An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as  Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and  Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

Galerkin finite element study for mixed convection (TiO2–SiO2/water) hybrid-nanofluidic flow in a triangular aperture heated beneath

Fluidity and thermal transport across the triangular aperture with lower lateral inlet and apply placed at the vertical outlet of the chamber which filled with efficient TiO2–SiO2/water hybrid nanofluid under the parametrical influence. Several parameters are tested like the numbers of Hartmann ($$0 \le Ha \le 100$$ 0 ≤ H a ≤ 100 ), Richardson ($$0 \le Ri \le 5$$ 0 ≤ R i ≤ 5 ), and Reynolds ($$10 \le Re \le 1000$$ 10 ≤ R e ≤ 1000 ) were critiqued through streamlines, isotherms, and Nusselt number ($$Nu$$ Nu ). Numerical model has to be developed and solved through the Galerkin finite element method (GFEM) by discretized with 13,569 triangular elements optimized through grid-independent analysis. The Hartmann number ($$Ha$$ Ha ), exerts minimal impact over the flow and thermal aspects while the other parameters significantly manipulate the physical nature of the flowing and thermal aspects behaviors.