In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.
It is shown that if is of bounded variation in the sense of Hardy-Krause on , then is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.
The paper is concerned with integrability of the Fourier sine transform function when , where is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of to be integrable in the Henstock-Kurzweil sense, it is necessary that . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.
The least concave majorant, , of a continuous function on a closed interval, , is defined by We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on . Given any function , it can be well-approximated on by a clamped cubic spline . We show that is then a good approximation to . We give two examples, one to illustrate, the other to apply our algorithm.
This paper considers a Volterra's population system of fractional order and describes a bi-parametric homotopy analysis method for solving this system. The homotopy method offers a possibility to increase the convergence region of the series solution. Two examples are presented to illustrate the convergence and accuracy of the method to the solution. Further, we define the averaged residual error to show that the obtained results have reasonable accuracy.