Abstract

An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied in this paper. The method, which is first order accurate in the time-step, is shown to convergence to an exact solution of the equations. By adequately splitting the viscous term, it allows to enforce full Dirichlet boundary conditions on the velocity in all substeps of the scheme, while needing no boundary condition at all for the pressure. It is also shown to be related to an iterative predictor-multicorrector algorithm for evolution equations, when this is applied to the incompressible Naiver-Stokes system. A new derivation of the algorithm in a general setting is provided. Two different finite element interpolations are considered for the implementation of the algorithm; numerical results obtained with them for standard benchmark cases are presented.

Abstract

An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied in this paper. The method, which is first order accurate in the time-step, is shown to convergence to an exact solution of the equations. By adequately splitting the viscous term, it allows to enforce full Dirichlet boundary conditions on the velocity in all substeps of the scheme, while needing no boundary condition at all for the pressure. It is also shown to be related to an iterative predictor-multicorrector algorithm for evolution equations, when this is applied to the incompressible Naiver-Stokes system. A new derivation of the algorithm in a general setting is provided. Two different finite element interpolations are considered for the implementation of the algorithm; numerical results obtained with them for standard benchmark cases are presented.

Abstract

The analysis of nonlinear events related to physical phenomena is a popular issue in the modern-day. The essential purpose of this work is to discover a novel approximate solution to the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation (BBMPB) utilizing the natural decomposition method (NDM) of fractional order. The suggested approach provides analytical solutions that are extremely near to the exact solution whereas obviating the complexities associated with many other approaches. The expected issue’s uniqueness theorem and convergence analysis are explored using Banach’s fixed-point theory. The reliability and accuracy of the recommended method were tested using numerical simulations. The graphs and tables reflect the results. The comparison of the suggested scheme’s solution with the exact solutions demonstrates that the scheme is efficient, methodical, and extremely exact in tackling nonlinear complicated phenomena.

Abstract

This article introduces and illustrates a novel approximation to the compound KdV-Burgers equation. For such a challenge, the q-homotopy analysis transform technique (q-HATM) is a potent approach. The suggested procedure avoids the complexity seen in many other methods and provides an approximation that is extremely near to the exact solution. The uniqueness theorem and convergence analysis of the expected problem are explored with the aid of Banach's fixed-point theory. Through a difference in the fractional derivative, the normal frequency for the fractional solution to this issue changes. All of the discovered solutions are illustrated in the figures and tables.

Abstract

In today’s world, analyzing nonlinear occurrences related to physical phenomena is a hot topic. The main goal of this research is to use the natural decomposition method (NDM) of fractional order to find an approximate solution to the fractional clannish random walker’s parabolic (CRWP) equation. The proposed method gives approximate solutions that are exceptionally near the exact solution without the complication that numerous other techniques imply. Banach’s fixed-point theory is used to investigate the anticipated issue’s convergence analysis and uniqueness theorem. To ensure that the suggested technique is trustworthy and precise, numerical simulations were conducted. The results are shown in the graphs and tables. When comparing the proposed scheme’s solution to the actual solutions, it becomes clear that the scheme is efficient, systematic, and very precise when dealing with nonlinear complex phenomena.