Nonlinear superposition formula for the Kaup–Kupershmidt partial differential equation

2000 ◽  
Vol 144 (1-2) ◽  
pp. 211-220 ◽  
Author(s):  
Micheline Musette ◽  
Caroline Verhoeven
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential ◽  
Nonlinear Superposition ◽  
Nonlinear Superposition Formula

scholarly journals A Class of PDEs with Nonlinear Superposition Principles

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Li Peng ◽  
Liu Keying ◽  
Pan Zuliang ◽  
Zhong Weizhou
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Group ◽  
Nonlinear Pde ◽  
Partial Differential ◽  
Nonlinear Superposition ◽  
Invariant Group ◽  
Group A ◽  
Superposition Principles

Through assuming that nonlinear superposition principles (NLSPs) are embedded in a Lie group, a class of 3rd-order PDEs is derived from a general determining equation that determine the invariant group. The corresponding NLSPs and transformation to linearize the nonlinear PDE are found, hence the governing PDE is provedC-integrable. In the end, some applications of the PDEs are explained, which shows that the result has very subtle relations with linearization of partial differential equation.

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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