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We propose a new version of potentially optimal intervals for the DIRECT algorithm. A two-points based sampling method is presented. The method starts from a distingished point (the peak point) by forming an initial triangle. The idea is to sample the midpoint of a specific interval: the basis of the resulting triangle. This specific interval is obtained by translating the initial interval towards the lowest function value : min{f(c i ),f(c i+1 )} and then overcoming the disadvantage if the global minimum lies at the boundaries. Two-dimensional version of our subdivision and sampling method is also discussed.

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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.