New Exact Operational Shifted Pell Matrices and Their Application in Astrophysics

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.