Related papers: Ordinary differential equations associated with th…
This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon…
We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ans\"atze reducing nonlinear evolution equations to system of ordinary differential equations. The ans\"atze are…
A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space…
We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are…
Using previous results and general thermodynamical formalism,an expression is obtained for the specific heat per particle under constant volume of a degenerate non-relativistic electron gas on a 1D lattice.The result is a non-linear…
Integral equation based numerical methods are directly applicable to homogeneous elliptic PDEs, and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, extensions to problems with inhomogeneous…
Difference equations have many applications and play an important role in numerical analysis, probability, statistics, combinatorics, computer science, quantum consciousness, etc. We first prove that the partial differential equation is…
We investigate the thermal responses of a harmonic oscillator chain coupled at its boundaries to heat baths held at different temperatures. This setup sustains a steady energy flux, continuously dissipating heat into both reservoirs. By…
A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for…
We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal…
We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely \mbox{$L^1$-coercivity}.…
In this paper, we study fractional order heat equation in higher space-time dimensions and offer specific role of heat flows in various fractional dimensions. We offer fractional solutions of the heat equations thus obtained, and examine…
We consider the heat equation with a superlinear absorption term $\partial_{t} u-\Delta u= -u^{p}$ in $\mathbb{R}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set,…
The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial…
We obtain necessary conditions and sufficient conditions for the solvability of the heat equation in a half-space of ${\bf R}^N$ with a nonlinear boundary condition. Furthermore, we study the relationship between the life span of the…
For an elliptic differential operator $D$ of order $h$ in $n$ dimensions, the spectral $\zeta$-function $\zeta_D(s)$ for $\Re s > \frac{n}{h}$ can be evaluated as an integral over the heat kernel $e^{-t D}$. Here, alternative expressions…
Starting from a master equation in a quantum Hamilton form we study analytically a nonequilibrium system which is coupled locally to two heat bathes at different temperatures. Based on a lattice gas description an evolution equation for the…
Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
The exact evolution in time and space of a distribution of the temperature (or density of diffusing matter) in an isotropic homogeneous medium is determined where the initial distribution is described by a piecewise polynomial. In two…