An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.

Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus

We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.