In this paper, Bernstein polynomial method applied to the solutions of generalized Emden-Fowler equations as singular initial value problems is presented. Firstly, the singular differential equations are converted to Volterra integro-differential equations and then solved by the Bernstein polynomials method. The properties of Bernstein polynomials via Gauss-Legendre rule are used to reduce the integral equations to a system of algebraic equations which can be solved numerically. Some illustrative examples are discussed to demonstrate the validity and applicability of the present method.

Some problems from modern physics and science can be described in terms of partial differential equations with nonlocal conditions. In this paper, a numerical method which employs the orthonormal Bernstein polynomials basis is implemented to give the approximate solution of integro-differential parabolic equation with purely nonlocal integral conditions. The properties of orthonormal Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given integro-differential parabolic equation to the solution of algebraic equations. An illustrative example is given to demonstrate the validity and applicability of the new technique.

In this paper the singular Emden-Fowler equation of fractional order is introduced and a computational method is proposed for its numerical solution. For the approximation of the solutions we have used Boubaker polynomials and defined the formulation for its fractional derivative operational matrix. This tool was not used yet, however, this area has not found many practical applications yet, and here introduced for the first time. The operational matrix of the Caputo fractional derivative tool converts these problems to a system of algebraic equations whose solutions are simple and easy to compute. Numerical examples are examined to prove the validity and the effectiveness of the proposed method to find approximate and precise solutions.