Green's Functions for Sturm-Liouville Problems on Directed Tree Graphs

Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\mathcal{L}[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed continuous on each edge, and $p$ strictly positive on $\Gamma$. The problem is to find a solution $f:\Gamma \to \mathbb{R}$ to the problem $\mathcal{L}[f] = h$ with $2m$ additional conditions at the nodes of $\Gamma$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $\rho$. Node conditions are given in the form of linear functionals $\l_1,\ldots,\l_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:\Gamma\times \Gamma \to \mathbb{R}$ associated to this problem. Namely, the solution to the semi-homogenous problem $\mathcal{L}[f] = h$, $\l_i[f] =0$ for $i=1,\ldots,2m$ is given by $f(x) = \int_\Gamma G(x,y) h(y)\,d\rho$.

Green's Functions for Sturm-Liouville Problems on Directed Tree Graphs Funciones de Green para problemas de Sturm-Liouville en árboles direccionales JORGE M. RAMIREZ1

1Universidad Nacional de Colombia, Medellín, Colombia. Email:jmramirezo@unal.edu.co 

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Abstract

Let Γ be geometric tree graph with m edges and consider the second order Sturm-Liouville operator L[u]=(-pu')'+qu acting on functions that are continuous on all of Γ, and twice continuously differentiable in the interior of each edge. The functions p and q are assumed continuous on each edge, and p strictly positive on Γ. The problem is to find a solution f:Γ → R to the problem L[f] = h with 2madditional conditions at the nodes of Γ. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of f with respect to a positive measure ρ. Node conditions are given in the form of linear functionals \l1,…,\l2m acting on the space of admissible functions. A novel formula is given for the Green's function G:Γ\times Γ → R associated to this problem. Namely, the solution to the semi-homogenous problem L[f] = h, \li[f] =0 for i=1,…,2m is given by f(x) = \intΓ G(x,y) h(y)\,dρ.

Key words: Problema Sturm-Liouville en grafo, función de Green.

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2000 Mathematics Subject Classification: 34B24, 35R02, 35J08.

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Resumen

Sea Γ un grafo tipo árbol con m aristas y considere el operador de Sturm-Liouville L[u]=(-pu')'+qu definido en el espacio de funciones continuas en Γ y continuamente diferenciables dos veces al interior de cada arista de Γ. Las funciones p y q se suponen continuas en cada arista, y p es estrictamente positiva en todo Γ. El problema consiste en hallar la solución f : Γ → R al problema dado por L[f] = h mas 2m condiciones en los nodos de Γ: en los nodos internos se especifican continuidad de f y condiciones de salto para las derivadas de f con respecto a una medida ρ. Estas condiciones de nodo se expresan en la forma de funcionales lineales \l1,…,\l2m actuando sobre el espacio de funciones admisibles para L. Se presenta una nueva fórmula para la función de Green G:Γ\times Γ → R asociada con este problema. Es decir, se expresa la solución del problema semi-homogéneo L[f] = h,\li[f] =0 para i=1,…,2m como f(x) = \intΓ G(x,y) h(y)\,dρ.

Palabras clave: Sturm-Liouville problems on graphs, Green's function.

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Texto completo disponible en PDF

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References

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[8] J. M. Ramirez, 'Population Persistence Under Advection-Diffusion in River Networks', Journal of Mathematical Biology, arXiv:1103.5488 (2011). To Appear.

[9] J. Roth, Le spectre du laplacien sur un graphe, 'Th\'eorie du potentiel (Orsay, 1983)', 1984, Vol. 1096 ofLecture Notes in Math., Springer, Berlin, Germany, p. 521-539.

[10] A. Zettl, Sturm-Liouville Theory, American Mathematical Soc., 2005.

(Recibido en mayo de 2011. Aceptado en mayo de 2012)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv46n1a02, 
AUTHOR = {Ramirez, Jorge M.}, 
TITLE = {{Green's Functions for Sturm-Liouville Problems on Directed Tree Graphs}}, 
JOURNAL = {Revista Colombiana de Matemáticas}, 
YEAR = {2012}, 
volume = {46}, 
number = {1}, 
pages = {15--25} 
}