%%%%%%% TEMPLATE for ARTICLE in FCAA Journal since 2022 %%%%%%%%%%%%%
%%                                                                 %%
%% Please do not use \input{...} to include other tex files.       %%
%% Submit your LaTeX manuscript as one .tex document ONLY.         %%
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%% All additional figures and files should be attached             %%
%% separately on Editorial manager site together with              %%
%% auxiliary ones as: svjour3.cls, svglov3.clo, etc.               %%
%%%%%%%%%%%%%%%%%%%%%%% file template-svjour3.tex %%%%%%%%%%%%%%%%%%%

% This is a general template file for the LaTeX package SVJour3
% for Springer journals. Springer Heidelberg 2010/09/16
%
% Copy it to a new file with a new name and use it as the basis
% for your article. Delete % signs as needed.
%
% This template includes a few options for different layouts and
% content for various journals. Please consult a previous issue of
% your journal as needed.
% As for FCAA, this below is adopted specially for our journal ! %%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Template for FCAA article in svjour3-style for Springer  %%%%%

 \RequirePackage{fix-cm}
    %\documentclass{svjour3}                 % onecolumn (standard format)
    %\documentclass[smallcondensed]{svjour3} % onecolumn (ditto)

 \documentclass[smallextended]{svjour3} %% onecolumn (second format) %% use this one for FCAA ! %%
 \smartqed  % flush right qed marks, e.g. at end of proof %

%%% publisher's and journal definitions come here: %%%%%%%%%%%%%%%%%%%%%%%
\def\theequation{\arabic{section}.\arabic{equation}}%%% for FCAA ! %%%%%%%
   % \numberwithin{equation}{section}
\usepackage{amssymb,amsmath,amsfonts,latexsym} %%
\usepackage{hyperref}
   \newtheorem{hypothesis}{Hypothesis}
   %%% similarly you can define other missing statements as Conjecture, etc.
   %%% how to make proofs so square to appear at ends: %%%%%%%%%%%%%%%%%%%%%
 \def\proofname{\smallskip \noindent\it Proof} %%
 \def\proofend{\hfill$\Box$} % same as: \def\qed{\hfill$\Box$} % for \square
 %% may add \smallskip afterwards...

%%%%%%%%%%%%%% additional, if necessary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{mathtools}
\usepackage{multirow,array}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{epstopdf}
\usepackage{float} %%% to use [H], [ht] etc. to fix places of figures
\usepackage{color, xcolor}
\allowdisplaybreaks

%%%%%% Specific Authors Defs come here: %%%%%%%%%%

%%% ........... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%% begin make title %%%%%%%%%%%%%%%%%

\journalname{Fract. Calc. Appl. Anal.} %%%%% and FCAA-logo will appear on the right %%

\begin{document}

 %%% Pls. specify the kind of the article:
 %%% This is: %% ORIGINAL PAPER %%%% 
 %%% or: REVIEW PAPER / SHORT PAPER, etc. %%%%%%%%

%%%% Title of article for FCAA %%%%%%%%%%%%%%%%%%%%%%%%
\title{Template for articles  in ``FCAA'' journal and instructions to authors}

\titlerunning{Template for articles  in ``FCAA'' journal and  \dots}
%% if too long for running head - use the text from 1st line !%

%%%%% authors:
\author{
        Virginia Kiryakova$^1$ %%% 1 %%% V. Kiryakova %% ORCID ID: 0000-0002-4591-6958 %% pls. add IF available %%
\and
        Stefan Kiryakov$^2$ %%% 2 %%% S. Kiryakov %% ORCID ID: N/A %%%
 }

\authorrunning{V. Kiryakova \and S. Kiryakov} %%  Short form of author list % if too long for running head

%%% affiliations:
\institute{Virgnia Kiryakova$^{1,*}$
\at
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, ``Acad. G. Bontchev" Str., Block 8,
Sofia -- 1113, Bulgaria \\
\email{virginia.kiryakova@gmail.com} $^*$ corresponding author %%% obligatory indicate who is corresp.author
 \and
Stefan Kiryakov$^{2}$
\at
Diogenes Co. Ltd., %%%  + postal address of affilaition ..., %%
 Sofia -- 1510, Bulgaria \\
\email{fcaa@math.bas.bg}
}

%%% Dates:
\date{Received: 1 December 2021 / Revised: 31 December 2021 / Accepted: ......}
% These dates will be entered by the Editor (EiC, V. Kiryakova) or authors/ system %

%%%%% For Production Dept.: variants of COPYRIGHT notice to appear: %%%%%%
% if Open Access option chosen, it is as:
   %% "\copyright The Author(s) 2022" % (or next year..) %
% if No Open Access, the TCA form signed to FCAA journal is used, and it appears:
   %% "\copyright Diogenes Co. Ltd. 2022" %% or 2021, or next year %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
{\color{blue}
Text of the abstract. \dots }
This text, up to half of page, should give a comprehensive idea about the article's subject and authors' results,
so to be enough informative and to attract the readers.
The text of Abstract, as well as References list, Citation information and Dates (Received/Revised, etc.)
will be seen for all readers of the article, at the
 {\color{blue} SpringerLink website of the journal}, \hfill \break
 {\color{blue}\href{https://www.springer.com/journal/13540}{https://www.springer.com/journal/13540}}.

Since Vol. 25 (2022) the journal is published by {\color{blue} Springer Nature}, while
Vols. 14--24 (2011--2021) were published by De Gruyter, \hfill \break
\href{https://www.degruyter.com/journal/key/fca/html}{https://www.degruyter.com/journal/key/fca/html},
and older Vols. 1--13 (1998--2010) are partly available at %% \hfill \break %%
\href{http://www.math.bas.bg/complan/fcaa/}{http://www.math.bas.bg/complan/fcaa/}.

{\color{blue} Recommended size} for Original Papers is up to 24 pages ($\max$ 28-30, in exceptional cases);
while Review Papers are {\color{blue} encouraged} and can be 30-50 pages (in format close to this one); Short Papers - approx. 10 pages. {\color{blue} To submit an article}, go to Editorial Manager site,
 \href{https://www.editorialmanager.com/fcaa}{https://www.editorialmanager.com/fcaa} (see details overleaf).


%%%%%%%%%% Enter suitable key words and phrases %%% examples:
\keywords{fractional calculus (primary) \and
 Mittag-Leffler type functions \and fractional ordinary and partial differential equations \and \dots}

\subclass{26A33 (primary) \and  33E12 \and 34A08 \and 34K37 \and 35R11 \and 60G22 \and \dots}
%%% These are examples only. Pls. use MSC 2020 for suitable topic numbers %%%%%%

\end{abstract} %%%%%%%%%%%%%

%%%%%%%% begin papers' body %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \label{sec:1}

\setcounter{section}{1} \setcounter{equation}{0} %% to have proper 2-digits numbers of eqs
%% Note that this style produces 1-digit numbering of definitons, statements, exmaples, etc.

FCAA is a {\color{blue} hybrid model journal}. There are no publication fees
(then articles' full texts are available only to subscribers or pay-per-view).
There is also the {\color{blue} option for Open Access (OA)}, then APC are to be paid by authors / their institutions,
 see details at Springer website and consult your funding institution. In cases of OA, the corresponding authors will be asked to sign a Springer OA license form {\color{blue} before} the proofs will be prepared.

After 30 November 2021 all submissions to FCAA should be done by the corresponding authors only
{\color{blue} via the Editorial Manager system} (EM), at \hfill \break
{\color{blue} \href{https://www.editorialmanager.com/fcaa}{https://www.editorialmanager.com/fcaa}}.
New corresponding authors need to register first.
Follow the instructions there. Initially, as {\color{blue} \texttt{*Manuscript}}, you can (better) upload a pdf-file only (made obligatory by \LaTeX\ in one-column, and with approximate page margins 13 $\times$ 19 cm and font size 11pt).
But for revised/final versions all source files made in the Springer style are necessary to be upload in the corresponding directories of EM
(the \LaTeX\ - as *Manuscript; the pdf - in Supplementary information; the style files - in \LaTeX\ Supporting files; figures - in Table/ Figure; the FCAA-TCA copyright form - in Supplementary information; etc.).
	
This FCAA template is based on the {\color{blue} \texttt{Springer style ``svjour3''}},
we use only {\color{blue} one-column model} and typesetting only by {\color{blue} \LaTeX\ }!
The basic auxiliary files are ``svjour3.cls'' and ``svglov3.clo'' (embedded at EM site for FCAA),
the rest style files need to be upload there if used.
For more details and to download full style package, go to \
\href{https://www.overleaf.com/latex/templates/a-general-template-file-for-the-latex-package-svjour3-for-springer-journals/pbbwqhxxvtbp}
{https://www.overleaf.com/latex/templates/} \\ % \hfill \break
{a-general-template-file-for-the-latex-package-svjour3-for-springer-journals/} \hfill \break {pbbwqhxxvtbp}, \
or to \ \href{https://latextemplates.github.io/svjour/}{https://latextemplates.github.io/svjour/}.

Please make all new paragraphs indented. To separate them, use empty lines and not $\backslash \backslash$.
Try to avoid footnotes.

Include {\color{blue} References} items directly between
\verb+\begin{thebibliography}{99}+ and \verb+\end{thebibliography}+,
arranged in Alphabetic order of authors' families
and following strictly the style of the examples in last section.
Check if each item is indeed cited in the text.

%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%
 \section{Second section of the paper} \label{sec:2}

\setcounter{section}{2} \setcounter{equation}{0} %% to have proper 2-digits numbers of eqs
%% Note that this style produces 1-digit numbering of definitons, statements, exmaples, etc.

Text text text ... (for details, see \cite{GasRah}, \cite{Rosbl}, \cite{Kir}, \cite{Moak}) ...

%%%% example of definition %%%%
 \begin{definition}\label{Def1} %%% appears as Definition 1 %
 Text of Definition~\ref{Def1} ..................\, .
 \end{definition}

 %%% example of subsection:
 \subsection{\bf Preliminary results} \label{subsec:2.1} %% 2.1. %%

%%%% example of theorem %%%%%%%%%%
 \begin{theorem}\label{Th1} %%% appears as  Theorem 1 %%
 Text of Theorem~\ref{Th1} ....
 \end{theorem}

 \proof %%%%%%%%%%%%%
 Give here the proof of Theorem~\ref{Th1}. Example for a numbered equation:
\begin{equation}\label{eq2.1} %%% appears as (2.1) %%%
ax^2+bx +c = 0.
\end{equation}
Provided the condition is fulfilled
%%% equation with no number: %%
\[
\Delta = b^2-4ac >0,
\] %% or use $$ ...... $$ %%
equation \eqref{eq2.1}, has two different solutions ...

 The details of proof follows as in Ref. \cite{Moak}. %% this is only as example! %
 \proofend %%%%%%%%%%
 \smallskip

%%%%% One can use also enviroments as \begin{lemma}... \end{lemma}, etc. %%%%%

%%%% example of corollary %%%%
 \begin{corollary}\label{Cor1} %%%  appears as Corollary 1 %%%
 Text of Corollary~\ref{Cor1} ....
 \end{corollary}

 \proof %%%
 Here comes the proof of Corollary~\ref{Cor1}. ................
 \proofend %%%
 %% \smallskip

%%%%%%%%  Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Next section of the paper} \label{sec:3}

\setcounter{section}{3} \setcounter{equation}{0} %% to have proper 2-digits numbers of eqs
%% Note that this style produces 1-digit numbering of definitons, statements, exmaples, etc.

 Text ... text ...

 As seen in Section~\ref{sec:2}, Theorem~\ref{Th1},
 provided $ a \neq 0$, the equation \eqref{eq2.1} %%% or (\ref{eq2.1}) %%%
 has the solutions
 \begin{equation}\label{eq3.1} %%%% appears as (3.1) %%%
 x_{1,2}= {\frac {-b \pm \sqrt{b^2-4ac}}{2a}}\,.
 \end{equation}

%%% example of proposition %%%%%%%%%%%
 \begin{proposition}\label{Pr1} %% appears as Proposition 1 %%%
 Text of Proposition ........
 \end{proposition}

 %%% example of example %%%%%
 \begin{example}\label{Ex1} %% appears as Example 1 %%%
 Let us take in \eqref{eq3.1} ... Then, by Theorem~\ref{Th1}, we have ...
 \end{example}

 \begin{example}\label{Ex2} %% appears as Example 2 %%%
 Under the same conditions as in Example~\ref{Ex1}, we consider the case ...
 \end{example}

%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%%%%
\section{This is an example for first level head--section head} \label{sec:4}

\setcounter{section}{4} \setcounter{equation}{0} %% do not forget! %%

\subsection{\bf This is an example for second level head--subsection head} \label{subsec:4.1} %% 4.1 %%

\subsubsection{This is an example for third level head--subsubsection head}\label{subsubsec:3.1.1} %% 4.1.1 %%

Sample body text. Sample body text. Sample body text. Sample body text.

%%% .............

%%%%%%%%%%%% Section 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Figures and tables} \label{sec:5}

\setcounter{section}{5} \setcounter{equation}{0} %% do not forget! %%

 The figures should be input in the \LaTeX\ file by eps-files, as below.

%%% example for Figure 1 %%%%%%%%%%%%%%%%%%%
\begin{figure}[H] % or [htbp] %% to fix the place %% with \usepackage{float} %%
  \begin{center}
  \includegraphics[scale=0.55]{springer.eps} %% or other suitable scale %%
      \caption{Description for the figure above }
    \label{fig:1}
    \end{center}
\end{figure} %%%%%%%%%%%%%%%%%%%%

 Often the figures include texts or Latin, Greek etc. letters, using fonts that are not basic for journal's style,
 e.g. Helvetica, etc. We kindly ask the authors to take care that no text fonts will need to be embedded.
 You may save first the figures as curves, or: make a figure's pdf-file and then save it as new eps-file.
 Anyway, along with the eps-files, please obligatory provide also equivalent pdf-files for all figures.
 Or may need to embed the used fonts (as Helvetica, etc.) %%%%%%%%%%%%%%

%%%%%%%%% example for Figure 2 %%%%%%%%%%%%%%%%%
\begin{figure}[H] % or [htbp] %%%%% with \usepackage{float} %%
  \begin{center}
  \includegraphics[scale=0.4]{half-derivative.eps}
\caption{Derivatives of function $f(x)=x$ of order 1 and order $1/2$}
    \label{fig:2}
 \end{center}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Tables can be inserted via the normal table and tabular environment.

% Example for a table %%% Table 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\begin{center}
\begin{tabular}{lll}
\hline\noalign{\smallskip}
first & second & third  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
number & number & number \\
number & number & number \\
\noalign{\smallskip}\hline
\end{tabular}
\smallskip
\caption{Please write your table caption here}
\label{tab:1} % Give a unique label %
\end{center}
\end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%% Section 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cross referencing} \label{sec:6}

Environments such as figure, table, equation and align can have a label
declared via the \verb+\label{#label}+ command.
For figures and table environments use the \verb+\label{}+ command
inside or just below the \verb+\caption{}+ command.
You can then use the \verb+\ref{#label}+ command to cross-reference them.
As an example, consider the label declared for Figure~\ref{fig:1}
which is \verb+\label{fig:1}+. To cross-reference it, use the command
\verb+Figure \ref{fig:1}+, for which it comes up as ``Figure~\ref{fig:1}''.

%%%%%%%%%% Section 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions} \label{sec:6}

Conclusions may be used to restate your hypothesis or research question, emphasize your major findings, explain the relevance and the added value of your work, highlight any limitations of your study, describe future directions for research and recommendations. But do not simply repeat Abstract or phrases from Introduction.

%%%%%%%%%%   Example for Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%   but please try to avoid, unless the article's structure needs so %%%%%%%%%%%

\section{First appendix} \label{secA1}

An appendix contains supplementary information that is not an essential part of the text itself but which may be helpful in providing a more comprehensive understanding of the research problem or it is information that is too cumbersome to be included in the body of the paper.

Pay attention to have proper 2-digits numbers of equations there.

%%=============================================================%%
%% Sample for another appendix section			       %%
%%=============================================================%%

%% \section{Example of another appendix section} \label{secA2}%

%%%%%%%%%%%%%%%%%%%%%%%%%% BACK MATTERS %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{acknowledgements}
 % Acknowledgments are not compulsory. Where included they should be brief.
 % Grant or contribution numbers may be acknowledged, or help by colleagues. Example:
  The first author thanks his institution for the support, under Grant No ...
 The other authors ...
 \end{acknowledgements}

 \section*{\small
 Conflict of interest} %%%%%%%%%%%%%%%%%%%%

 {\small
 The authors declare that they have no conflict of interest.}

%%%%%%%%%%%%%%%%%% REFERENCES: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BibTeX users: please use \bibliographystyle{spmpsci} %% for math. and phys. sci.
%% Non-BibTeX users: please use the model as below !!! %%

%%%% for FCAA - pls. include directly the Refs items here ! %%%
%%%% following STRICTLY the models below %%%%%
%%%% and ARRANGE the items in ALPHABETIC ORDER for authors' family names !!!

\begin{thebibliography} {99}
 \normalsize %% to be better seen by reviewers, until final proofs ... %%

% Format for Journal Reference
\bibitem{RefJ}
Author, I.: Article title. Journal Name \textbf{Vol}(Issue), page numbers (year)
%% always separate pages by  double -- %%

% Format for Books
\bibitem{RefB}
Author, II.: Book Title. % , page numbers. %
Publisher, Place (year)

% example for a Book %% Title of Book should be in Initial Capitals ! %
\bibitem{GasRah}
 Gasper, G., Rahman, M.:
 {Basic Hypergeometric Series}.
 Cambridge University Press, Cambridge (1990)

%%% example for article published in FCAA journal - obligatory add DOI ! %%%%%%%%%%%%%%
\bibitem{Kir1}
 Machado, J.A.T., Kiryakova, V.:
 The chronicles of fractional calculus.
{Fract. Calc. Appl. Anal.} \textbf{20}(2), 307--336 (2017). DOI: 10.1515/fca-2017-0017

%% example for article in Book with Editors %%%
 \bibitem{Kir2}
 Machado, J.A.T., Kiryakova, V.:
 Recent history of the fractional calculus: data and statistics.
 In: Kochubei,  A., Luchko, Yu. (eds.) Handbook of Fractional Calculus with Applications:
 Basic Theory, vol. 1, pp. 1--21. Walter de GruyterGmbH, Berlin (2019).
 DOI: 10.1515/9783110571622-001

 %%%% example for article in other journals %%%% pages are to be separated by double -- %%
\bibitem{Moak}
 Moak, D.S.:
 The $q$-analogue of the Laguerre polynomials.
{J. Math. Anal. Appl.} \textbf{81}(1), 20--47 (1981) % . DOI: ........ is optional %

%%%% example for article in Proceedings %%%%%
\bibitem{Rosbl}
 Rosenblum, M.:
 Generalized Hermite polynomials and the Bose-like oscillator calculus.
 In: {Operator Theory: Advances and Applications}, %% Vol. #,
 369--396, Birkh\"auser, Basel (1994)

 %%%% example for Internet sources %%%%
 \bibitem{TeX}
 \TeX\ Users Group (TUG), \href{http://www.tug.org}{http://www.tug.org}
 %% possibly add year. or date of access %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{thebibliography} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\small %%%
\noindent
{\bf Publisher's Note}
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


\end{document} %%% Thank you for following strictly this FCAA template ! %%
