Mixed Problems for Linear Systems of first Order Equations

1958 ◽  
Vol 10 ◽  
pp. 127-160 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Auxiliary Data ◽  
First Order ◽  
Independent Variables ◽  
Mixed Problems ◽  
Linear Partial Differential Equations ◽  
Data Problem ◽  
Hyperbolic Differential Equations

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

scholarly journals VII. On the theory of the solution of a system of simultaneous non-linear partial ddifferential equations of the first order

1875 ◽  
Vol 23 (156-163) ◽  
pp. 510-510
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Arbitrary Constant ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
The Given

Given an equation of the form z = ϕ ( x 1 , x 2 , ..... x n+m , a 1 , a 2 ,. . . . a n ), we obtain by differentiation with respect to each of the n + m independent variables x 1 , x 2 , ..... x n+m , and elimination of the n arbitrary constant a 1 , a 2 ,. . . . a n a system of m +1 non-linear partial differential equations of the first order. Of this system the given equation may be said to be "complete primitive.”

scholarly journals On the theory of the solution of a system of simultaneous non-linear partial differential equations of the first order

1876 ◽  
Vol 24 (164-170) ◽  
pp. 337-344
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Non Linear

Given an equation of the form z = ϕ ( x 1 , x 2 , . . . x n+r , a 1 , a 2 ,... a r , a + r + 1 ), we obtain by differentiation with respect to each of the n + r variables n + r equations, together with the original equation n + r + 1 equations, from which, eliminating the r + 1 constants, we have a system of n nonlinear partial differential equations. Conversely, given a system of n non-linear partial differential equations with n + r independent variables, if there exists an equation

III. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

1863 ◽  
Vol 12 ◽  
pp. 420-424
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamical Problem ◽  
General Character ◽  
Partial Differential ◽  
Central Function ◽  
First Order ◽  
Linear Partial Differential Equations

Jacobi in a posthumous memoir, which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the differential equations of dynamics which was established by Sir W. R. Hamilton in the 'Philosophical Transactions’ for 1834-35. The knowledge, indeed, that the solution of the equation of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results.

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