Analytical approximate solutions for two-dimensional incompressible Navier-Stokes equations

Analytical approximate solutions of the two-dimensional incompressible Navier-Stokes equations by means of Adomian decomposition method are presented. The power of this manageable method is confirmed by applying it for two selected  flow problems: The first is the Taylor decaying vortices, and the second is the flow behind a grid, comparison with High-order upwind compact finite-difference method is made. The numerical results that are obtained for two incompressible flow problems  showed that the proposed method is less time consuming, quite accurate and easily implemented. In addition, we prove the convergence of this method when it is applied to the flow problems, which are describing them by  unsteady two-dimensional incompressible Navier-Stokes equations.   Keywords: Navier-Stokes equations, Adomian decomposition, upwind compact difference, Accuracy, Convergence analysis,Taylor's decay vortices, flow behind a grid

Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations

The purpose of this paper is to extend the homotopy perturbation method to fractional heat transfer and porous media equations with the help of the Laplace transform. The fractional derivatives described in this paper are in the Caputo sense. The algorithm is demonstrated to be direct and straightforward, and can be used for many other non-linear fractional differential equations

Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

We have proposed in this  research a new scheme to find analytical  approximating solutions for Navier-Stokes equation  of  one  dimension. The  new  methodology depends on combining  Adomian  decomposition  and Homotopy perturbation methods  with the splitting time scheme for differential operators . The new methodology is applied on two problems of  the test: The first has an exact solution  while  the other one has no  exact solution. The numerical results we  obtained  from solutions of two problems, have good convergent  and high  accuracy   in comparison with the two traditional Adomian  decomposition  and Homotopy  perturbationmethods .&nbsp

The Existence and Uniqueness of the Solution of Partial Differential Equation by Using the Non-linear Transform Function

The development of Burger equation through the transform function is studied and we prove the existence and uniqueness of solution , also we give some applications

On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He\u27s Homotopy Perturbation Method

In this paper, we obtain the approximate solution for 2-dimensional Boussinesq equation with initial condition by Adomian\u27s decomposition and homotopy perturbation methods and numerical results are compared with exact solutions

Conformable Derivative Operator in Modelling Neuronal Dynamics

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense

DECOMPOSITION METHOD IN COMPARISON WITH NUMERICAL SOLUTIONS OF BURGERS EQUATION

This paper presents a solution of the one-dimension Burgers equation using Decomposition Method and compares this solution to the analytic solution [Cole] and solutions obtained with other numerical methods. Even though decomposition method is a non-numerical method, it can be adapted for solving nonlinear differential equations. The advantage of this methodology is that it leads to an analytical continuous approximated solution that is very rapidly convergent [2,7,8]. This method does not take any help of linearization or any other simplifications for handling the non-linear terms. Since the decomposition parameter, in general, is not a perturbation parameter, it follows that the non-linearities in the operator equation can be handled easily, and accurate solution may be obtained for any physical problem

Approximate Analytical Solutions for Fractional Space- and Time- Partial Differential Equations using Homotopy Analysis Method

This article presents the approximate analytical solutions of first order linear partial differential equations (PDEs) with fractional time- and space- derivatives. With the aid of initial values, the explicit solutions of the equations are solved making use of reliable algorithm like homotopy analysis method (HAM). The speed of convergence of the method is based on a rapidly convergent series with easily computable components. The fractional derivatives are described in Caputo sense. Numerical results show that the HAM is easy to implement and accurate when applied to space- time- fractional PDEs

Absolute Value Boundedness, Operator Decomposition, and Stochastic Media and Equations

The research accomplished during this period is reported. Published abstracts and technical reports are listed. Articles presented include: boundedness of absolute values of generalized Fourier coefficients, propagation in stochastic media, and stationary conditions for stochastic differential equations