Nabil Mlaiki

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions.

In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we have included some examples and a section on the existence of solutions for the systems of Volterra–Hammerstein integral equations and Urysohn integral equations, respectively.