Related papers: Group Classification of a generalization of the He…
A heat equation with non-constant diffusivity depending as a power law on the spatial variable is analysed using Lie's method to identify classical point symmetries. It is shown that the group invariant solutions of a four-dimensional…
The heat equation is considered in the complex system consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies a Newton-type boundary condition is imposed. An equation for the limiting…
We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method…
We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
The heat kernel method is extended to the case of finite temperature. Special emphasis is given to the study of gauge theories. Due to the compactness of space in the Euclidean time direction (inverse temperature) the field strength cannot…
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of…
In this paper, we consider the following indefinite fully fractional heat equation involving the master operator . Under certain assumptions of the indefinite nonlinearity and its weight, we prove that there is no positive bounded solution,…
We present a new analytic expression for the heat current through a general harmonic network coupled with Ohmic reservoirs. We use a new method that enables us to express the stationary state of the network in terms of the eigenvectors and…
Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including…
Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form…
Batch normalization has been widely used to improve optimization in deep neural networks. While the uncertainty in batch statistics can act as a regularizer, using these dataset statistics specific to the training set impairs generalization…
In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is…
The aim of this note is twofold. The first one is to find conditions on the asymptotic sequence which ensures differentiation of a general asymptotic expansion with respect to it. Our method results from the classical one but generalizes…
We propose a generalized model of a heat engine and calculate the minimum and maximum bounds on the efficiency at maximum power. We obtain a universal form of generalized extreme bounds on the efficiency at maximum power. Our model unifies…
We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes,…
Equations of motion corresponding to the H\'{e}non - Heiles system are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential…
Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of…
We report on a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisenberg chain we derive, for arbitrary values of the anysotropy, a single non-linear…
We study nodal lines of solutions to the heat equations. We are interested in the global geometry of nodal sets, in the whole domain of definition of the solution. The local structure of nodal sets is a well understander subject, while the…