By Michael Renardy Robert C. Rogers

Partial differential equations are basic to the modeling of typical phenomena. the will to appreciate the options of those equations has continually had a favourite position within the efforts of mathematicians and has encouraged such diversified fields as complicated functionality conception, useful research, and algebraic topology. This publication, intended for a starting graduate viewers, presents a radical creation to partial differential equations.

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**Sample text**

Show that this system is elliptic. 5. The Euler equations in R3 are (u · ∇)u − ∇p = 0, div u = 0. 43) Here the unknowns are the vector function u : R3 → R3 and the scalar function p : R3 → R. Show that this system is neither elliptic nor hyperbolic. 46 2. 6. Consider the system ut + Aux + Buy = 0. 44) What condition must A and B satisfy for the system to be hyperbolic in the t direction? The condition which you will ﬁnd is, in general, diﬃcult to verify. Can you give a simple special case? Consider now the special case where A and B are diagonal.

We will generalize these notions to many other types of equations in later chapters. Note that every strong solution of Laplace’s equation is also a weak solution. 33) by an arbitrary function 22 1. Introduction v ∈ A0 , integrate by parts (use Green’s identity) and use the fact that v ≡ 0 on ∂Ω. This gives (∆u)v dx = − 0= Ω ∇u · ∇v dx + Ω v∇u · n dS = − ∂Ω ∇u · ∇v dx. 75) However, as we noted above when we showed that a solution of the minimum energy problem was a weak solution of Laplace’s equation, unless we know more about the continuity of a weak solution we cannot show it is a strong solution.

13 to display partial sums of the cosine series. 16. Both the Fourier sine and cosine series given above converge not only in the interval [0, 1], but on the entire real line. If one computed both the sine and cosine series for the functions graphed below, what would you expect the respective graphs of the limits of the series to be on the whole real line. 2. 17. Solve Laplace’s equation on the square [0, 1] × [0, 1] for the following boundary conditions: (a) uy (x, 0) = 0, u(x, 1) = x2 − x, u(0, y) = 0, u(1, y) = 0.