In this paper, the continuous classical boundary optimal control problem (CCBOCP) for triple linear partial differential equations of parabolic type (TLPDEPAR) with initial and boundary conditions (ICs & BCs) is studied. The Galerkin method (GM) is used to prove the existence and uniqueness theorem of the state vector solution (SVS) for given continuous classical boundary control vector (CCBCV). The proof of the existence theorem of a continuous classical boundary optimal control vector (CCBOCV) associated with the TLPDEPAR is proved. The derivation of the Fréchet derivative (FrD) for the cost function (CoF) is obtained. At the end, the theorem of the necessary conditions for optimality (NCsThOP) of this problem is stated and prov

In this research we have been studied the 3rd order spherical aberration for an optical system consisted of obscured circular aperture with non central circular obscuration through the calculation of point spread function (P.S.F) in presence of the obscuration in the center and comparing the obtained results with that results of moving obscuration far away from the center, where the results showed significant improvement for(P.S.F) value. The study was done of different obscurities ratios in addition to the different 3rd order spherical aberration values (W40=0.25 ,0.5 ,0.75 ,1 ).

This paper is attempt to study the nonlinear second order delay multi-value problems. We want to say that the properties of such kind of problems are the same as the properties of those with out delay just more technically involved. Our results discuss several known properties, introduce some notations and definitions. We also give an approximate solution to the coined problems using the Galerkin's method.

An efficient combination of Adomian Decomposition iterative technique coupled with Laplace transformation to solve non-linear Random Integro differential equation (NRIDE) is introduced in a novel way to get an accurate analytical solution. This technique is an elegant combination of theLaplace transform, and the Adomian polynomial. The suggested method will convert differential equations into iterative algebraic equations, thus reducing processing and analytical work. The technique solves the problem of calculating the Adomian polynomials. The method’s efficiency was investigated using some numerical instances, and the findings demonstrate that it is easier to use than many other numerical procedures. It has also been established that (LT

In this paper, the dynamic behaviour of the stage-structure prey-predator fractional-order derivative system is considered and discussed. In this model, the Crowley–Martin functional response describes the interaction between mature preys with a predator.  e existence, uniqueness, non-negativity, and the boundedness of solutions are proved. All possible equilibrium points of this system are investigated.  e su‰cient conditions of local stability of equilibrium points for the considered system are determined. Finally, numerical simulation results are carried out to con‹rm the theoretical results.

     An Alternating Directions Implicit method is presented to solve the homogeneous heat diffusion equation when the governing equation is a bi-harmonic equation (X) based on Alternative Direction Implicit (ADI). Numerical results are compared with other results obtained by other numerical (explicit and implicit) methods. We apply these methods it two examples (X): the first one, we apply explicit when the temperature .

This study relates to  the estimation of  a simultaneous equations system for the Tobit model where the dependent variables  ( )  are limited, and this will affect the method to choose the good estimator. So, we will use new estimations methods  different from the classical methods, which if used in such a case, will produce biased and inconsistent estimators which is (Nelson-Olson) method  and  Two- Stage limited dependent variables(2SLDV) method  to get of estimators that hold characteristics the good estimator .

That is , parameters will be estim

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.