Related papers: Alternative probabilistic representations of Baren…

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x)…

Conductivity of unsaturated porous media to fluids is of theoretical and applied interest to mathematicians, physicists, and chemical, petroleum, civil and agricultural engineers. We explore the expression of unsaturated relative…

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation $\partial_tu-\Delta u=|u|^\alpha u,\;\alpha>0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular,…

We consider the following quasi-linear parabolic system of backward partial differential equations: $(\partial_t+L)u+f(\cdot,\cdot,u, \nabla u\sigma)=0$ on $[0,T]\times \mathbb{R}^d\qquad u_T=\phi$, where $L$ is a possibly degenerate second…

We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the…

We introduce a nonlinear degenerate parabolic equation containing a nonlocal term. The equation serves as a replicator dynamics model where the set of strategies is a continuum. In our model the payoff operator (which is the continuous…

The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x)…

A representation for a solution $u(\omega,x)$ of the equation $-u"+q(x)u=\omega^2 u$, satisfying the initial conditions $u(\omega,0)=1$, $u'(\omega,0)=i\omega$ is derived in the form \[ u(\omega,x)=e^{i\omega x}\left(…

We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $ \mathbb{R}^N $, whose weight $\rho(x)$ behaves at spatial infinity like $ |x|^{-\gamma} $ with subcritical power, namely $ \gamma…

In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…

The paper proposes a model of optical transmittance of ultra diluted gas taking into account gas particles non-locality, the quantum effect of wave function spreading derived from solving the Schr\"odinger equation for a free particle. A…

A fairly general expression for a light beam is found as a solution of the paraxial Helmholtz equation. It is achieved by exploiting appropriately chosen complex variables which entail the separability of the equation. Next, the expression…

Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the mobility. For…

We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…

The uniform longitudinal flow is characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$ is the strain rate, a uniform density $n(t)\propto a(t)$, and a uniform granular temperature $T(t)$.…

We propose a simple, intuitive alternative method of deriving the rule for connecting asymptotic wave function amplitudes to scattering probabilities. This is illustrated using the standard example of a 1-D particle reflecting or…

The variable change w=exp(u) is applied to establish novel integral representations of the incomplete gamma-function, hypergeometric F-function,confluent hypergeometric /Phi-function and beta-function, and to analyze these functionsas as…

In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^\lambda}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient…

We consider a random process $Y(t)=\exp\{X(t)\}$, where $X(t)$ is a centered second-order process which correlation function $R(t,s)$ can be represented as $\int_{\mathbb{R}} u(t,y)\overline{u(s,y)} dy.$ A multiplicative wavelet-based…