Reducing labor intensity when computing optimal technical characteristics of aerial ropeways

The article develops a procedure for optimizing the technical characteristics of ropeways - the step and the height of intermediate towers, and carrying ropes tension force. The optimization problem was based on the minimization of the tower structures cost. The reduction of computing labor intensity is based on the fact that the position of the minimum point of the objective function will be tied to one of the optimization restrictions. This allowed us to propose two ways to reduce the labor intensity of computing: a) reduction in the dimension of the optimization problem; b) replacement of the search for the minimum of the objective function with the solution of the nonlinear algebraic equation. The article shows that the proposed algorithm has increased computational efficiency. The algorithm allows us to obtain the same optimal values of technical characteristics of ropeways as in the solution of the previously developed optimization task but using simpler mathematical methods.

A Nonlinear System Related to Investment under Uncertainty Solved using the Fractional Pseudo-Newton Method

A nonlinear algebraic equation system of two variables is numerically solved, which is derived from a nonlinear algebraic equation system of four variables, that corresponds to a mathematical model related to investment under conditions of uncertainty. The theory of investment under uncertainty scenarios proposes a model to determine when a producer must expand or close, depending on his income. The system mentioned above is solved using a fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems.

A Collocation Method for Solving Fractional Riccati Differential Equation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation is derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.

A Collocation Method for Solving Fractional Riccati Differential Equation

We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

A Comparison of the Performance of Nonlinear Equation Solvers for Machine Design Problems

Often, the mathematical modeling of mechanical components leads to a set of nonlinear equations which have to be solved in order to compute the design variables. This paper discusses the performance of six different nonlinear, algebraic-equation solvers useful for machine design problems. Problems from spring design are taken as test cases for the purpose of evaluation.