scholarly journals On the arbitrariness of the general solution of an involutive partial differential equation

1994 ◽  
Vol 35 (1) ◽  
pp. 486-498 ◽  
Author(s):  
Werner M. Seiler
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
General Solution ◽  
Partial Differential

The spectrum of the self-adjoint partial differential equation in any domain

1933 ◽  
Vol 29 (1) ◽  
pp. 23-44
Author(s):  
S. W. P. Steen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemann Surface ◽  
General Solution ◽  
Boundary Conditions ◽  
Continuous Functions ◽  
The Self ◽  
Partial Differential ◽  
Image Position ◽  
Dimensional Domain

The object of this paper is to obtain the general solution to the self-adjoint partial differential equation in n dimensionswhere pij, q and ρ and bounded, continuous functions of (x1,…, xn) in a domain D and on its boundary, and where ∑pijXiXj≥0 for all (x1,…, xn) of D and all X1,…, Xn. The domain D is an n-dimensional domain and may be either the whole or part of a Riemann surface space of n dimensions. Its boundary is to consist of any number, zero, finite or enumerable, of continuous continua of n − 1 dimensions. These terms will be explained in paragraph II. The solution u = u(x1,…, xn; t) will be valid for (x1,…, xn) in D and t ≥ 0, and will satisfy boundary conditions of the type or similar, these conditions becoming identical at any part of the boundary of D that lies at infinity.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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