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a b s t r a c tIn this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.

This paper presents an analysis of the features of magnetohydrodynamic Darcy–Forchheimer nanofluid flow over a nonlinear stretching sheet. A viscous incompressible nanofluid saturates the porous medium via the Darcy–Forchheimer relation. Heat and mass transfer is analyzed through the Brownian motion factor and thermophoresis. A non-uniform magnetic field is induced to enhance the electric conductivity of the nanofluid. The model emphasizes a small magnetic Reynolds number and boundary layer formulation. The governing partial differential equations are converted into nonlinear ordinary differential equations using similarity transformations and thereafter solved utilizing the homotopy analysis method. Graphs are sketched for different values of various fluid parameters. Drag force, heat flux, and mass flux are interpreted numerically. The porosity factor results in increasing the skin friction due to the high resistance offered by the porous medium. The heat and mass flux are reduced for a stronger porosity factor, which is an important finding for industrial applications of nanofluids. The temperature profile shows increasing values of both Brownian diffusion and thermophoresis.

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