Browsing by Author "Dolapci İ.T."

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    A nonlinear curve equation for an object moving with constant acceleration components
    (North Atlantic University Union NAUN, 2016) Pakdemirli M.; Dolapci İ.T.
    An ordinary differential equation describing a curve for which the tangential and normal acceleration components of the object remains constant is derived. The equation and initial conditions are expressed in dimensionless form. In its dimensionless form, the curves are effected only by a parameter which represents the ratio of the tangential acceleration to the normal acceleration. For constant velocity case, the equation can be solved analytically yielding a circular arc solution as expected. For nonzero tangential acceleration, closed form solutions are not available. Using a series solution, the curve is approximated by polynomials of arbitrary order. The general recursion relation for the polynomial coefficients are given. Two different perturbation solutions are also presented. In the first perturbation approach, the curve parameter is selected as the perturbation parameter. In the second approach, the depending variable is assumed to be small by introducing an alternative perturbation parameter. It is found that the second perturbation solution yields identical results with the series solutions. The approximate solutions and the numerical solutions are contrasted and within the range of validity, the curves can be successfully approximated by the analytical solutions. Potential application areas can be the design of highway curves, highway exits, railroads, route selection for ships and aircrafts. A practical application to highway exits is considered as an example. © 2016, North Atlantic University Union NAUN. All rights reserved.
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    Solution Curves of Equations in the Differential Space
    (Birkhauser, 2024) Pakdemirli M.; Dolapci İ.T.
    Solutions of ordinary differential equations are considered. Differential Space is defined as the three-dimensional space with coordinates being the solution function and its first and second derivatives. Solution curves are represented as parametric three-dimensional curves in the differential space with the curve parameter being the independent variable. For various sample differential equations, the solution curves and their properties are depicted. The solution curves may converge to a point, may blow up and diverge to infinity, may be periodic, may end up with a limit cycle periodic solution or may be chaotic. Local solutions with a given initial condition set is treated in this introductory study. Differential space is a generalization of 2-D state space to 3-D’s. Global solutions with phase diagrams, basins of attraction are left for further studies. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.