### Keywords : Finite difference methods

##### Alternating Direction Implicit Method for Solving Parabolic Partial Differential Equations in Three Dimensions

Abdulghafor M. Al-Rozbayani; Mahmood H. Yahya

AL-Rafidain Journal of Computer Sciences and Mathematics, 2012, Volume 9, Issue 2, Pages 79-97
DOI: 10.33899/csmj.2012.163703

In this paper, the parabolic partial differential equations in three-dimensions are solved by two types of finite differences, such as, Alternating Direction Explicit (ADE) method and Alternating Direction Implicit (ADI) method. By the comparison of the numerical results for the previous two methods with the Exact solution, we observe that the results of Alternating Direction Implicit (ADI) method is better and nearest to the exact solution compared with the results of Alternating Direction Explicit (ADE) method. we also studied the numerical stability of both methods by Von-Neumann Method.

##### Numerical solution and stability analysis of the Sine-Gordon system in two dimensions

Saad A. Manna; Haneen T. Jassim

AL-Rafidain Journal of Computer Sciences and Mathematics, 2012, Volume 9, Issue 1, Pages 13-25
DOI: 10.33899/csmj.2012.163684

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 .

##### Explicit- Implicit Runge-Kutta Methods to Solve the System of Nonlinear PPDEs in Two Dimensions

Abbas Y. Al-Bayati; Saad Manaa; Abdulghafor M. Al-Rozbayani

AL-Rafidain Journal of Computer Sciences and Mathematics, 2012, Volume 9, Issue 1, Pages 13-21
DOI: 10.33899/csmj.2012.163667

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.

##### Comparison of Finite Difference Solution Methods for Reaction Diffusion System in Two Dimensions

Abbas Y. Al-Bayati; Saad A. Manaa; Abdulghafor M. Al-Rozbayani

AL-Rafidain Journal of Computer Sciences and Mathematics, 2011, Volume 8, Issue 1, Pages 21-36
DOI: 10.33899/csmj.2011.163605

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.

##### Finite Differences Methods for solving Korteweg-de Vries-Burger's Equation

Ekhlass S. Al-Rawi; Almutasim Albaker

AL-Rafidain Journal of Computer Sciences and Mathematics, 2011, Volume 8, Issue 1, Pages 65-80
DOI: 10.33899/csmj.2011.163624

In this paper we solved the Korteweg-de Vries-Burger's equation numerically by finite difference methods, using two different schemes which are the Fully Implicit scheme and the Exponential finite difference scheme, because of the existence of the third derivative in the equation we suggested a treatment for the  numerical solution by parting the mesh grid into five regions, the first region represents the first boundary condition, the second one at the grid point , while the third represents the grid points , the fourth represents the grid point  and the fifth is for the second boundary condition .
We also studied the numerical stability, using Fourier (Von-Neumann) method for the two schemes which used in the solution on all mesh points to ensure the stability of the point which had been treated in the suggested style. Numerical results obtained by using these schemes are compared with existing analytical results. Excellent agreement was found between the exact solution and approximate solutions obtained by these schemes. The obtained approximate numerical solutions maintain good accuracy compared with exact solution specially for small values of the viscosity parameter.

##### Numerical Solution for Sine-Gordon System in One Dimension

Saad A. Manna; Haneen T. Jassim

AL-Rafidain Journal of Computer Sciences and Mathematics, 2010, Volume 7, Issue 2, Pages 47-59
DOI: 10.33899/csmj.2010.163896

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  .

##### The Finite Difference Methods for φ4 Klein-Gordon Equation

Saad A. Manaa; Laya Y. Hawell

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008, Volume 5, Issue 2, Pages 47-63
DOI: 10.33899/csmj.2008.163970

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.

##### Numerical Solution of Non-Linear Prey-Predator System using Finite Elements Method

Saad A. Manaa; Ahmed F. Qasem

AL-Rafidain Journal of Computer Sciences and Mathematics, 2007, Volume 4, Issue 2, Pages 113-133
DOI: 10.33899/csmj.2007.164020

A non-linear prey-predator system solved numerically by Galerkin method, and we compare these results with the results of Pius Peter Nyaanga 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 Finite Difference Methods for Hyperbolic – Parabolic Equations

Abbas Y. Al-Bayati; Saad A. Manaa; Ekhlass S. Al-Rawi

AL-Rafidain Journal of Computer Sciences and Mathematics, 2005, Volume 2, Issue 2, Pages 57-71
DOI: 10.33899/csmj.2005.164084

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.