In the present study, high order COMPACT FINITE DIFFERENCE methods are used to solve one-dimensional Bratutype equations numerically. The convergence analysis of the methods is discussed and it is shown that the theoretical order of the methods are consistent with their numerical rate of convergence. The maximum absolute errors in the solution at grid points are calculated and it is shown that the presented methods are efficient and applicable for lower and upper solutions.

In this study, we solve the Fokker-Planck equation by a COMPACT FINITE DIFFERENCE method, By the FINITE DIFFERENCE method the computation of Fokker-Planck equation is reduced to a system of ordinary differential equations. Two different methods, boundary value method and cubic C1-spline collocation method, for solving the resulting system are proposed. Both methods have fourth-order accuracy in time variable. By the boundary value method, some pointwise approximate solutions are only obtained. But, C1-spline method gives a closed-form approximation in each space step, too. Illustrative examples are included to demonstrate the validity and efficiency of the methods. A comparison is made with existing results.

In this paper, we propose a new method for solving the stochastic advection-diffusion equation of Ito type. In this work, we use a COMPACT FINITE DIFFERENCE approximation for discretizing spatial derivatives of the mentioned equation and semi-implicit Milstein scheme for the resulting linear stochastic system of differential equation. The main purpose of this paper is the stability investigation of the applied method. Finally, some numerical examples are provided to show the accuracy and efficiency of the proposed technique.

This paper is devoted to applying the sixth-order COMPACT FINITE DIFFERENCE approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the COMPACT FINITE DIFFERENCE method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.