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Nonlinear evolution equations with exponential nonlinearities: conditional symmetries and exact solutions

100%
Cherniha R., Pliukhin O.
Banach Center Publications
|
2011
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tom 93
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nr 1
105-115
EN
New Q-conditional symmetries for a class of reaction-diffusion-convection equations with exponential diffusivities are derived. It is shown that the known results for reaction-diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained are applied to construct exact solutions of the relevant nonlinear equations. An application of exact solutions to solving a boundary-value problem with constant Dirichlet conditions is presented.
2
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Lie symmetry of a class of nonlinear boundary value problems with free boundaries

100%
Cherniha R., Kovalenko S.
Banach Center Publications
|
2011
|
tom 93
|
nr 1
95-104
EN
A class of (1 + 1)-dimensional nonlinear boundary value problems (BVPs), modeling the process of melting and evaporation of solid materials, is studied by means of the classical Lie symmetry method. A new definition of invariance in Lie's sense for BVP is presented and applied to the class of BVPs in question.
3
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A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

81%
Cherniha R., Stachowska-Piętka J., Waniewski J.
International Journal of Applied Mathematics and Computer Science
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2014
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tom 24
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nr 4
837-851
EN
A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.
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