Download e-book for kindle: An Introduction to Partial Differential Equations by Y. Pinchover, J. Rubenstein

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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Extra resources for An Introduction to Partial Differential Equations

Sample text

Returning to (i) we obtain x(t, s) = (1 + s)et − e−t and u(t, s) = set + e−t . Observing that x − y = set − e−t , we finally get u = 2/y + (x − y). The solution is not global (it becomes singular on the x axis), but it is well defined near the initial curve. 5 The existence and uniqueness theorem We shall summarize the discussion on linear and quasilinear equations into a general theorem. For this purpose we need the following definition. 16) defining an initial curve for the integral surface. e. J |t=0 = xt (0, s)ys (0, s) − yt (0, s)xs (0, s) = a b = 0.

It remains to compute γ . For this purpose we write the weak formulation in the form γ (y) ∂y a u(ξ, y)dξ + b 1 u(ξ, y)dξ + [(u 2 (b, y) − u 2 (a, y)] = 0. 2 γ (y) Differentiating the integrals with respect to y and using the PDE itself leads to γ y (y)u − − γ y (y)u + − 1 2 γ (y) (u 2 (ξ, y))ξ dξ + a b (u 2 (ξ, y))ξ dξ γ (y) 1 + [u 2 (b, y) − u 2 (a, y)] = 0. 2 Here we used u − and u + to denote the values of u when we approach the curve γ from the left and from the right, respectively. 50) γ y (y) = (u − + u + ), 2 namely, the curve γ moves at a speed that is the average of the speeds on the left and right ends of it.

But we have already seen that this is not the case. What, therefore, are the obstacles we might face? 14) well-posed? For simplicity we shall discuss in this chapter two aspects of well-posedness: existence and uniqueness. 3) that contains the initial curve. (1) Notice that even if the PDE is linear, the characteristic equations are nonlinear! We know from the theory of ODEs that in general one can only establish local existence of a unique solution (assuming that the coefficients of the equation are smooth functions).

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


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