In this article, we study the existence and uniqueness of distributional solution for semilinear fractional problems of Dirichlet form involving new operator. By means of the Leray–Schauder degree theory, we establish the existence results, with suitable assumptions on the semilinear term g and the contraction principle to prove the uniqueness in a particular case, then the numerical study of this problem using the finite difference method.
The characterization of both volatiles and fatty acids of Tunisian olive varieties (Ouslati and Chemlali) is achieved in order to understand their correlation with the aroma accumulation via the lipoxygenase (LOX) pathway. The main identified volatiles in both crushed pulps and crushed stones are hexanal, E‐(2)‐hexenal, (Z)‐3‐hexen‐1‐ol, 1‐hexanol, and hexyl acetate. With regard to their fatty acid composition, results show that both de‐stoned (DOO) and conventional (COO) olive oils contain mainly unsaturated fatty acids. The results also show that oleic acid (C18:1) is the main fatty acid in both DOO and COO. Meanwhile, linoleic acid (C18:2) is present in a relatively higher percentage (5.2% and 19.8%, respectively, for Ouslati and Chemlali DOI) than linolenic fatty acid (C18:3) (<2%). The analysis of volatiles shows that more than 80% of total LOX‐compounds from both crushed pulps and crushed stones have linolenic acid as synthesis precursor despite its lower level (2%). Moreover, the contribution of olive parts in generating aroma is much higher in crushed pulps (more than 80%) than in crushed stones, which explains the fact that DOO, extracted from crushed pulps, contains the dominant amount of volatiles. Therefore, the obtained results promote both the consumption and the marketing of DOO.
Practical Applications: The aroma and the organoleptic properties of olive oil are strictly correlated with both the olives quality and the extraction techniques used. Many technologies are developed to extract olive oil with highest quality. De‐stoned olive oil extraction process is one of these technical procedures. In the present work, the characterization of volatiles in pulp and stone separately provides important information about the contribution of olive constitutive parts on the accumulation of olive oil aroma during the oil extraction process.
The main goal of this manuscript is to study the existence results in the Bessel Potential space for the following convection‐reaction fractional problem involving by distributional Riesz fractional derivative,
{array−Dα.(Dαu(x))−divαbφ(u(x))=f(x,u(x),Dαu(x))arrayinΩ,arrayu=0arrayonℝd/Ω,$$ \left\{\begin{array}{cc}-{D}&amp;#x0005E;{\alpha }.\left({D}&amp;#x0005E;{\alpha }u(x)\right)- di{v}&amp;#x0005E;{\alpha}\left( b\varphi \left(u(x)\right)\right)&amp;#x0003D;f\left(x,u(x),{D}&amp;#x0005E;{\alpha }u(x)\right)\kern0.60em &amp; \kern0.1em \mathrm{in}\kern0.51em \Omega, \\ {}u&amp;#x0003D;0\kern0.30em &amp; \kern0.1em \mathrm{on}\kern0.51em {\mathbb{R}}&amp;#x0005E;d/\Omega, \end{array}\right. $$
where
normalΩ⊂ℝd$$ \Omega \subset {\mathbb{R}}&amp;#x0005E;d $$ is a bounded open set with a Lipschitz boundary,
α∈false(0,1false)$$ \alpha \in \left(0,1\right) $$ and
d>2α$$ d&gt;2\alpha $$. To reach our goal, we use the application of the Schauder fixed point theory with some assumption on the nonlinear terms
φ$$ \varphi $$ and
f$$ f $$.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.