Two Approaches to Solving the Potential Equation in Self-Similar Variables

Abstract

The authors, using the methodpreviously proposedby them, investigatethe general velocity potential equation for the case ofthree self-similar variables. Two approachesof this method are used. The first approach assumesthat the solutiondepends only onone variable, which, in turn, isan unknown functionof all independent variables, and thuspotential equation is reduced to theODE. Finding unknown functionis basedon a study ofthe correspondingoverdetermined systemof partial differential equations. A numberofcompatibility conditionsforthesystem are found. Some exact solutions are constructed.It is shown howthe solutions obtainedcan be usedin considering the problemof shock-freecompression of the gas. Thesecondapproachassumesthatthefunctionisknownandcoincideswiththefunctionthatgivesasolutionofthepotential equation. It is alsoreceived a numberof exact solutionsthat can be usedto solve someinitialandboundary value problems.