By Y. Pinchover, J. Rubenstein

ISBN-10: 0511111576

ISBN-13: 9780511111570

ISBN-10: 052161323X

ISBN-13: 9780521613231

ISBN-10: 0521848865

ISBN-13: 9780521848862

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**Additional info for An Introduction to Partial Differential Equations**

**Sample text**

Sometimes it is also called the Robin condition. 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.

4), and write the initial condition parameterically: = (s) = (x0 (s), y0 (s), u 0 (s)), s ∈ I = (α, β). 11) The curve will be called the initial curve. 4) can be rewritten as (a, b, c0 u + c1 ) · (u x , u y , −1) = 0. 3 The method of characteristics 27 tangent plane. 13) dt du (t) = c(x(t), y(t)))u(t) + c1 (x(t), y(t)) dt deﬁnes spatial curves lying on the solution surface (conditioned so that the curves start on the surface). This is a system of ﬁrst-order ODEs. They are called the system of characteristic equations or, for short, the characteristic equations.

6 Find a solution of the wave equation u tt − 4u x x = sin t + x 2000 . Notice that we are asked to ﬁnd a solution, and not the most general solution. We shall exploit the linearity of the wave equation. According to the superposition principle, we can split u = v + w, such that v and w are solutions of vtt − 4vx x = sin t, wtt − 4wx x = x 2000 . 53) The advantage gained by this step is that solutions for each of these equations can be easily obtained: v(x, t) = − sin t, w(x, t) = − 1 x 2002 . 4 × 2001 × 2002 Thus u(x, t) = − sin t − 1 x 2002 .

### An Introduction to Partial Differential Equations by Y. Pinchover, J. Rubenstein

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