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.

The non-dimensional Navier stokes equations in rotational form for the boundary layer flow is solved using direct numerical simulation. The length scale and velocity scale of the base flow the boundary layer thickness and the inviscid velocity outside the layer are used as the length and velocity scales at the inlet boundary of the computational domain are used as two characteristics to define the flow Reynolds number. The governing equations are discritised in the streamwise direction using a sixth order COMPACT FINITE DIFFERENCE scheme, and in the cross-direction using a mapped COMPACT FINITE DIFFERENCE scheme. An algebraic mapping is used to map the physical domain into the computational domain .The COMPACT third order of Runge-kutta method is used for the time-advancement purpose. The convective outflow boundary condition is employed to create a non-reflective type boundary condition at the outlet. The simulation resuhs of this flow were compared by Blasius solution that show accuracy program. In this study, also, characteristics of laminar boundary layer flow verification for accuracy program with divided the lengths and velocity by length of plane and uniform velocity of environment respectively. Profiles and contours of velocity and vorticity have planed in flow arrow and self- similar have seen.

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.