We present a solution method for a general Riccati differential equation by imposing relationships between

the coefficients of the Riccati differential equation and explaining them through proofs and examples, we can find the

general solution to the various forms of Riccati's equation by integration directly after transforming it into a separable

differential equation.

The Hill cipher is considered the first encryption technology that has some

advantages in encrypting data. The aim of this research is to modify the Hill

cipher to process letters of the Arabic alphabet. We have proven the success of

the Hill encryption on letters of the Arabic alphabet and the research was

supported by a number of different examples which were then processed using

MATLAB.

Hill Cipher is one of the symmetric key algorithms in cryptography, we discussed Encryption

and decryption for the alphabet Arabic by Hill cipher encryption and descriptions using the

invertible matrix as a key. In this study, we introduce a new method on the Hill Cipher for the

alphabet Arabic using a rectangular matrix as a key.

Derivative-free optimization techniques are widely used for solving optimization problems. The

focus in this work is given to some variants of Nelder-Mead and particle swarm methods to tackle two

unconstrained optimization problems originated from optimal control, namely the pole assignment problem for

discrete and continuous-time positive systems. we present the Nelder-Mead and Particle Swarm optimization

methods to solve problems. Moreover comparing our results with benchmarks in the literature.

In this work, we consider Nelder-Mead method to tackle the output

feedbackeigenvalue assignment problem for discrete–time control

systems. First, we reformulate this problem as a constrained minimization

problem. Then by applying apenalty-method approach we find a local

solution to the problem using Nelder-Meadmethod. The performance of

the method is illustrated through some test problemsfrom the literature.

In this paper, the discrete–time periodic static output feedback control design problem is considered. A

hybrid conjugate gradient methods are analyzed and studied to tackle an equivalent optimization problem of this

optimal control problem. Finally, the proposed algorithms are tested numerically through several test problems

from the benchmark collection.

In this article, we consider the output feedback eigenvalue assignment problem for

continuous and discrete–time control systems. This problem is formulated as unconstrained matrix optimization problems and tackled by the Nelder–Mead simplex

method and particle swarm optimization method. The two methods are extended to

compute the approximate solutions of the eigenvalue assignment problem for the particular cases of decentralized and periodic control systems. The performance of the

methods is demonstrated numerically on several test problems from the benchmark

collection [22] as well as other test examples from the system and control literature.

Te output feedback eigenvalue assignment problem for discrete-time systems is considered. Te problem is formulated frst as

an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to fnd a local

solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a

logarithmic barrier method is proposed for fnding the local solution. Te conjugate gradient method is further extended to tackle

the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. Te

performance of the methods is illustrated through various test examples.

We are concerned here with the existence of at least one positive

continuous solution of the quadratic integral equation of fractional orders

x(t) = a(t) + Iαf(t; x(t)) · Iβg(t; x(t)); α; β 2 (0; 1]:

The maximal and minimal solutions are also proved. Some applications are

given.