Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and  engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function  is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

    In this work, we use the explicit and the implicit finite-difference methods to solve the nonlocal problem that consists of the diffusion equations together with nonlocal conditions. The nonlocal conditions for these partial differential equations are approximated by using the composite trapezoidal rule, the composite Simpson's 1/3 and 3/8 rules. Also, some numerical examples are presented to show the efficiency of these methods.

In this paper the Galerkin method is used to prove the existence and uniqueness theorem for the solution of the state vector of the triple linear elliptic partial differential equations for fixed continuous classical optimal control vector. Also, the existence theorem of a continuous classical optimal control vector related with the triple linear equations of elliptic types is proved. The existence of a unique solution for the triple adjoint equations related with the considered triple of the state equations is studied. The Fréchet derivative of the cost function is derived. Finally the theorem of necessary conditions for optimality of the considered problem is proved.

Background; Hyperlipoproteinemia (HLP)  are divided in primary and secondary subtypes. Primary HLP is usually due to genetic causes. Secondary HLP is resulting from another underlying disorder such as diabetes mellitus that leads to alterations in plasma lipid and lipoprotein metabolism, HLP may be idiopathic .

Patients and Methods; Ninety male individuals (age 30-45)years were enrolled in this study which were divided into three groups as follows:- (G1) consist of 30 healthy male individuals as a control group, (G2) consist of  30 male patients with (HLP) without any other disease  (15 of them were HLP1,the other were HLP11), (G3) consist of 30 male patients with DM2 without any o

      In this paper, a new class of ordinary differential equations is designed for some functions such as probability density function, cumulative distribution function, survival function and hazard function of power function distribution, these functions are used of the class under the study. The benefit of our work is that the equations ,which are generated from some probability distributions, are used to model and find  the  solutions  of problems in our lives, and that the solutions of these equations are a solution to these problems, as the solutions of the equations under the study are the closest and the most reliable to reality. The existence and uniqueness of solutions the obtained equations in the current study are dis

     Optimization is the task of minimizing or maximizing an objective function f(x) parameterized by x. A series of effective numerical optimization methods have become popular for improving the performance and efficiency of other methods characterized by high-quality solutions and high convergence speed. In recent years, there are a lot of interest in hybrid metaheuristics, where more than one method is ideally combined into one new method that has the ability to solve many problems rapidly and efficiently. The basic concept of the proposed method is based on the addition of the acceleration part of the Gravity Search Algorithm (GSA) model in the Firefly Algorithm (FA) model and creating new individuals. Some stan

     The focus of this article is to add a new class of rank one of  modified Quasi-Newton techniques to solve the problem of unconstrained optimization by updating the inverse Hessian matrix with an update of rank 1, where a diagonal matrix is the first component of the next inverse Hessian approximation, The inverse Hessian matrix is  generated by the method proposed which is symmetric and it satisfies the condition of modified quasi-Newton, so the global convergence is retained. In addition, it is positive definite that  guarantees the existence of the minimizer at every iteration of the objective function. We use  the program MATLAB to solve an algorithm function to introduce the feasibility of

       In this work, we prove that the triple linear partial differential equations (PDEs) of elliptic type (TLEPDEs) with a given classical continuous boundary control vector (CCBCVr) has a unique "state" solution vector (SSV)  by utilizing the Galerkin's method (GME). Also, we prove the existence of a classical continuous boundary optimal control vector (CCBOCVr) ruled by the TLEPDEs. We study the existence solution for the triple adjoint equations (TAJEs) related with the triple state equations (TSEs). The Fréchet derivative (FDe) for the objective function is derived. At the end we prove the necessary "conditions" theorem (NCTh) for optimality for the problem.

In this paper, three approximate methods namely the Bernoulli, the Bernstein, and the shifted Legendre polynomials operational matrices are presented to solve two important nonlinear ordinary differential equations that appeared in engineering and applied science. The Riccati and the Darcy-Brinkman-Forchheimer moment equations are solved and the approximate solutions are obtained. The methods are summarized by converting the nonlinear differential equations into a nonlinear system of algebraic equations that is solved using Mathematica®12. The efficiency of these methods was investigated by calculating the root mean square error (RMS) and the maximum error remainder (𝑀𝐸𝑅n) and it was found that the accuracy increases with increasi

This paper aims to study the fractional differential systems arising in warm plasma, which exhibits traveling wave-type solutions. Time-fractional Korteweg-De Vries (KdV) and time-fractional Kawahara equations are used to analyze cold collision-free plasma, which exhibits magnet-acoustic waves and shock wave formation respectively. The decomposition method is used to solve the proposed equations. Also, the convergence and uniqueness of the obtained solution are discussed. To illuminate the effectiveness of the presented method, the solutions of these equations are obtained and compared with the exact solution. Furthermore, solutions are obtained for different values of time-fractional order and represented graphically.