Tension spline-quasilinearization scheme for generalized nonlinear time-fractional convection-diffusion equations

This article presents an efficient numerical technique based on non-polynomial tension spline for solving generalized nonlinear time fractional convection-diffusion equation with polynomial nonlinear convection terms. The proposed method uses non-polynomial tension spline spatial discretization with the Crank-Nicolson time-stepping scheme, and employs the quasi-linearization technique to handle the nonlinear term $p^{v}\frac{\partial p}{\partial x}$ for $v=1,2,...,10$. The tension spline introduces a controllable smoothness parameter, which can improve accuracy and reduce spurious oscillations. Additionally, the order of convergence for the suggested scheme is determined by analyzing the truncation error, and we theoretically presented the stability of the developed scheme using the von Neumann method. The method is shown to be conditionally stable. The method is validated on two representative test problems, showing good agreement with expected solutions and improved accuracy compared with existing numerical methods. These results illustrates the potential of the approach for solving nonlinear time-fractional convection-diffusion problems.