In this paper a modified approach have been used to find the approximate solution of ordinary delay differential equations with constant delay using the collocation method based on Bernstien polynomials.

Rehabilitation robots are widely recognized as vital for restoring motor function in patients with lower-limb impairments. A Modified Fractional-Order Proportional-Integral-Derivative (MFOPID) controller is proposed to improve trajectory tracking of a 2-DoF Lower Limb Rehabilitation Exoskeleton Robot (LLRER). The classical FOPID is augmented with a modified control formulation by which steady-state error is reduced and the transient response is sharpened. Controller gains and fractional orders were tuned offline using a hybrid metaheuristic Improved Elk Herd Optimization hybridized with Grey Wolf and Multi-Verse Optimization algorithms (IElk-GM) so that exploration and exploitation are balanced. Superiority over the classical FOPID

The Boltzmann equation has been solved using (EEDF) package for a pure sulfur hexafluoride (SF6) gas and its mixtures with buffer Helium (He) gas to study the electron energy distribution function EEDF and then the corresponding transport coefficients for various ratios of SF6 and the mixtures. The calculations are graphically represented and discussed for the sake of comparison between the various mixtures. It is found that the various SF6 – He content mixtures have a considerable effect on EEDF and the transport coefficients of the mixtures

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

 In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.

 In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.

The Wang-Ball polynomials operational matrices of the derivatives are used in this study to solve singular perturbed second-order differential equations (SPSODEs) with boundary conditions. Using the matrix of Wang-Ball polynomials, the main singular perturbation problem is converted into linear algebraic equation systems. The coefficients of the required approximate solution are obtained from the solution of this system. The residual correction approach was also used to improve an error, and the results were compared to other reported numerical methods. Several examples are used to illustrate both the reliability and usefulness of the Wang-Ball operational matrices. The Wang Ball approach has the ability to improve the outcomes by minimi