A dynamic multigrid algorithm suitable for partial differential equations with singular solutions

2005 ◽  
pp. 190-199 ◽  
Author(s):  
Maria Cecilia Rivara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Singular Solutions ◽  
Partial Differential ◽  
Multigrid Algorithm

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.

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