هناك دائما حاجة إلى طريقة فعالة لتوليد حل عددي أكثر دقة للمعادلات التكاملية ذات النواة المفردة أو المفردة الضعيفة لأن الطرق العددية لها محدودة. في هذه الدراسة ، تم حل المعادلات التكاملية ذات النواة المفردة أو المفردة الضعيفة باستخدام طريقة متعددة حدود برنولي. الهدف الرئيسي من هذه الدراسة هو ايجاد حل تقريبي لمثل هذه المشاكل في شكل متعددة الحدود في سلسلة من الخطوات المباشرة. أيضا ، تم افتراض أن مقام النواة

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 paper the modified trapezoidal rule is presented for solving Volterra linear Integral Equations (V.I.E) of the second kind and we noticed that this procedure is effective in solving the equations. Two examples are given with their comparison tables to answer the validity of the procedure.

The aim of this paper is to propose a reliable iterative method for resolving many types of Volterra - Fredholm Integro - Differential Equations of the second kind with initial conditions. The series solutions of the problems under consideration are obtained by means of the iterative method. Four various problems are resolved with high accuracy to make evident the enforcement of the iterative method on such type of integro differential equations. Results were compared with the exact solution which exhibits that this technique was compatible with the right solutions, simple, effective and easy for solving such problems. To evaluate the results in an iterative process the MATLAB is used as a math program for the calculations.

In this paper, a method based on modified adomian decomposition method for solving Seventh order integro-differential equations (MADM). The distinctive feature of the method is that it can be used to find the analytic solution without transformation of boundary value problems. To test the efficiency of the method presented two examples are solved by proposed method.

Aromaticity reversals between the electronic ground (S0) and low-lying singlet (S1, S2) and triplet (T1, T2, T3) states of naphthalene and anthracene are investigated by calculating the respective off-nucleus isotropic magnetic shielding distributions using complete-active-space self-consistent field (CASSCF) wavefunctions involving gauge-including atomic orbitals (GIAOs). The shielding distributions around the aromatic S0, antiaromatic S1 (1Lb), and aromatic S2 (1La) states in naphthalene are found to resemble the outcomes of fusing together the respective S0, S1, and S2 shielding distributions of two benzene rings. In anthracene, 1La is lower in energy than 1Lb, and as a result, the S1 state becomes aromatic, and the S2 state becomes anti

Pure and Fe-doped zinc oxide nanocrystalline films were prepared
via a sol–gel method using -
C for 2 h.
The thin films were prepared and characterized by X-ray diffraction
(XRD), atomic force microscopy (AFM), field emission scanning
electron microscopy (FE-SEM) and UV- visible spectroscopy. The
XRD results showed that ZnO has hexagonal wurtzite structure and
the Fe ions were well incorporated into the ZnO structure. As the Fe
level increased from 2 wt% to 8 wt%, the crystallite size reduced in
comparison with the pure ZnO. The transmittance spectra were then
recorded at wavelengths ranging from 300 nm to 1000 nm. The
optical band gap energy of spin-coated films also decreased as Fe
doping concentra

In this paper, third order non-polynomial spline function is used to solve 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of this method, and to compare the computed results with other known methods.

        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.