Analytic Methods for Nonlinear Equations: A Compendium

While the general theory of linear equations (both integral and differential) is well understood regrettably the same cannot be said of nonlinear equations. The basic problem is one of finding methods for obtaining exact solutions for nonlinear equations, assuming there are such methods, or of showing there is no method and so no exact analytical solution. Because of high speed and powerful computers there is a regrettable tendency to slam any old equation directly into the computer and claim the resulting output is the solution desired. But such feats of legerdemain do not consider several factors: How accurate is the numerical procedure? That it is precise is clear for a computer produces precise answers but that they are accurate and represent the desired solution to the equation programmed is rarely tested, if at all. Accuracy can best be tested against a known analytic solution and there, of course, lies the rub for one goes full circle if not careful. A further, often major, problem is that nonlinear equations often contain parameters, and so numerical methods must usually undergo trial and error searches for values (or domains) of parameters where solutions can be obtained (if at all) and this problem can be extremely time-consuming with numerical procedures particularly when several or many parameters are involved. There are, to be sure, nonlinear minimization procedures available to help but such involve running a program many times that can be difficult if the program takes, say, one hour for one iteration. But it can also be that there are no acceptable solutions when a parameter is in some domain and to find that domain numerically is often itself difficult. Even if one has found such a domain there is no guarantee that solutions exist satisfying required boundary or initial conditions in the residual domains. One has only the knowledge that there are no solutions in the specific domain. For all these reasons it is more than relevant to see if analytic solutions are available to nonlinear equations. There is the point also that should one find a method for effecting a solution then it can be that the method itself is useful for a much broader class of problems than the original equation it was devised to solve. There is also no guarantee, of course, that a chosen method will be effective in obtaining analytic solutions. The hope is that by identifying enough transformations for enough nonlinear equations some sort of pattern will emerge to provide a framework .Regrettably, as of the date of writing no such pattern is known to the authors.