This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.

A numerical method is developed for calculation of the wake geometry and aerodynamic forces on two-dimensional airfoil under going an arbitrary unsteady motion in an inviscid incompressible flow (panel method). The method is applied to sudden change in airfoil incidence angle and airfoil oscillations at high reduced frequency. The effect of non-linear wake on the unsteady aerodynamic properties and oscillatory amplitude on wake rollup and aerodynamic forces has been studied. The results of the present method shows good accuracy as compared with flat plate and for unsteady motion with heaving and pitching oscillation the present method also shows good trend with the experimental results taken from published data. The method shows good result

In this paper, a computational method for solving optimal problem is presented, using indirect method (spectral methodtechnique) which is based on Boubaker polynomial. By this method the state and the adjoint variables are approximated by Boubaker polynomial with unknown coefficients, thus an optimal control problem is transformed to algebraic equations which can be solved easily, and then the numerical value of the performance index is obtained. Also the operational matrices of differentiation and integration have been deduced for the same polynomial to help solving the problems easier. A numerical example was given to show the applicability and efficiency of the method. Some characteristics of this polynomial which can be used for solvin

Removal of heavy metals from waste water has received a great deal of attention. The compare Cr
(VI) adsorption characteristics removing from wastewater by using thermally modified and non-modified
eggshells were examined

Recently, the financial mathematics has been emerged to interpret and predict the underlying mechanism that generates an incident of concern. A system of differential equations can reveal a dynamical development of financial mechanism across time. Multivariate wiener process represents the stochastic term in a system of stochastic differential equations (SDE). The standard wiener process follows a Markov chain, and hence it is a martingale (kind of Markov chain), which is a good integrator. Though, the fractional Wiener process does not follow a Markov chain, hence it is not a good integrator. This problem will produce an Arbitrage (non-equilibrium in the market) in the predicted series. It is undesired property that leads to erroneous conc

 In this paper, we introduce and discuss an algorithm for the numerical solution of some kinds of fractional integral and fractional integrodifferential equations. The algorithm for the numerical solution of these equations is based on iterative approach. The stability and convergence of the fractional order numerical method are described. Finally, some numerical examples are provided to show that the numerical method for solving the fractional integral and fractional integrodifferential equations is an effective solution method.

This work is concerned with the vibration attenuation of a smart beam interacting with fluid using proportional-derivative PD control and adaptive approximation compensator AAC. The role of the AAC is to improve the PD performance by compensating for unmodelled dynamics using the concept of function approximation technique FAT. The key idea is to represent the unknown parameters using the weighting coefficient and basis function matrices/vectors. The weighting coefficient vector is updated using Lyapunov theory. This controller is applied to a flexible beam provided with surface bonded piezo-patches while the vibrating beam system is submerged in a fluid. Two main effects are considered: 1) axial stretching of the vibrating beam that leads