A Treatise on Linear Differential Equations, Volume I: by Thomas Craig

By Thomas Craig

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44) corresponds to multiplying, at each point of the tangent plane ΣT , the length of the gradient vector ∇ζ by a conformal factor exp[η]. For Q < 1, which in the case G = 0 corresponds geometrically to a space-like hypersurface, the system of Eqs. 44) is elliptic. 4 An Alternative: Explicit Solutions for a Class of Quasilinear Vector Fields 45 For Q > 1, which in the case G = 0 corresponds geometrically to a time-like hypersurface, the Eqs. 44) are a hyperbolic system. In this example, Q , Q = 1.

6) on the subdomain Ω + . But that Tricomi problem has become the free boundary problem u x x + u yy = 0 in Ω + , u = β(x) on Γ0 , u = 0 and u x − u y = φ x−y 2 on Ω0 . The unknown sonic line Ω0 of the original elliptic–hyperbolic boundary value problem has become the free boundary of a degenerately elliptic boundary value problem. In order to fix the boundary, we now apply the partial hodograph transformation (x, y) → (ξ, z), where ξ = x, z = u (x, y) . 57) In terms of the inverse transformation x = ξ, y = h (ξ, z), the original Eq.

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