In this paper, cubic trigonometric spline is used to solve nonlinear Volterra integral equations of second kind. Examples are illustrated to show the presented method’s efficiency and convenience.

In this paper, we obtain a complete characterization for the norm and the minimum norm attainment sets of bounded linear operators on a real Banach spaces at a vector in the unit sphere, using approximate 𝜖-Birkhoff-James orthogonality techniques. As an application of the results, we obtained a useful characterization of
bounded linear operators on a real Banach spaces. Also, using approximate 𝜖-Birkhoff -James orthogonality proved that a Banach space is a reflexive if and only if for any closed hyperspace of , there exists a rank one linear operator such that , for some vectors in and such that 𝜖 .Mathematics subject classification (2010): 46B20, 46B04, 47L05.

In this research, the influence of the distance factor on the optimal working
frequency (FOT) parameter has been studied theoretically for the ionosphere layer
over the Middle East Zone. The datasets of the (FOT) parameter have been
generated using the (VOACAP) model which considers as one of the recommended
modern international communication models that used to calculate the ionosphere
parameters. The calculations have been made for the connection links between the
capital Baghdad and many other locations that distributed on different distances and
directions over the Middle East region. The years (2011-2013) of the solar cycle 24
have been adopted to study the influence of the distance factor on the FOT
param

The solar photocatalytic degradation of diuron, which is one of the herbicides, has been studied by a solar pilot plant in heterogeneous solar photocatalysis with titanium dioxide. The pilot plant was made up of compound parabolic collectors specially designed for solar photocatalytic applications. The influence of different variables such as, H2O2 initial concentration, TiO2 initial concentration, and diuron initial concentration with their relationship to the degradation efficiency were studied. Hydrogen peroxide (H2O2) found to increase the rate of diuron degradation. The best removal efficiency of heterogeneous solar photocatalytic TiO2 system was found to be 46.65 % and for heterogeneous solar photocatalytic TiO2/ H2O2 system was fo

The DC electrical conductivity properties of Ge60Se40-xTex alloy with x = 0, 5, 10, 15 and 20). The samples were formed in the form of discs with the thickness of 0.25–0.30 cm and the diameter of 1.5 cm. Samples were pressed under a pressure of 6 tons per cm2 , using a ton hydraulic press. They were prepared after being pressed using a ton hydraulic press using a hydraulic press. Melting point technology use to preper the samples. Continuous electrical conductivity properties were recorded from room temperature to 475 K. Experimental data indicates that glass containing 15% Te has the highest electrical conductivity allowing maximum current through the sample compared to Lu with other samples. Therefore, it is found that the DC co

       In this paper, the classical continuous triple optimal control problem (CCTOCP) for the triple nonlinear parabolic boundary value problem (TNLPBVP) with state vector constraints (SVCs) is studied.  The solvability theorem for the classical continuous triple optimal control vector CCTOCV with the SVCs is stated and proved. This is done under suitable conditions. The mathematical formulation of the adjoint triple boundary value problem (ATHBVP) associated with TNLPBVP is discovered. The Fréchet derivative of the Hamiltonian" is derived.  Under suitable conditions, theorems of necessary  and sufficient conditions for the optimality of the TNLPBVP with the SVCs are stated and proved.