Solution of Nonlinear Equations in Science through Lagrange’s Inversion Theorem

Abstract

Nonlinear problems arise in most of the scientific fields. In general, such behavior is represented by a nonlinear equation, whose solution is sought. Analytical and numerical methods have been applied to the solution of this class of equations, notwithstanding, in cases where highly nonlinear phenomena are analyzed, the number of iterations and computational effort necessary to achieve the minimum required accuracy is very high. Lagrange´s Inversion Theorem (LIT) has been applied to solve this kind of problems analytically, giving the solution as an infinite power series. This way, the accuracy can be as high as necessary by taking more terms from the series solution, which is easily computationally implemented. Also, in some cases it is possible to relate the series obtained to the expansion of special and elementary functions, which enables one to exactly solve the desired equation. In the present review paper, a total of eleven applications have been discussed in order to show the role of LIT in various areas of nonlinear sciences.