By Shuxing Chen

ISBN-10: 9814304832

ISBN-13: 9789814304832

The booklet presents a finished evaluation at the conception on research of singularities for partial differential equations (PDEs). It encompasses a summarization of the formation, improvement and major effects in this subject. a number of the author's discoveries and unique contributions also are integrated, corresponding to the propagation of singularities of recommendations to nonlinear equations, singularity index and formation of shocks

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**Additional info for Analysis of Singularities for Partial Differential Equations**

**Example text**

Generalized bicharacteristic strip is a map γ from I \ B to Γ ⊂ Σb , where I is an inteval in R1 , B is a set of isolate points. The map satisfies (1) If γ(t0 ) ∈ Σ0b ∪Σ2− b , then γ(t) = (x(t), y(t), ξ(t), η(t)) is differentiable at t0 , and γ (t0 ) = Hp (γ(t0 )); (3) (2) If γ(t0 ) ∈ Σ2+ b ∪ Σb , then the projection ((x(t), y(t), η(t)) of γ(t) is differentiable at t = t0 , and x (t0 ) = 0, (y (t0 ), η (t0 )) = Hr0 (y(t0 ), η(t0 )). (3) If t1 ∈ B, then γ(t) ∈ Σ0b as |t − t0 | > 0 small enough. Moreover, γ(t1 ± 0) exist, which are the different points on a same fiber based on a given point of {x = 0}.

1 + tan2 θ 1 + tan2 θ √ 2 −1/2 For (ξ, η, ζ) = (1, c−1 , 0), the four eigenvalues of the + tan θ( 1 + tan θ) matrix a is ξ = −√ √ ±c−1 + c−1 + , −2 ± c−2 − − c+ tan2 θ 1 + tan2 θ 1/2 . 1 + tan2 θ When c2+ > c2− tan2 θ(1 + tan2 θ)−1/2 , these four eigenvalues are all real. Then the projection of bicharacteristic strips through the origin on the space (x, y, z) is L1 : y = x tan θ, L2 : y = −x tan θ, c−1 + x tan θ L3 : y = , −2 −2 2 c− + (c−2 − − c+ ) tan θ L4 : y = −c−1 + x tan θ −2 −2 2 c−2 − + (c− − c+ ) tan θ , August 12, 2010 42 15:49 World Scientific Book - 9in x 6in singularities Analysis of Singularities for Partial Differential Equations where L2 is the equation of the incident ray.

The proof mainly consists of two steps. The first step is to decompose the operator P , so that to reduce the problem to the case for an operator of first order, while the second step is to prove the theorem for the reduced operator of first order. August 12, 2010 15:49 World Scientific Book - 9in x 6in singularities Singularity analysis for linear equations 25 Since ∇ξ pm (x, ξ) = 0, we may assume ∂pm /∂ξn = 0. Hence we may assume that pm (x, ξ) takes the form m−1 am−k (x, ξ )ξnk . 27) where λ(x, ξ ) is a homogeneous function of ξ with degree 1, qm−1 (x, ξ) is a homogeneous function of ξ with degree m − 1, and qm−1 (x0 , ξ0 ) = 0.

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