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José Villa-Morales

Publications and source records attributed to José Villa-Morales.

12 recordsLinked to original sources
arXiv2025

Some stochastic process techniques applied to deterministic models

Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the distribution of exit times of the stochastic process within a bounded domain. These quantities are obtained by solving elliptic and parabolic partial differential equations (PDEs), respectively. To support practical applications, we propose a numerical scheme implemented in FreeFEM, emphasizing its effectiveness in two- and three-dimensional cases due to the software's limitations in higher dimensions. The examples provided illustrate the theoretical results, which extend known one-dimensional solutions to higher-dimensional settings. This contribution bridges theoretical and computational approaches for analyzing stochastic processes in multidimensional domains, offering insights into their behavior and potential applications.

math.PR↗
arXiv2025

An Application of Fractional Calculus to Column Theory

In this article, we employ a fractional version of the radius of curvature in Euler's equation for column buckling, enabling us to derive a fractional differential equation in the Caputo sense. We solve this equation and demonstrate that for certain values of the fractional parameter, there exists a critical buckling force. Additionally, we provide a numerical scheme for accurately approximating this critical force.

math.GM↗
arXiv2019

Blow-up at space infinity for solutions of a system of non-autonomous semilinear heat equations

In this paper we will see that the global or local existence of solutions to \begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t) Δu_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial u_{2}}{\partial t} & = & \mathit{k}_{2} (t) Δu_{2} + h_{2}(t) u_{2}^{p_{22}} u_{1}^{p_{21}}, \end{eqnarray*} depends on the initial datums and the global or local existence of solutions to \begin{eqnarray*} \dfrac{dy_{1}}{dt} & = & h_{1}(t) y_{1}^{p_{11}}(t) y_{2}^{p_{12}}(t),\\ \dfrac{dy_{2}}{dt} & = & h_{2}(t) y_{2}^{p_{22}}(t) y_{1}^{p_{21}}(t). \end{eqnarray*} We also give some bounds for the maximal existence time of the partial differential system. Moreover, if such existence time is finite and $\min\{p_{11} + p_{12},p_{22}+p_{21}\} > 1$ then we will prove the partial differential system has solutions that blows-up at space infinite.

math.AP↗
arXiv2019

Certain fractional Laplacian equations that do not have smooth solutions

Let $f$ be a real-valued function defined on $\mathbb{R}$, with $f(0) \neq 0$ and which is not constant in non empty open intervals. We prove the equations \begin{equation}\label{edif} \left\{ \begin{array}{rcll} (-Δ)^{s}u & = & f(u), & \text{in }B_{1}, \\ u & = & 0, & \text{in }B_{1}^{c}, \end{array} \right. \end{equation} where $(-Δ)^{s}$ is the $s$-fractional Laplacian, $0< s <1$, have no solutions in $C^{2}(\overline{B_{1}})$, if $d>2s$. The proof is based on the moving plane method and in the approximation of $C^{2}(\overline{B_{1}})$ functions by $s$-harmonic functions.

math.AP↗
arXiv2016

Solution of the Dirichlet problem for the equation $aΔu+b\cdot \nabla u=0$ by the Monte Carlo method

In this paper we study the Dirichlet problem corresponding to an open bounded set $D\subset \mathbb{R}^{d}$ and the operator \begin{equation*} A=\sum_{i=1}^{d}a\frac{\partial ^{2}}{\partial x_{i}^{2}} +\sum_{i=1}^{d}b_{i}\frac{\partial }{\partial x_{i}}, \end{equation*} where $a>0$ and $b\in \mathbb{R}^{d}$. We define a mean value property and prove that a function $u$ has such property in $D$ if and only if $Au=0$ in $D$. Using this characterization, and a drifted Brownian motion, we define a family of random variables that converges almost surely and the limit is used to give an explicit representation for the solutions to the Dirichlet problem, this immediately implies the uniqueness. On the other hand, the existence of the solution is proved imposing a regular condition on the boundary of $D$.

math.PR↗
arXiv2016

Global existence and persistence of mass for a nonlinear equation with fractional Laplacian

In this paper we study the partial differential equation \begin{equation} \begin{split} \partial_tu &= k(t)Δ_αu - h(t)φ(u), u(0) &= u_0. \end{split} \end{equation} Here $Δ_α$ is the fractional Laplacian, $k,h:[0,\infty)\to[0,\infty)$ are continuous functions and $φ:\mathbb{R}\to[0,\infty)$ is a convex differentiable function. If $u_0\in C_b(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$ we prove that the above equation has a classical global solution and is non-negative if $u_0\geq0$. Imposing some restrictions on the parameters we prove that the mass $M(t)=\int_{\mathbb{R}^d}u(t,x)dx$, $t>0$, of the system $u$ does not vanish in finite time, moreover we see that $\lim_{t\to\infty}M(t)>0$, under the restriction $\int_0^\infty h(s)ds<\infty$. A comparison result is also obtained for non-negative solutions, and as an application we get a better condition when $φ$ is a power function.

math.AP↗
arXiv2014

An Osgood's criterion for a semilinear stochastic differential equation

The purpose of this paper is to give an Osgood's criterion for solutions of semilinear stochastic differential equations of the form $X_{t}=ξ+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}σ(s)X_{s}dW_{s},\ t\geq 0$. Here, $b$ is a non-negative, non-decreasing by components and continuous random field and $σ$ is a predictable and continuous process. Also we present a generalization of the so-called Feller's test whenever $σ\equiv 1$.

math.PR↗
arXiv2013

Existence of mild solutions for a system of partial differential equations with time-dependent generators

We give sufficient conditions for global existence of positive mild solutions for the weak coupled system: \begin{eqnarray*} \frac{\partial u_{1}}{\partial t} &=&ρ_{1}t^{ρ_{1}-1}Δ_{α_{1}}u_{1}+t^{σ_{1}}u_{2}^{β_{1}},\ \ u_{1}\left(0\right) =φ_{1}, \\ \frac{\partial u_{2}}{\partial t} &=&ρ_{2}t^{ρ_{2}-1}Δ_{α_{2}}u_{2}+t^{σ_{2}}u_{1}^{β_{2}},\ \ u_{2}\left(0\right) =φ_{2}, \end{eqnarray*} where $Δ_{α_{i}}$ is a fractional Laplacian, $0<α_{i}\leq 2,\ β_{i}>1,\ ρ_{i}>0,\ σ_{i}>-1\ $are constants and the initial data $φ_{i}$ are positive, bounded and integrable functions.

math.AP↗
arXiv2013

On the Distribution of Explosion Time of Stochastic Differential Equations

In this paper we use the Itô's formula and comparison theorems to study the blow-up in finite time of stochastic differential equations driven by a Brownian motion. In particular, we obtain an extension of Osgood criterion, which can be applied to some nonautonomous stochastic differential equations with additive Wiener integral noise. In most cases we are able to provide with a method to figure out the distribution of the explosion time of the involved equation.

math.PR↗