Analytical Methods for Solving Nonlinear Partial Differential Equations

An introductory course in differential equations can easily give students the false impression that all differential equations have closed-form solutions. After all, all the differential equations given in homework assignments do have closed-form solutions. An instructor can say that this is not the norm, but homework assignments speak louder than words.

 

A more advanced course in differential equations will shift from finding exact solutions to proving existence and uniqueness theorems, finding numerical solutions, etc. Now students start to believe that differential equations with closed-form solutions really are exceptional. In particular, the equations with closed-form solutions are linear, ordinary differential equations (ODEs). Nonlinear equations and partial differential equations (PDEs) rarely can be solved in elementary terms. And for a nonlinear PDE it would be hopeless to find such a solution.

 

With these expectations, Daniel Arrigo’s book Analytical Methods for Solving Nonlinear Partial Differential Equations comes as a shock. There are analytical methods for solving PDEs? Enough to fill a book?!

 

It’s wrong to assume differential equations always have closed-form solutions. It’s also wrong, though not as wrong, to assume that they never have closed-form solutions. The truth is somewhere in between, though much closer to the latter.  There are indeed differential equations, even nonlinear partial differential equations, with closed-form solutions. And not just artificial examples, but equations that came from modeling physical phenomena. Arrigo gives nine examples in the exercises at the end of his first chapter. Solitons, solutions to the Korteweg–de Vries (KdV) equation, are the most well-known example, but not the only example.

 

Closed-form solutions are very convenient, even if you have to simplify your model a bit in order to have such solutions. If your equation does not have a closed-form solution, but its solutions can be approximated by an equation that does, the simplified model can give you a good idea what to expect of the more elaborate model.  Arrigo gives numerous techniques for finding exact solutions to nonlinear PDEs. But these techniques feel isolated. They do not cohere into a systematic theory. Someone said that a technique is a trick that works twice. The techniques in Arrigo’s book are somewhere along the continuum between tricks and a general theory, applicable to a range of equations but still specialized.

 

There are a lot of lengthy calculations in Arrigo’s book, which is to be expected from a book about calculating solutions. The book includes a generous number of examples. It has more examples than theorems, which is appropriate to its subject matter.

 

Closed-form solutions to nonlinear PDEs are indeed rare, but very useful when they can be found. And if there are known techniques for finding a particular solution, they are likely to be found in Arrigo’s book.

---

John D. Cook is an independent consultant applying mathematics to data privacy.