Abstract
An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or 180 ° rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations.
Highlights
Nonlinear partial differential equations (NLPDEs) have been difficult to solve analytically.analytic solutions may serve as bench marks for numerical solutions where the computations can be tricky
The algebraic equations are determined by equating the coefficients of each power of U in the NLPDE series expansion to zero
An example of a NLPDE that has no solution in terms of the five nonlinear functions is the Blasius equation [4,21] for laminar flow of a fluid past a plate u xxx + γuu xx = 0 (11)
Summary
Nonlinear partial differential equations (NLPDEs) have been difficult to solve analytically. There is no one method that works for all the cases of analytic solutions. Scattering Method for solitons [1] is restricted to certain NLPDEs, but the inclusion of initial conditions is very advantageous. The Method of Characteristics [2] applies to quasi-linear partial differential equations (PDEs) that have first-order derivatives. Lie symmetries [3,4,5,6,7] are employed to reduce the NLPDEs to nonlinear ordinary differential equations (NLODEs) that may be reduced to quadradures if sufficient symmetries exist. Various Expansion Methods such as F, G’/G [11,12] require auxiliary differential equations (ODEs). As a consequence traveling waves are a common form of solutions
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