In this study, we have considered the linear classes of differential-(difference), integro-differential-(difference) and integral equations by constituting a generalized form, which contains proportional delay, difference, differentiable difference or delay. To solve the generalized form numerically, we use the efficient matrix technique based on Dickson polynomials with the parameter-[Formula: see text] along with the collocation points. We also encode the useful computer program for susceptibility of the technique. The residual error analysis is implemented by using the residual function. The consistency of the technique is analyzed. Also, the numerical results illustrated in tables and figures are compared.

Abstract:In this article, an improved collocation method based on the Morgan-Voyce polynomials for the approximates solution of multi-pantograph equations is introduced. The method is based upon the improvement of Morgan-Voyce polynomial solutions with the aid of the residual error function. First, the Morgan-Voyce collocation method is applied to the multi-pantograph equations and then Morgan-Voyce polynomial solutions are obtained. Second, an error problem is constructed by means of the residual error function and this error problem is solved by using the Morgan-Voyce collocation method. By summing the Morgan-Voyce polynomial solutions of the original problem and the error problem, we have the improved Morgan-Voyce polynomial solutions. When the exact solution of problem is not known, the absolute error can then be approximately computed by the Morgan-Voyce polynomial solution of the error problem. Numerical examples that the pertinent features of the method are presented. We have applied all of the numerical computations on computer using a program written in MATLAB.

Large eddy simulation (LES) seeks to predict the dynamics of the organized structures in the flow, that is, local spatial averages u ̄ $\bar{u}$ of the velocity u of the fluid. Although LES has been extensively used to model turbulent flows, very often, the model has difficulty predicting turbulence generated by interactions of a flow with a boundary. A critical problem in LES is to find appropriate boundary conditions for the flow averages, which depend on the behavior of the unknown flow near the wall. In the light of the works of Navier and Maxwell, we use boundary conditions on the wall. We compute the appropriate friction coefficient β for channel flows and investigate its asymptotic behavior as the averaging radius δ → 0 and as the Reynolds number Re → ∞. No-slip conditions are recovered in the first limit, and free-slip conditions are recovered in the second limit. This study is not intended to develop new theories of the turbulent boundary layer; we use available boundary layer theories to improve numerical boundary conditions for flow averages.

In this paper, a numerical matrix method is used to solve the systems of high-order linear Fredholm integro-differential equations with variable coefficients under mixed conditions. The technique consists of collocation points and the Morgan-Voyce polynomials. The residual error functions of numerical solutions of the method are also presented. Firstly, the approximate solutions are formed and secondly, an error problem is constituted in favor of the residual error function. The numerical solutions are computed for this error problem by using the present method. The approximate solutions of the original problem and the error problem are the corrected Morgan-Voyce polynomial solutions. When the exact solutions of the problem are not known, the absolute errors can be approximately constructed through the approximate solutions of the error problem. Numerical examples are included to demonstrate the validity and the applicability of the technique, and also the results are compared with the different methods. All numerical computations have been performed using MATLAB v7.11.0 (R2010b).

It is difficult and essential to determine appropriate boundary conditions for the flow averages because they depend on the behavior of the unknown flow near the wall. Large-eddy simulation (LES) is one of the promising approaches. LES estimates local spatial averages u of the velocity u of the fluid. The main problem is modeling near-wall turbulence in complex geometries. Inspired by the works of Navier and Maxwell, the boundary conditions are developed on the wall. In this study, the appropriate friction coefficient for 2-D laminar flows is computed, and existing boundary layer theories are used to improve numerical boundary conditions for flow averages. The slip with friction and penetration with resistance boundary conditions are considered. Numerical tests on two-dimensional channel flow across a step using this boundary condition on the top and bottom wall and the step are performed.