I am in graduate school for chemical engineering and the method of characteristics appears in many journal articles. Does anyone have a good reference: either online or a book that provides a good explanation with examples of the method of characteristics.
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Off the top of my head, I've found both of these to be well-written and easy to follow. Cavet: I don't have them in front of me, so my recommendation is based on general impressions rather than their particular treatment of characteristics.
The External Links listed at the end of the wikipedia article on characteristics also look like they might be worth your time.
Since you have more of an engineering background, I would suggest the textbook of Robert McOwen, Partial Differential Equations: Methods and Applications. Chapter 1 is about solving first order partial differential equations, where the method of characteristics is most applicable. Chapters 2 and 3 also contains some application of the notion of characteristics to higher order equations and to the wave equation.
Compare to the treatments in other standard, more theory-oriented textbooks, I find McOwen's treatment of the method of characteristics to be more detailed and complete, and easier to teach from. As a remark: section 1.3, the application to fully non-linear first order equations, is very difficult to wrap your head around. So don't worry if you feel like you suddenly hit a wall when you get to that section.