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

Effects of different thermal modes of obstacles on the natural convective Al2O3-water nanofluidic transport inside a triangular cavity

This study investigates the Al2O3-water nanofluidic transport within an isosceles triangular compartment with top vertex downwards. The top wall is maintained isothermally cooled and left as well as right inclined walls are made uniformly heated. Two diamond-shaped obstacles are positioned inside the enclosure. The nanofluidic motion is supposed to be magnetically influenced. This investigation includes a fine analysis of how various thermal modes of obstacles affect the velocity and thermal profiles of the nanofluid. Appropriate similarity conversion leads to having a non-dimensional flow profile and is treated with Galerkin finite element scheme. The grid independency, experimental verification, and comparison assessments are directed to explore the model accuracy. The dynamic parameters like Rayleigh number [Formula: see text], nanoparticle volume fraction [Formula: see text], and Hartmann number [Formula: see text] are varied to perceive the noteworthy changes in isotherms, velocity, streamlines, and Nusselt number. The consequences specify average Nusselt number deteriorates for Hartmann number but escalates for nanoparticle concentration and Rayleigh number. Both heated and adiabatic obstacles exhibit high heat transport, while cold obstacles reveal the lowest magnitude in heat transmission. For Rayleigh number, cold obstacles reveal 34.51% heat transport enhancement, whereas it is 52.72% for heated obstacles compared to cold one. mathematics subject classification: 76W05

The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.