449 research outputs found
Reduction Operators of Linear Second-Order Parabolic Equations
The reduction operators, i.e., the operators of nonclassical (conditional)
symmetry, of (1+1)-dimensional second order linear parabolic partial
differential equations and all the possible reductions of these equations to
ordinary differential ones are exhaustively described. This problem proves to
be equivalent, in some sense, to solving the initial equations. The ``no-go''
result is extended to the investigation of point transformations (admissible
transformations, equivalence transformations, Lie symmetries) and Lie
reductions of the determining equations for the nonclassical symmetries.
Transformations linearizing the determining equations are obtained in the
general case and under different additional constraints. A nontrivial example
illustrating applications of reduction operators to finding exact solutions of
equations from the class under consideration is presented. An observed
connection between reduction operators and Darboux transformations is
discussed.Comment: 31 pages, minor misprints are correcte
Group Analysis of Nonlinear Fin Equations
Group classification of a class of nonlinear fin equations is carried out
exhaustively. Additional equivalence transformations and conditional
equivalence groups are also found. They allow to simplify results of
classification and further applications of them. The derived Lie symmetries are
used to construct exact solutions of truly nonlinear equations for the class
under consideration. Nonclassical symmetries of the fin equations are
discussed. Adduced results amend and essentially generalize recent works on the
subject [M. Pakdemirli and A.Z. Sahin, Appl. Math. Lett., 2006, V.19, 378-384;
A.H. Bokhari, A.H. Kara and F.D. Zaman, Appl. Math. Lett., 2006, V.19,
1356-1340].Comment: 6 page
Improving physics-informed DeepONets with hard constraints
Current physics-informed (standard or operator) neural networks still rely on
accurately learning the initial conditions of the system they are solving. In
contrast, standard numerical methods evolve such initial conditions without
needing to learn these. In this study, we propose to improve current
physics-informed deep learning strategies such that initial conditions do not
need to be learned and are represented exactly in the predicted solution.
Moreover, this method guarantees that when a DeepONet is applied multiple times
to time step a solution, the resulting function is continuous.Comment: 15 pages, 5 figures, 4 tables; release versio
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
ТЕРМІНОЛОГІЧНА НОМІНАЦІЯ У СФЕРІ ВОДНОГО ГОСПОДАРСТВА: СПІВСТАВНИЙ АСПЕКТ (Terminology nomination in a sphere of water economy: proportional aspect)
Здійснюється аналіз термінологічної номінації водогосподарських понять, представлених засобами різних
мов. виявляються спільні характеристики та специфічні для окремих мов ознаки, розглядаються пов’язані з цим проблеми перекладу водогосподарських термінів.
(The analysis of terminological nomination of water notions presented by, different languages is made.common
characteristics and specific for some languages features are shown, problems connected with water terms translation
are considered.
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