Abstract: A decomposition of a graph G is a collection ψ of
graphs H1,H2, . . . , Hr of G such that every edge of G belongs
to exactly one Hi. If each Hi is either an induced path in G,
then ψ is called an induced acyclic path decomposition of G and
if each Hi is a (induced) cycle in G then ψ is called a (induced)
cycle decomposition of G. The minimum cardinality of an induced
acyclic path decomposition of G is called the induced acyclic path
decomposition number of G and is denoted by ¤Çia(G). Similarly
the cyclic decomposition number ¤Çc(G) is defined. In this paper we
begin an investigation of these parameters.
Abstract: This paper investigates the inverse problem of determining
the unknown time-dependent leading coefficient in the parabolic
equation using the usual conditions of the direct problem and an additional
condition. An algorithm is developed for solving numerically
the inverse problem using the technique of space decomposition in a
reproducing kernel space. The leading coefficients can be solved by a
lower triangular linear system. Numerical experiments are presented
to show the efficiency of the proposed methods.