Abstract
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.
Highlights
It is probably not an overstatement to say that almost all partial differential equations (PDEs) that arise in a practical setting are solved numerically on a computer
The basic idea of the method of finite differences is to cast the continuous problem described by the PDE and auxiliary conditions into a discrete problem that can be solved by a computer in finitely many steps
We concluded from the comparison between the two schemes that the explicit scheme is easier and has faster convergence than the Crank
Summary
It is probably not an overstatement to say that almost all partial differential equations (PDEs) that arise in a practical setting are solved numerically on a computer. Since the development of high-speed computing devices, the numerical solution of PDEs has been in active state with the invention of new algorithms and the examination of the underlying theory This is one of the most active areas in applied mathematics and it has a great impact on science and engineering because of the ease and efficiency it has shown in solving even the most complicated problems. The numerical solution of Huxley equation by using two finite difference methods and stability analysis of these two methods are analyzed. Equation (12) is employed to create ( j + 1)th row across the grid, assuming that approximations in the jth row are known Notice that this formula explicitly gives the value ui, j+1 in terms of ui−1, j , ui, j , and ui+1, j. This precisely the conditions imposed on the explicit scheme to be stable
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