By Yehuda Pinchover and Jacob Rubinstein

ISBN-10: 0511111576

ISBN-13: 9780511111570

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**Extra resources for An introduction to partial differential equations**

**Sample text**

5 Let p : R → R be a differentiable function. Prove that the equation u t = p(u)u x t >0 has a solution satisfying the functional relation u = f (x + p(u)t), where f is a differentiable function. In particular ﬁnd such solutions for the following equations: 22 Introduction (a) u t = ku x . (b) u t = uu x . (c) u t = u sin(u)u x . 34), and compute the average time for which the broker holds the stock. Analyze the result in light of the ﬁnancial interpretation of the parameters (m 1 , m 2 , k). 7 (a) Consider the equation u x x + 2u x y + u yy = 0.

After the critical time yc when the solution becomes singular we need to deﬁne a weak solution. We seek a solution with a single discontinuity. 6 Several snapshots in the development of a shock wave. implies that the discontinuity moves with a speed 12 . Therefore the following weak solution is compatible with the integral balance even for y > α: u(x, y) = 1 x < α + 12 (y − α), 0 x > α + 12 (y − α). 53) The solution thus constructed has the structure of a moving jump discontinuity. It describes a step function moving at a constant speed.

Prove that the equation u t = p(u)u x t >0 has a solution satisfying the functional relation u = f (x + p(u)t), where f is a differentiable function. In particular ﬁnd such solutions for the following equations: 22 Introduction (a) u t = ku x . (b) u t = uu x . (c) u t = u sin(u)u x . 34), and compute the average time for which the broker holds the stock. Analyze the result in light of the ﬁnancial interpretation of the parameters (m 1 , m 2 , k). 7 (a) Consider the equation u x x + 2u x y + u yy = 0.

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

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