Probability methods for the convergence of finite difference approximations to partial differential equations

Finite difference method for solving partial integro-differential equations

In this paper, we have introduced a new method for solving a class of the partial integro-differential equation with the singular kernel by using the finite difference method. First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results ob...

Finite Difference Approximations for Nonlinear First Order Partial Differential Equations

Classical solutions of nonlinear partial differential equations are approximated in the paper by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. This new approach to the numerical solving of nonlinear equations is generated by a linea...

Difference Approximations to Partial Differential Equations

Nonstandard finite difference schemes for differential equations

In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with ...

On Error Bounds of Finite Difference Approximations to Partial Differential Equations - Temporal Behavior and Rate of Convergence

The question of the role of numerically imposed boundary conditions in the solution of parabolic and hyperbolic PDE's has been with us for many years. Many investigators have studied the effect of boundary conditions on the stability of the overall scheme, (i.e., the ``inner algorithm''+boundary conditions), see for example [8, 11, 12, 17, 6, 20, 7, 13 16, 18, 19]. In this context stability imp...