Minimising Truncation Error in Finite Difference Approximations to Ordinary Differential Equations

Rate of convergence of finite difference approximations for degenerate ordinary differential equations

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations: 1 2 σ2(x)u′′(x) + b(x)u′(x)− u(x) = −f(x), x ∈ R, where σ is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain c...

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 ...

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

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...