Keywords : Finite Differences Methods


Numerical Solution and Stability Analysis of the Sine-Gordon Equation

Saad Abdullah Manna; Norjan Hasan Juma

AL-Rafidain Journal of Computer Sciences and Mathematics, 2007, Volume 4, Issue 1, Pages 39-56
DOI: 10.33899/csmj.2007.164002

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 .
 

Numerical Analysis of Fisher Equation

Saad A. Manna; Ahmed F. Qassim

AL-Rafidain Journal of Computer Sciences and Mathematics, 2006, Volume 3, Issue 1, Pages 85-100
DOI: 10.33899/csmj.2006.164046

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.