In this work, the exact operational matrices for shifted Pell polynomials are achievable; so one can integrate and product the vector of basic functions s. The general form of the matrix of integration P is established, the dual matrix of integration Q is derived with general formulation, and the general form of the matrix derived from the product of two shifted Pell polynomials has been given. This idea is implemented on shifted Pell basis vector. Using such exact matrices, then the resident function of the equation is reached which can be written as R.P(x), where R is an algebraic equation vector and P(x) is the shifted Pell basis vector. The presented matrices can be utilized to find the approximate solution of differential equations, integral equation and the calculus of variations problems. An investigation for the convergence and error analysis of the proposed shifted Pell expansion is performed. Numerical treatment for problems in physics are included in this work to demonstrate the accuracy, easy to implement as well as accurate and satisfactory results with a small number of shifted Pell basis. Using operational matrices and the spectral technique are used together for solving Lane-Emden equation.

The aim of this paper is to state and prove the uniform convergence theorem and accuracy estimation for shifted fourth kind Chebyshev wavelets.

In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms of their original expansion coefficients, is also included in the present article. The main advantage of the presented formulas is that the new properties of Pell polynomials in two variables greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion. A direct spectral method along with the presented two variables Pell polynomials is proposed to solve the partial differential equation. Illustrated examples are included to demonstrate the validity of the technique.

Background: Caffeine boosts metabolism and the neurological system. When extreme weakness or sleepiness occurs, it is used recreationally and medicinally to reduce physical and mental fatigue. Caffeine initially stimulates the central nervous system, increasing intellect, speed, accuracy, focus, and coordination. Objective: To evaluate how caffeine nanoparticles affect potassium, calcium, zinc, and magnesium levels, in addition, the anti-bacterial activity of the samples has been employed. Methods: Eighteen male albino rats were divided into three separate groups. The first group (G1) was made up of six animals that served as a control group. The second group (G2) was made up of six animals that were given caffeine, and the third group (G3) was made up of six animals that were given silver nanoparticles from a caffeine solution. The particle size and structural morphology of Caffeine and silver nanoparticles were analyzed using Brookhaven Instruments Corp., XRD and SEM respectively. Results: The structural results showed after addition, caffeine was tube-shaped and silver was spherical granular. Caffeine has more silver nanoparticles than caffeine solution. Caffeine solution affects potassium, calcium, zinc, and magnesium levels. Additionally, the solution has antibacterial activity against the following bacteria: Staphylococcus aureus (Gram +ve), E. coli (Gram –ve), and Pseudomonas aeruginosa (Gram-ve), but it has no effect against yeast (Candida albicans). Conclusions: To determine the biological effects of caffeine-synthesized Ag NPs on zinc, magnesium, calcium, and potassium levels in male albino mice's serum, this is antibacterial against staphylococcus.