A novel definition of the caputo fractional finite difference approach for Maxwell fluid

This paper focuses on modeling and analyzing a nonlinear fractional Maxwell fluid across a vertical plate using a novel definition of the Caputo fractional derivative. Specifically, the newly formulated fractional definition is tailored for the finite difference method. Buoyancy effects are considered to bring the model closer to the natural occurrences of the magnetic field. The novel fractional derivative definition of the designed model has been implemented, and the results have been further analyzed using theoretical and numerical methods. The range of fractional parameters \(\alpha \) $\alpha$ and \(\beta \) $\beta$ is \(0<\alpha \le 1\) $0<\alpha \le 1$ and \(0<\beta \le 1\) $0<\beta \le 1$ , respectively. Significant theorems are also included in the study. A simulation was performed to determine which fractional derivative operator is more accurate than the others. The designed definition demonstrates a high convergence rate and decreased computational cost, both of which are primary objectives. Graphs have also been plotted to illustrate the velocity and temperature distributions as the physical parameters vary.