By Guo Chun Wen
ISBN-10: 0415269717
ISBN-13: 9780415269711
This quantity bargains with first and moment order complicated equations of hyperbolic and combined (elliptic hyperbolic) sort within which a number of normal boundary price difficulties for first and moment order linear and quasilinear complicated equations of hyperbolic and combined style are investigated intimately.
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Additional info for Linear and quasilinear complex equations of hyperbolic and mixed type
Example text
6) in D. 8) where Φ(χ) is a hyperbolic regular function in the domain G = χ(D). Proof Suppose that z(χ) is the inverse function of χ(z), we can find {w[z(χ)]}χ¯ = wz zχ¯ + wz¯z¯χ¯ = wz [zχ¯ + Q¯ zχ¯ ], and zχ ], χχ = 1 = χz zχ + χz¯z¯χ = χz [zχ + Q¯ zχ¯ ]. χχ¯ = 0 = χz zχ¯ + χz¯z¯χ¯ = χz [zχ¯ + Q¯ zχ¯ = 0, consequently Φ(χ) = w[z(χ)] From the above equalities, we see χz = 0, zχ¯ + Q¯ satisfies [Φ(χ)]χ¯ = 0, χ ∈ G = χ(D). 8) holds. 6). 6) can be written in the form ξν = (a + b)ξµ , ηµ = (a − b)ην .
2) in which α(0 < α < 1), k0 , k2 are non-negative constants. 16) with A3 (z, u, w) = 0, z ∈ D, u ∈ IR, w ∈ C I and r(z) = b0 = b1 = 0, z ∈ L will be called Problem P0 . 3) where λ(z) = (a + b)e1 + (a − b)e2 . 1). We can assume that w(z3 ) = 0, otherwise through the transformation W (z) = {w(z) − [a(z3 ) − jb(z3 )][r(z3 ) + jb1 ]} /[a2 (z3 ) − b2 (z3 )], the requirement can be realized. It is not difficult to see that the oblique derivative boundary value problem (Problem P ) includes the Dirichlet boundary value problem (Problem D) as a special case.
5), zj (j = 0, 1, 3) are as stated in Section 3, R2 − 2R1 , 0 ≤ x ≤ R2 }, and λ(z) = 1 + j, z ∈ L5 , λ(z) = 1 − j, z ∈ L6 . L6 = {y = R2 x For corresponding boundary value problems of hyperbolic systems of first order complex equations, whether there are similar results as before? The problem needs to be investigated. 5 Hyperbolic Mappings and Quasi-hyperbolic Mappings Now, we introduce the definitions of hyperbolic mappings and quasi-hyperbolic mappings, and prove some properties of quasi-hyperbolic mappings.