40 research outputs found
Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations
Q-conditional symmetries (nonclassical symmetries) for a general class of
two-component reaction-diffusion systems with constant diffusivities are
studied. Using the recently introduced notion of Q-conditional symmetries of
the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207),
an exhaustive list of reaction-diffusion systems admitting such symmetry is
derived. The form-preserving transformations for this class of systems are
constructed and it is shown that this list contains only non-equivalent
systems. The obtained symmetries permit to reduce the reaction-diffusion
systems under study to two-dimensional systems of ordinary differential
equations and to find exact solutions. As a non-trivial example, multiparameter
families of exact solutions are explicitly constructed for two nonlinear
reaction-diffusion systems. A possible interpretation to a biologically
motivated model is presented
Lie and conditional symmetries of a class of nonlinear (1+2)-dimensional boundary value problems
A new definition of conditional invariance for boundary value problems
involving a wide range of boundary conditions (including initial value problems
as a special case) is proposed. It is shown that other definitions worked out
in order to find Lie symmetries of boundary value problems with standard
boundary conditions, follow as particular cases from our definition. Simple
examples of direct applicability to the nonlinear problems arising in
applications are demonstrated. Moreover, the successful application of the
definition for the Lie and conditional symmetry classification of a class of
(1+2)-dimensional nonlinear boundary value problems governed by the nonlinear
diffusion equation in a semi-infinite domain is realised. In particular, it is
proved that there is a special exponent, , for the power diffusivity
when the problem in question with non-vanishing flux on the boundary
admits additional Lie symmetry operators compared to the case . In
order to demonstrate the applicability of the symmetries derived, they are used
for reducing the nonlinear problems with power diffusivity and a constant
non-zero flux on the boundary (such problems are common in applications and
describing a wide range of phenomena) to (1+1)-dimensional problems. The
structure and properties of the problems obtained are briefly analysed.
Finally, some results demonstrating how Lie invariance of the boundary value
problem in question depends on geometry of the domain are presented.Comment: 25 pages; the main results were presented at the Conference Symmetry,
Methods, Applications and Related Fields, Vancouver, Canada, May 13-16, 201
A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis
A mathematical model for fluid and solute transport in peritoneal dialysis is
constructed. The model is based on a three-component nonlinear system of
two-dimensional partial differential equations for fluid, glucose and albumin
transport with the relevant boundary and initial conditions. Its aim is to
model ultrafiltration of water combined with inflow of glucose to the tissue
and removal of albumin from the body during dialysis, and it does this by
finding the spatial distributions of glucose and albumin concentrations and
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 certain assumptions the model are simplified
with the aim of obtaining exact formulae for spatially non-uniform steady-state
solutions.
As the result, the exact formulae for the fluid fluxes from blood to 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.Comment: 28 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1110.128
Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour
New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. II
In the first part of this paper math-ph/0612078, a complete description of
Q-conditional symmetries for two classes of reaction-diffusion-convection
equations with power diffusivities is derived. It was shown that all the known
results for reaction-diffusion equations with power diffusivities follow as
particular cases from those obtained in math-ph/0612078 but not vise versa. In
the second part the symmetries obtained in are successfully applied for
constructing exact solutions of the relevant equations. In the particular case,
new exact solutions of nonlinear reaction-diffusion-convection (RDC) equations
arising in application and their natural generalizations are found
Conservation laws for self-adjoint first order evolution equations
In this work we consider the problem on group classification and conservation
laws of the general first order evolution equations. We obtain the subclasses
of these general equations which are quasi-self-adjoint and self-adjoint. By
using the recent Ibragimov's Theorem on conservation laws, we establish the
conservation laws of the equations admiting self-adjoint equations. We
illustrate our results applying them to the inviscid Burgers' equation. In
particular an infinite number of new symmetries of these equations are found
and their corresponding conservation laws are established.Comment: This manuscript has been accepted for publication in Journal of
Nonlinear Mathematical Physic
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
A new approach to group classification problems and more general
investigations on transformational properties of classes of differential
equations is proposed. It is based on mappings between classes of differential
equations, generated by families of point transformations. A class of variable
coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the
general form () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with is singled out. The group classifications
of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and
all point transformations. The set of admissible transformations of the imaged
class is exhaustively described in the general case . The procedure of
classification of nonclassical symmetries, which involves mappings between
classes of differential equations, is discussed. Wide families of new exact
solutions are also constructed for equations from the classes under
consideration by the classical method of Lie reductions and by generation of
new solutions from known ones for other equations with point transformations of
different kinds (such as additional equivalence transformations and mappings
between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica
New variable separation approach: application to nonlinear diffusion equations
The concept of the derivative-dependent functional separable solution, as a
generalization to the functional separable solution, is proposed. As an
application, it is used to discuss the generalized nonlinear diffusion
equations based on the generalized conditional symmetry approach. As a
consequence, a complete list of canonical forms for such equations which admit
the derivative-dependent functional separable solutions is obtained and some
exact solutions to the resulting equations are described.Comment: 19 pages, 2 fig
Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
We discuss the classical statement of group classification problem and some
its extensions in the general case. After that, we carry out the complete
extended group classification for a class of (1+1)-dimensional nonlinear
diffusion--convection equations with coefficients depending on the space
variable. At first, we construct the usual equivalence group and the extended
one including transformations which are nonlocal with respect to arbitrary
elements. The extended equivalence group has interesting structure since it
contains a non-trivial subgroup of non-local gauge equivalence transformations.
The complete group classification of the class under consideration is carried
out with respect to the extended equivalence group and with respect to the set
of all point transformations. Usage of extended equivalence and correct choice
of gauges of arbitrary elements play the major role for simple and clear
formulation of the final results. The set of admissible transformations of this
class is preliminary investigated.Comment: 25 page