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In this paper, we will find the numerical solution of Gray-Scott model in two dimensions space, this method is a system of non-linear parabolic partial differential equations. Then transforming the original model (system of non-linear PPDEs), by using the method of lines to a system of ODEs. Therefore we used Runge-Kutta methods (Explicit RK method and Implicit RK method) to find the numerical solutions of the new systems, and we compared between these methods, we saw that the numerical results of IRK methods is more accurate than the numerical results of ERK method.

This paper deals with the numerical solution for Sine-Gordon system in two dimensions using two finite difference methods the (ADE) and (ADI) methods .A comparison between the two methods has been done and we have obtained that the (ADE) method is the easer while the (ADI) method is more accurate than the (ADE) method. We also studied the stability analysis for each method by using Fourier (Von Neumann) method and we have obtained that the (ADI) method is unconditionally stable while the (ADE) method is stable under the condition and where is the ratio of the waves speed u , w and .

The numerical stability analysis of Brusselator system has been done in one and two dimensional space. For one dimension we studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we found that explicit method for solving Brusselator system is stable under the conditions , and While the implicit method is unconditionally stable. For two dimensional space we found that ADE method is stable under condition , and , while ADI is unconditionally stable.

In this paper, we study three types of finite difference methods, to find the numerical solution of reaction difference systems of PDEs in two dimensions. These methods are ADE, ADI and Hopscotch, where Gray-Scott model in two dimensions has been considered. Our numerical results show that the ADI method produces more accurate and stable solution than ADE method and Hopscotch method is the best because does not involve any tridiagonal matrix. Also we studied the consistency, stability and convergence of the above methods.

In this paper, we solved the Newell-Whitehead equation approximately using Adomain Decomposition method and we have compared this solution with the exact solution; we found that the solution of this method is so close to the exact solution and this solution is slower to converge to the exact solution when we increase t however, this method is effective for this kind of problems.

This paper has studied the numerical solution for Sine-Gordon system in one dimensions using finite difference methods. We have used Explicit method and Crank-Nicholson method.A comparison between results of the two methods has been done and we obtained that Crank-Nicholson method is more accurate than the Explicit method but the Explicit method is easer . We also studied the stability analysis for each method by using Fourier(Von-Neumann) method and obtained that Crank-Nicholson method is unconditionally stable while the Explicit method is stable under the condition and .

Stability analysis for steady state solution of reaction-diffusion equations with double diffusivity discuss and arise in the solution of problems of flow of homogeneous liquids and heat conduction involving air-temperature and a grain-temperature , the resulting of this analysis shows that the system is stable when:

The Stability Analysis of Steady State Solution of Burger equation by using Fourier mode Stability analysis in two cases has been considered , the first one when the amplitude is constant and the second one when the amplitude is variable . In the first case the steady state solution is always stable and the second case is conditionally stable . In the second case the comparison between the analytical solution and numerical solution of Galerkin technique are the same.

The Burger’s-Huxley equation has been solved numerically by using two finite difference methods, the explicit scheme and the Crank-Nicholson scheme. A comparison between the two schemes has been made and it has been found that, the first scheme is simpler while the second scheme is more accurate and has faster convergent. Also, the stability analysis of the two methods by using Fourier (Von Neumann) method has been done and the results were that, the explicit scheme is stable under the condition and the Crank-Nicholson is unconditionally stable.

The Stability Analysis of Convection& Diffusion equation by using Fourier mode Stability analysis in two cases has been considered , the first one when the amplitude is constant and the second one when the amplitude is variable . In the first case, the solution is always stable and in the second case the solution is conditionally stable .

A non-linear parabolic system is derived to describe compressible nuclear waste disposal contamination in porous media . Galerkin method is applied for the pressure equation . For the concentration of the brine of the fluid, a kind of partial upwind finite element scheme is constructed. A numerical application is included to demonstrate certain aspects of the theory and illustrate the capabilities of the kind of partial upwind finite element approach.

We solved φ4 Klein-Gordon equation numerically by using two finite difference methods: The first is the explicit method and the second is the implicit (Crank-Nicholson) method. Also, we studied the numerical stability of the two methods using Fourier (Von Neumann) method and it has been found that the first method is simpler and has faster convergence while the second method is more accurate, and the explicit method is conditionally stable while the implicit method is unconditionally stable.

In this Paper, we studied the stability analysis of steady state solutions of Gray-Scott Model in one-dimension using Fourier mode and we showed that the solutions are conditionally stable.

The stability analysis of steady state solutions of Sine–Gordon equation using Fourier mode stability analysis in two cases has been considered : Firstly when the amplitude is constant and secondly when the amplitude is variable in the two cases the results were found to be : The steady state solutions and are unconditionally stable . In the second case the comparison between the analytical solution and the numerical solution of Galerkin technique has been done . This comparison showed that the analytical solution and the numerical solution of Galerkin technique are the same.

A non-linear prey-predator system solved numerically by Galerkin method, and we compare these results with the results of Pius Peter Nyaanga[6] who used finite difference methods, we found that Galerkin finite elements method is faster than finite difference method to reach equilibrium state where the density for the prey and the predator are equals for all the values for and , also we found that Galerkin method converges towards the steady state solutions faster than finite difference method with less steps in time.

The Burger equation had been solved numerically by using two finite differences methods. The first is explicit scheme method and the second is Crank–Nicholson method. A comparison had been made between the two and we find that the Crank–Nicholson method is the best and most accurate than the explicit scheme method, as it is shown in (Table 1). Also the numerical stability for both methods has been made, the explicit scheme method is conditionally stable and the condition is , while Crank–Nicholson method is unconditionally stable.

The Sine – Gordon equation has been solved numerically by using two finite differences methods: The first is the explicit scheme and the second is the Crank – Nicholson scheme. A comparison between the two schemes has been made and the results were found to be : the first scheme is simpler and has faster convergence while the second scheme is more accurate . Also , the stability analysis of the two methods by the use of Fourier (Von Neumann) method has been done and the results were found to be : The explicit scheme is conditionally stable if and the Crank–Nicholson is unconditionally stable .

We studied the stability of the steady state solutions for Fisher Equation in two cases, the First one with constant amplitude and we show that the steady state solution is always stable under any condition, but the other two solutions and are conditionally stable. In the Second case, we studied the steady state solutions for various amplitude by using two Methods. The First is analytically by direct Method and the second is numerical method using Galerkin technique which shows the same results, that is the steady state solution is always stable under any conditions, but the other two solutions and are conditionally stable.

In this paper we used two numerical methods to investigate propagating heat solutions of PDEs. The explicit and Crank-Nicolson methods and the results show that Crank-Nicolson method is more accurate than the explicit method. As an illustration, we used the above method to an autocatalytic reaction diffusion equations involving two diffusing chemicals in one dimension.

The Fisher Equation had been solved numerically by using two Methods of Finite Differences Methods. The First is Explicit Scheme Method and the Second is Crank–Nicholson Method. A Comparison had been made between these two methods and we find that the Crank–Nicholson Method converges towards saturation state u=1 faster than the Explicit Scheme Method (Table 1). Also the numerical stability for both Methods had been made, the Explicit Scheme Method is conditionally stable and the condition is , while Crank–Nicholson Method has the condition for step size, but time step is unconditionally stable.

The objective of this paper is to construct numerical schemes using finite difference methods for the one-dimensional general hyperbolic- parabolic- reaction problem. The finite difference method with the exponential transformation form is used to solve the problem, and employs difference approximation technique to obtain the numerical solutions. Computational examples are presented and compared with the exact solutions. We obtained that the Crank-Nicholson scheme is more accurate than Forward scheme. Therefore the form of exponential transformation for the problem yields a stable solution compared with exact solution.

The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme is conditionally stable if and the second scheme is unconditionally stable.

Stability analysis of steady state solutions of Huxley equation using Fourier mode stability analysis in two cases is investigated. Firstly when the amplitude is constant and secondly when the amplitude is variable and the results were found to be: The solutions and are always stable while the solutions and are conditionally stable. In the second case, a comparison between the analytical solution and the numerical solution of Galerkin method is done and the results are the same.

Anon-linear parabolic system is derived to describe incompressible nuclear waste disposal contamination in porous media. Galerkin method is applied for the pressure equation. For the concentration, a kind of partial upwind finite element scheme is constructed. The finite element solution satisfies discrete maximum principle and converges to the solution in norm .