scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

On a Linear Partial Differential Equation of Hyperbolic Type

1922 ◽  
Vol 41 ◽  
pp. 76-81
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Partial Differential Equation ◽  
Sound Waves ◽  
Partial Differential ◽  
Method Of Solution ◽  
Image Position

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equationwhere α, β, γ are functions of x and y.

scholarly journals Decomposition theorems for the generalized metaharmonic equation in several independent variables

1972 ◽  
Vol 13 (1) ◽  
pp. 35-46
Author(s):  
David Colton
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Radiation Condition ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Entire Space ◽  
Partial Differential ◽  
Independent Variables ◽  
Image Position ◽  
Decomposition Theorems

In this paper solutions of the generalized metaharmonic equation in several independent variables where λ > 0 are uniquely decomposed into the sum of a solution regular in the entire space and one satisfying a generalized Sommerfeld radiation condition. Due to the singular nature of the partial differential equation under investigation it is shown that the radiation condition in general must hold uniformly in a domain lying in the space of several complex variables. This result indicates that function theoretic methods are not only the correct and natural avenue of approach in the study of singular ordinary differential equations, but are basic in the investigation of singular partial differential equations as well.

scholarly journals XX.—A Demonstration of Lagrange's Rule for the Solution of a Linear Partial Differential Equation, with some Historical Remarks on Defective Demonstrations hitherto Current

1892 ◽  
Vol 36 (2) ◽  
pp. 551-562 ◽  
Author(s):  
G. Chrystal
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
English Text ◽  
Exact Nature ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Historical Remarks

It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

scholarly journals Galois theory of aigebraic and differential equations

1996 ◽  
Vol 144 ◽  
pp. 1-58 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galois Theory ◽  
Linear Partial Differential Equation ◽  
Linear Algebraic Group ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Infinite Dimensional ◽  
Linear Partial Differential Equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

III. On the calculus of symbols (fifth memoir), with applica­tions to linear partial differential equations, and the calculus of functions

1864 ◽  
Vol 13 ◽  
pp. 227-228
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Functional Equations ◽  
Partial Differential ◽  
Independent Variables ◽  
Partial Differentiation ◽  
Linear Partial Differential Equations ◽  
Constant Coefficients ◽  
Extensive Class

In applying the calculus of symbols to partial differential equations, we find an extensive class with coefficients involving the independent variables which may in fact, like differential equations with constant coefficients, be solved by the rules which apply to ordinary algebraical equations; for there are certain functions of the symbols of partial differentiation which com­bine with certain functions of the independent variables according to the laws of combination of common algebraical quantities. In the first part of this memoir I have investigated the nature of these symbols, and applied them to the solution of partial differential equations. In the second part I have applied the calculus of symbols to the solution of functional equa­tions. For this purpose I have worked out some cases of symbolical division on a modified type, so that the symbols may embrace a greater range. I have then shown how certain functional equations may be expressed in a symbolical form, and have solved them by methods analo­gous to those already explained.

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