Constructing a method for solving the riccati equations to describe objects parameters in an analytical form
DOI:
https://doi.org/10.15587/1729-4061.2020.205107Keywords:
Riccati equation, special points, linearization, nondimensionalization, analytical solution, elementary functions.Abstract
This paper reports the established feature of non-linear differential equations as those that most adequately describe the properties of objects. Possible methods of their linearization have been analyzed. The issues related to solving the original equations in a linearized form have been defined. The Riccati equation has been given as an example.
For a special type Riccati equation, a method to solve it has been constructed, whereby the results are represented in an analytical form. It is based on the use of linearization and a special method of nondimensionalization.
A special feature of the constructed method is determined by its application not to the original equation but to its discrete analog. The result of solving it is an analytical expression based on elementary functions. It is derived from using the existing analytical solution (supporting, basic) to one of the equations of the examined type. All the original equations of the examined type have the same type of solution. This also applies to equations that had no previous analytical solution.
A formalized procedure for implementing the devised method has been developed. It makes it possible to link the analytical type of solution to the examined equation and known analytical solution to the basic one. The link is possible due to the equality of discrete analogs of the considered and basic equations. The equality of discrete analogs is provided by using a special nondimensionalization method.
The applicability of the method and the adequacy of the results obtained have been shown by comparing them with existing analytical solutions to two special type Riccati equations. In one case, the solution has movable special points. In the second case, a known solution has an asymptote but, at the positive values of the argument, has no special points.
The possibility of using the constructed method to solve the general Riccati equation has been indicated.
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