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.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

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.

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.

scholarly journals Conservation laws for second-order parabolic partial differential equations

2004 ◽  
Vol 45 (3) ◽  
pp. 333-348 ◽  
Author(s):  
B. Van Brunt ◽  
D. Pidgeon ◽  
M. Vlieg-Hulstman ◽  
W. D. Halford
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Necessary And Sufficient ◽  
Derivatives Of

AbstractConservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Fréchet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

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