On the Solution of the Diffusion Equations by Numerical Methods

1966 ◽  
Vol 88 (4) ◽  
pp. 421-427 ◽  
Author(s):  
H. Z. Barakat ◽  
J. A. Clark
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Present Method ◽  
Finite Difference Methods ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Procedure ◽  
Implicit Methods ◽  
Difference Methods ◽  
Explicit Finite Difference

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

scholarly journals Finite Difference Methods for Option Pricing under Lévy Processes: Wiener-Hopf Factorization Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Oleg Kudryavtsev
Keyword(s):  
Finite Difference ◽  
Lévy Processes ◽  
Finite Difference Methods ◽  
Factorization Method ◽  
Levy Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Pricing Options ◽  
Black Scholes ◽  
Difference Methods

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

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