In this paper, we present an approximate method for solving integro-differential equations of multi-fractional order by using the variational iteration method.
First, we derive the variational iteration formula related to the considered problem, then prove its convergence to the exact solution. Also we give some illustrative examples of linear and nonlinear equations.

       In this paper, we apply a new technique combined by a Sumudu transform and iterative method called the Sumudu iterative method for resolving non-linear partial differential equations to compute analytic solutions. The aim of this paper is to construct the efficacious frequent relation to resolve these problems. The suggested technique is tested on four problems. So the results of this study are debated to show how useful this method is in terms of being a powerful, accurate and fast tool with a little effort compared to other iterative methods.

Recently digital mammography is most widely used technology for early detection of breast cancer. The main diagnosing elements such as lesion or masses in digital mammograms are with law contrast. The purpose of this paper is to enhance the mammogram images by increasing its contrast. Different enhancement method are used for this purpose such as histogram equalization (HE), Contrast Limited Adaptive Histogram Equalization (CLAHE), Morphological, and Retinex. The Retinex method also implement by combining it with HE once, and with CLAHE to improve its performance. The experimental results show that using Retinex with CLAHE can produce an image with enhancement in contrast better than using it with HE method and better than other methods

In this paper, a computational method for solving optimal problem is presented, using indirect method (spectral methodtechnique) which is based on Boubaker polynomial. By this method the state and the adjoint variables are approximated by Boubaker polynomial with unknown coefficients, thus an optimal control problem is transformed to algebraic equations which can be solved easily, and then the numerical value of the performance index is obtained. Also the operational matrices of differentiation and integration have been deduced for the same polynomial to help solving the problems easier. A numerical example was given to show the applicability and efficiency of the method. Some characteristics of this polynomial which can be used for solvin

    The main purpose of the work is to apply a new method, so-called LTAM, which couples the Tamimi and Ansari iterative method (TAM) with the Laplace transform (LT). This method involves solving a problem of non-fatal disease spread in a society that is assumed to have a fixed size during the epidemic period. We apply the method to give an approximate analytic solution to the nonlinear system of the intended model. Moreover, the absolute error resulting from the numerical solutions and the ten iterations of LTAM approximations of the epidemic model, along with the maximum error remainder, were calculated by using MATHEMATICA® 11.3 program to illustrate the effectiveness of the method.

The aim of the current  study is to determine the elastic properties  of carbonate rocks using ultrasonic method.  Forty rock samples of  Anah formation  were collected at  different depths from  four wells drilled at the study area . The relationship between wave velocities and elastic properties of rocks was defined. Regression analyses to define these relations were applied. The results indicate that the elastic properties of the rocks show a linear relationship with both P- and S-wave velocities. The best relationship was obtained between both Young's modulus and Shear modulus with Vs in the determination of the coefficient ( R²  ), with values of 0.91 and 0.94,  respectively.&n

In this paper, we present an approximate analytical and numerical solutions for the differential equations with multiple delay using the extend differential transform method (DTM). This method is used to solve many linear and non linear problems.

In this paper, the series solutions of a non-linear delay integral equations are considered by a modified approach of homotopy analysis method (MAHAM). We split the function   into infinite sums. The outcomes of the illustrated examples are included to confirm the accuracy and efficiency of the MAHAM. The exact solution can be obtained using special values of the convergence parameter.

In this paper Volterra Runge-Kutta methods which include: method of order two and four will be applied to general nonlinear Volterra integral equations of the second kind. Moreover we study the convergent of the algorithms of Volterra Runge-Kutta methods. Finally, programs for each method are written in MATLAB language and a comparison between the two types has been made depending on the least square errors.

In this work ,the modified williamos-Hall method was used to analysis the x-ray diffraction lines for powder of magnesium oxide nanoparticles (Mgo) .and for diffraction lines (111),(200),(220),(311) and (222).where by used special programs such as origin pro Lab and Get Data Graph ,to calculate the Full width at half maximum (FWHM) and integral breadth (B) to calculate the area under the curve for each of the lines of diffraction .After that , by using modified Williamson –Hall equations to determin the values of crystallite size (D),lattice strain (ε),stress( σ ) and energy (U) , where was the results are , D=17.639 nm ,ε =0.002205 , σ=0.517 and U=0.000678 respectively. And then using the scherrer method can by calculated the crystal