By Guo Chun Wen

Within the fresh half-century, many mathematicians have investigated a variety of difficulties on numerous equations of combined variety and received fascinating effects, with vital purposes to gasoline dynamics. in spite of the fact that, the Tricomi challenge of normal combined kind equations of moment order with parabolic degeneracy has now not been thoroughly solved, quite the Tricomi and Frankl difficulties for common Chaplygin equation in multiply hooked up domain names posed via L Bers, and the life, regularity of ideas of the above difficulties for combined equations with non-smooth degenerate curve in numerous domain names posed by way of J M Rassias.

the tactic printed during this booklet is not like the other, within which the hyperbolic quantity and hyperbolic complicated functionality in hyperbolic domain names, and the advanced quantity and complicated functionality in elliptic domain names are used. The corresponding difficulties for first order complicated equations with singular coefficients are first mentioned, after which the issues for moment order complicated equations are thought of, the place we pose the hot partial spinoff notations and intricate analytic tools such that the varieties of the above first order advanced equations in hyperbolic and elliptic domain names are thoroughly exact. meanwhile, the estimates of suggestions for the above difficulties are bought, for that reason many open difficulties together with the above Tricomi Bers and Tricomi Frankl Rassias difficulties might be solved.

Contents: Elliptic complicated Equations of First Order; Elliptic advanced Equations of moment Order; Hyperbolic complicated Equations of First and moment Orders; First Order advanced Equations of combined sort; moment Order Linear Equations of combined style; moment Order Quasilinear Equations of combined kind

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**Additional info for Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy: Including Tricomi-Bers and Tricomi-Frankl-Rrassias Problems**

**Example text**

3) satisfies Condition C. 3) has a solution. e. wZ = F (z, w), F (z, w) = [A1 w+A2 w+A3 ]/H(y) in DZ . 23) in which the function S(z) satisfies the condition H(y)X(Z)S(z) ∈ L∞ (DZ ). 24) This problem is called Problem Bt . 24). 25) where Φ(Z) is an analytic function in DZ . Hence T is non-empty. If we can prove that T is both open and closed in [0,1], then we can derive that T = [0, 1]. e. 22) is solvable. In order to prove that T is a open set in [0,1], let t0 ∈ T . 26) ˆ then has a solution w1 (z) (w1 (z) ∈ C(D)).

24) with k = 0 is derived. 2) satisfies Condition C. 5) has a unique solution in D. 5). 5). It is easy to see that w(z) = w1 (z) − w2 (z) satisfy the homogeneous equation and boundary conditions wZ = [A1 w + A2 w]/H(y) in DZ , Re[λ((Z))w(z(Z))] = 0 in ∂D ∗, Im[λ(z0 )Φ(Z0 )] = 0. 1, Chapter IV, [87]1), it is not difficult to derive that Φ(Z) = 0 in DZ , hence w(z) = w1 (z) − w2 (z) = 0 in D. 5), we can prove by using the method of parameter extension. 5) can be rewritten as wZ = F (Z, w), F (Z, w) = 1 {A1 [z(Z)]w+A2 [z(Z)]w+A3 [z(Z)]} in DZ .

Chapter I Elliptic Complex Equations of First Order 39 3. 19), we see that w = T [W, t] (0 ≤ t ≤ 1) does not have a solution w(z) on the boundary ∂BM = BM \BM . 1). 4 the following result can be derived. 4, the following statements hold. 1) is solvable. (2) If 0 ≤ K < N , then the total number of solvability conditions for Problem A does not exceed N − K. (3) If K < 0, then Problem A has N − 2K − 1 solvability conditions. In latter chapters the notations Mj = Mj (p0 , δ, k, D), Mj = Mj (p0 , δ, k, D) (j is a positive integer) mean all non-negative constants dependent on p0 , δ, k, D.