fractional partial differential equations arising in diverse applications.