In this work, the modified Lyapunov-Schmidt reduction is used to find a nonlinear Ritz approximation of Fredholm functional defined by the nonhomogeneous Camassa-Holm equation and Benjamin-Bona-Mahony. We introduced the modified Lyapunov-Schmidt reduction for nonhomogeneous problems when the dimension of the null space is equal to two.  The nonlinear Ritz approximation for the nonhomogeneous Camassa-Holm equation has been found as a function of codimension twenty-four.

     In this article, we aim to define a universal set consisting of the subscripts of the fuzzy differential equation (5) except the two elements  and , subsets of that universal set are defined according to certain conditions. Then, we use the constructed universal set with its subsets for suggesting an analytical method which facilitates solving fuzzy initial value problems of any order by using the strongly generalized H-differentiability. Also, valid sets with graphs for solutions of fuzzy initial value problems of higher orders are found.

        In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using  Mathcad 15.and graphic in Matlab R2015a.

The dye–semiconductor interface between N749 sensitized and zinc semiconductor (ZnSe) has been investigated and studied according to quantum transition theory with focusing on the electron transfer processes from the N749 sensitized (donor) to the ZnSe semiconductor (acceptor). The electron transfer rate constant and the orientation energy were studied and evaluated depended on the polarity of solvents according to refractive index and dielectric constant coefficient of solvents and ZnSe semiconductor. Attention focusing on the influence of orientation energies on the behavior of electron transfer rate constant. Differentdata of rate constant was discussion with orientation energy and effective driving energy for N749-ZnSe system.

In this work, Elzaki transform (ET) introduced by Tarig Elzaki is applied to solve linear Volterra fractional integro-differential equations (LVFIDE). The fractional derivative is considered in the Riemman-Liouville sense. The procedure is based on the application of (ET) to (LVFIDE) and using properties of (ET) and its inverse. Finally, some examples are solved to show that this is computationally efficient and accurate.

       In this work, Elzaki transform (ET) introduced by Tarig Elzaki is applied to solve linear Volterra fractional integro-differential equations (LVFIDE). The fractional derivative is considered in the Riemman-Liouville sense. The procedure is based on the application of (ET) to (LVFIDE) and using properties of (ET) and its inverse. Finally, some examples are solved to show that this is computationally efficient and accurate.

Watermarking operation can be defined as a process of embedding special wanted and reversible information in important secure files to protect the ownership or information of the wanted cover file based on the proposed singular value decomposition (SVD) watermark. The proposed method for digital watermark has very huge domain for constructing final number and this mean protecting watermark from conflict. The cover file is the important image need to be protected. A hidden watermark is a unique number extracted from the cover file by performing proposed related and successive operations, starting by dividing the original image into four various parts with unequal size. Each part of these four treated as a separate matrix and applying SVD

This paper assesses the impact of changes and fluctuations in bank deposits on the money supply in Iraq. Employing  the research constructs an Error Correction Model (ECM) using  monthly time series data from 2010 to 2015. The analysis begins with  the Phillips-Perron unit root test to ascertain the stationarity of the  time series and the Engle and Granger cointegration test to examine  the existence of a long-term relationship. Nonparametric regression functions are estimated using two methods: Smoothing Spline and M-smoothing. The results indicate that the  M-smoothing approach is the most effective, achieving the shortest adjustment period and the highest adjustment ratio for short-term disturbances, thereby facilitating a return