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.

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.

This paper aims to apply and investigate the COMPACT FINITE DIFFERENCE METHODs for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of COMPACT FINITE DIFFERENCE. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the convergence analysis of the proposed approach for the fractional and nonfractional cases and prove that the METHODs are convergent under some suitable conditions. Examples are also given to illustrate the efficiency of our METHOD compared to other METHODs.

The dimensionless form of Navier-Stokes equations for two dimensional jet flows are solved using direct numerical simulation. The length scale and the velocity scale of jet flow at the inlet boundary of computational domain are used as two characteristics to define the jet Reynolds number. These two characteristics are jet half-width and centerline velocity. Governing equations are discretized in streamwise and cross stream directions using a sixth order COMPACT FINITE DIFFERENCE scheme and a mapped COMPACT FINITE DIFFERENCE METHOD, respectively. Cotangent mapping of y=-b cot (pz) is used to relate the physical domain of y to the computational domain of z. The COMPACT third order Runge-Kutta METHOD is used for time-advancement of the simulation. convective outflow boundary condition is employed to create a non-reflective type boundary condition at the outlet. An inviscid Stuart flow and a completely viscose solution of Navier Stokes equations are used for the verification of numerical simulations. Results for perturbed jet flow in self-similar coordinates were also investigated which indicate that the time-averaged statistics for velocity, vorticity, turbulence intensities and Reynolds stress distribution tend to collapse on top of each other at flow downstream locations.

This paper presents an efficient METHOD for solving free vibration problems for an Euler-Bernoulli beam under various supporting conditions. The METHOD is based on the implementation of the sixth-order COMPACT FINITE DIFFERENCE METHOD (CFDM) for discretizing the governing differential equation to obtain the natural frequencies of beam corresponding to two commonly used boundary conditions namely simply supported-simply supported and clamped-free. A very good agreement is found between the natural frequencies obtained using the sixth-order COMPACT FINITE DIFFERENCE scheme and exact natural frequencies, which confirms the validity of the present sixth-order discretization.

The main objective of this paper is to introduce the fourth and sixth-order COMPACT FINITE DIFFERENCE METHODs for solving anti-periodic boundary value problems. COMPACT FINITE DIFFERENCE formulas can approximate the derivatives of a function more accurately than the standard FINITE DIFFERENCE formulas for the same number of grid points. The convergence analysis of the proposed METHOD is also investigated. This analysis shows how the error between the approximate and exact solutions decreases as the grid space is reduced.  To validate the proposed METHOD's accuracy and efficiency, some computational experiments are provided. Moreover, a comparison is performed between the standard and COMPACT FINITE DIFFERENCE METHODs.  The experiments indicate that the COMPACT FINITE DIFFERENCE METHOD is more accurate and efficient than the standard one.