This paper presents seven modified Adomian Decomposition Method (ADM) techniques for efficiently solving initial value problems, especially those involving non-homogeneous and nonlinear differential equations. While the classical ADM is effective for linear homogeneous cases, it has difficulties solving more complex problems. The proposed modifications—from MADM1 to MLADM—include Maclaurin and Taylor expansions, Laplace transforms, and single-step iterations.• These modifications enhance convergence, reduce complexity, and improve accuracy.• Each method offers specific advantages, such as accelerating convergence (MADM2, RADM4), simplifying computation (TSADM5), and achieving higher accuracy (MLADM).• Numerical examples confirm th

We investigate the interaction of proton with a solid target, describing the wake effects by taking fitted parameters with experimental values of energy loss function ELF for copper using the dielectric function of random phase approximation (RPA). The results exhibited a damped oscillatory behavior in the longitudinal direction behind the projectile. In addition, the wake potential becomes asymmetric around the z-axis with proton velocity values higher than Fermi velocity, as well as it depends on the position of projectile in cylindrical coordinates.

We propose two simple, rapid, and convenient spectrophotometric methods which are described for the determination of cephalexin in bulk and its pharmaceutical preparations. They are based on the measurement of the flame atomic emission of potassium ion (in the first method) and colorimetric determination of the green colored solution at 610 nm formed after the reaction of cephalexin with potassium permanganate as an oxidant agent (in the second method) in basic medium. The working conditions of the methods are investigated and optimized. Beer's law plot shows a good correlation in the concentration range of 5-40?g ml-1. The detection limits are 2.573,2.814 ?g ml-1 for the flame emission photometric method and 1.844,2.016 ?g ml-1 for colo

Background: Prophylaxis methods are used to mechanically remove plaque and stain from tooth surfaces; such methods give rise to loss of superficial structure and roughen the surface of composites as a result of their abrasive action. This study was done to assess the effect of three polishing systems on surface texture of new anterior composites after storage in artificial saliva. Materials and methods: A total of 40 Giomer and Tetric®N-Ceram composite discs of 12 mm internal diameter and 3mm height were prepared using a specially designed cylindrical mold and were stored in artificial saliva for one month and then samples were divided into four groups according to surface treatment: Group A (control group):10 specimens received no surfa

Topology and its applications occupy the interest of many researching centers in the advanced world. From this point of view and because the near open sets play a very important role in general topology and they are now the research topics of many topologists worldwide and its sets doesn’t enter in fibrewise topology yet. Therefore, we use some of the near open sets to be model for introduce results and new spaces in fibrewise topological spaces. Also, there is a very important role of closure operators in constructing a topological spaces, so we introduce a new closure operators on the power set of vertices on graphs and conclusion theorems and new spaces from it. Furthermore, we discuss the relationships of connectedness between some ty