By Y. Pinchover, J. Rubenstein
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Additional info for An Introduction to Partial Differential Equations
In the theory of ODEs one solves ﬁrst-order equations by the method of integration factors. While this method is always feasible in theory, it involves great technical difﬁculties. In fact, it is possible to ﬁnd integration factors only in special cases. 35). Hence it is not surprising that the method is applicable only in special cases. 40 First-order equations We proceed by introducing a method for computing the surfaces ψ and φ, and then apply the method to a speciﬁc example. 13 Recall that by deﬁnition, the surfaces ψ = α, φ = β contain the characteristic curves.
Although the three types of boundary conditions deﬁned above are by far the most common conditions seen in applications, there are exceptions. For example, we can supply the values of u at some parts of the boundary, and the values of its normal derivative at the rest of the boundary. This is called a mixed boundary condition. Another possibility is to generalize the condition of the third kind and replace the normal derivative by a (smoothly dependent) directional derivative of u in any direction that is not tangent to the boundary.
For example, it is easy to check that if we add to the solution above a function of the form f (x − 2t), where f (s) is an arbitrary twice differentiable function, a new solution is obtained. 7 Exercises 21 Unfortunately one rarely encounters real problems described by such simple equations. Nevertheless, we can draw a few useful conclusions from these examples. For instance, a commonly used method is to seek a transformation from the original variables to new variables in which the equation takes a simpler form.
An Introduction to Partial Differential Equations by Y. Pinchover, J. Rubenstein