By Yehuda Pinchover and Jacob Rubinstein

ISBN-10: 0511111576

ISBN-13: 9780511111570

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**Additional info for An introduction to partial differential equations**

**Sample text**

To illustrate further the physical meaning of boundary conditions, let us consider again the wave equation for a string: u tt − c2 u x x = f (x, t) a < x < b, t > 0. 1(a)): u(a, t) = β1 (t), u(b, t) = β2 (t), t > 0. 50) Another possibility is that the tension at the end points is given. 1 Illustrating boundary conditions for a string. 20 Introduction Neumann condition: u x (a, t) = β1 (t), u x (b, t) = β2 (t), t > 0. e. β1 = β2 = 0. 6 Simple examples Before proceeding to develop general solution methods, let us warm up with a few very simple examples.

It is difﬁcult to formulate universal statements here. Instead, each such case has to be analyzed separately. 6 The Lagrange method First-order quasilinear equations were in fact studied by Lagrange even before Hamilton. Lagrange developed a solution method that is also geometric in nature, albeit less general than Hamilton’s method. The main advantage of Lagrange’s method is that it provides general solutions for the equation, regardless of the initial data. 15). The set of all solutions to this system forms a twoparameter set of curves.

34) But the initial condition implies (0) = 0. 34). Since that equation has smooth coefﬁcients, it has a unique solution. Thus ≡ 0 is the only solution, and the curve (x(t), y(t), u(t)) indeed lies on the integral surface. Therefore the integral surface we constructed earlier through the parametric representation induced by the characteristic equations is unique. When the transversality condition does not hold along an interval of s values, the characteristic there is the same as the projection of .

### An introduction to partial differential equations by Yehuda Pinchover and Jacob Rubinstein

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