Math 220: First Order Scalar Quasilinear Equations
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چکیده
(1) a(x, y, u)ux + b(x, y, u)uy = c(x, y, u), with a, b, c at least C, given real valued functions. There is an immediate difference between semilinear and quasilinear equations at this point: since a and b depend on u, we cannot associate a vector field on R to the equation: we need to work on R at least to account for the (x, y, u) dependence. To achieve this, we proceed as follows. We consider the graph S of u in R ×R, given by z = u(x, y). If we know the solution u, we know the graph, and conversely we can recover u if we find its graph, a surface in R. So let U(x, y, z) = u(x, y)− z;
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Math 220: First Order Scalar Semilinear Equations
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