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In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

The aim of the study is to analyse the effect of two different thermal boundary conditions on natural convection in a porous triangular enclosure. Sinusoidal and linearly varying temperature profiles on the left wall are considered. The top wall of the enclosure is kept to be adiabatic. The Darcy model is adopted in this study and the governing equations are solved using the finite difference method for various values of inclination angle and Darcy-Rayleigh number. The results are presented in the form of streamlines, isotherms, and Nusselt numbers. The heat transfer rates are calculated and compared between two thermal boundary conditions. The heat transfer is more enhanced in the case of sinusoidal wall temperature than linearly varying wall temperature. [ABSTRACT FROM AUTHOR]