A new approach to find an approximate solution of linear initial value problems with high degree of accuracy

This work investigates a new approach to find closed form solution to linear initial value problems (IVP). Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite operational matrix to simplify derivatives in IVP. These orthonormal polynomials together with the operational matrix of relevant order provides a robust approximation to the solution of a linear initial value problem by converting the IVP into a set of algebraic equations. Depending upon the nature of a problem, a polynomial of degree n or numerical approximation can be obtained. The technique has been demonstrated through four examples. In each example, obtained solution has been compared with available exact or numerical solution. High degree of accuracy has been observed in numerical values of solutions for considered problems

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable