A New Analytical Method for Solving General Riccati Equation

Abstract

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method doesn't require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at a first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

A new transformation technique to find the analytical solution of general second order linear ordinary differential equation

“…More interestingly, the Riccati equation can be transformed into second order OLDE by using the transformation = − ′ ( ) as discussed by Busawon and Johnson (2005). The study made by Busawon and Johnson (2005) and Pala and Ertas (2017) showed that Eq. 1 and 2 are inter-convertible.…”

“…Many attempts have also been made to solve Riccati Eq. 2 in the form of special cases in the literature Al Bastami et al (2010), Batiha (2015), Busawon and Johnson (2005), Pala and Ertas (2017), and Sarafian (2011) but once again there is no general method to solve (2) in its general form Al Bastami et al (2010).…”

“…All the above literature and discussion shows the importance of Eqs. 1 and 2 for their applications but at the same time we can observe that every method to find the general solution of these equations has its own limitations and specific area of implementation based on the nature of coefficient functions, see Al Bastami et al (2010), Batiha (2015), Busawon and Johnson (2005), Johnson et al (2008), Kim (2016), Mohammed and Zeleke (2015), Pala and Ertas (2017), Robin (2007), Sarafian (2011), Saravi (2012), and Wilmer III and Costa (2008). In the next section, we have discussed the special case of second order OLDE on the basis of a new transformation to obtain its general solution.…”

“…In the next section, we have discussed the special case of second order OLDE on the basis of a new transformation to obtain its general solution. We have modified the transformation used by (Pala and Ertas, 2017) to solve second order OLDEs.…”

Generalized Riccati Wick differential equation and applications

“…As soon as differential equations were invented, the Riccati differential equation was the first one to be investigated extensively since the end of the 17th century [22]. This equation appears widely in large variety of applications in engineering and applied science such as random processes, diffusion problems, optimal control etc., [1,11,12,15]. It has the form where a(x), b(x) and c(x) are continuous functions of x.…”