Solution Techniques for Models Producing PDEs

In Chapter 1, we showed how conservation laws often lead to situations wherein more than one independent variable is needed. This gave rise to so-called partial derivatives, defined for example in the distance-time domain as 

The first implies holding time constant while differentiating with respect to x, while the second implies holding x constant while differentiating with respect to t. These implicit properties must be kept in mind when integrating partial derivatives. Thus, for ordinary derivatives, the integral of dy/dx = 0 yields y(x) = constant. However, for the partial derivatives, we must account for the implied property 

instead of adding an arbitrary constant, we must in general add an arbitrary function of the independent variable, which was held constant during integration. 

The total differential of u can be written using the chain rule 

For continuously differentiable functions, the order of differentiation is immaterial 

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Chapter 10 Solution Techniques for Models Producing PDEs... 

Chapter 10 Solution Techniques for Models Producing PDEs Suppose there exists a combined variable such that... 

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