Envelopes of families of framed surfaces and singular solutions of first-order partial differential equations

2020 ◽  
pp. 1-28
Author(s):  
Masatomo Takahashi ◽  
Haiou Yu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Singular Solutions ◽  
Uniqueness Theorems ◽  
Singular Surfaces ◽  
Partial Differential ◽  
First Order ◽  
Existence Conditions ◽  
Two Parameter

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

XIII.—“On the Singular Solutions of Partial Differential Equations of the First Order.”

1913 ◽  
Vol 32 ◽  
pp. 150-163
Author(s):  
H. Levy
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Theorem ◽  
Infinite Series ◽  
Singular Solutions ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Original Case

The complete integral of the differential equationφ(xyzpq) = 0is a relation among the variables, which includes as many arbitrary constants as there are independent variables. But it is important to distinguish carefully between differential equations which have been formed by the elimination of constants from some complete primitive, and those whose origin is quite unknown, or which may have been constructed by some method totally different from the first.In the original case, the differential equation can always be integrated in finite terms, while in the latter, only under the conditions laid down in Cauchy's Existence Theorem can an integral be obtained, and even then usually as an infinite series.

ON SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS OF BRIOT–BOUQUET TYPE

2018 ◽  
Vol 98 (1) ◽  
pp. 122-133
Author(s):  
FENGBAI LI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Singular Solutions ◽  
Partial Differential ◽  
First Order ◽  
Associated Matrix ◽  
Necessary And Sufficient

We study systems of partial differential equations of Briot–Bouquet type. The existence of holomorphic solutions to such systems largely depends on the eigenvalues of an associated matrix. For the noninteger case, we generalise the well-known result of Gérard and Tahara [‘Holomorphic and singular solutions of nonlinear singular first order partial differential equations’, Publ. Res. Inst. Math. Sci.26 (1990), 979–1000] for Briot–Bouquet type equations to Briot–Bouquet type systems. For the integer case, we introduce a sequence of blow-up like changes of variables and give necessary and sufficient conditions for the existence of holomorphic solutions. We also give some examples to illustrate our results.

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