{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T21:08:12Z","timestamp":1778188092662,"version":"3.51.4"},"reference-count":21,"publisher":"ASME International","issue":"6","content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2016,11,1]]},"abstract":"<jats:p>In this paper, we present a new type of fractional operator, the Caputo\u2013Katugampola derivative. The Caputo and the Caputo\u2013Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy-type problem, with dependence on the Caputo\u2013Katugampola derivative, is proved. A decomposition formula for the Caputo\u2013Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation (FDE).<\/jats:p>","DOI":"10.1115\/1.4034432","type":"journal-article","created":{"date-parts":[[2016,8,18]],"date-time":"2016-08-18T16:33:33Z","timestamp":1471538013000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":104,"title":["Fractional Differential Equations With Dependence on the Caputo\u2013Katugampola Derivative"],"prefix":"10.1115","volume":"11","author":[{"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810\u2013193, Portugal e-mail:"}]},{"given":"Agnieszka B.","family":"Malinowska","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science, Bialystok University of Technology, Bia\u0142ystok 15-351, Poland e-mail:"}]},{"given":"Tatiana","family":"Odzijewicz","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Mathematical Economics, Warsaw School of Economics, Warsaw 02-554, Poland e-mail:"}]}],"member":"33","published-online":{"date-parts":[[2016,9,16]]},"reference":[{"key":"2019100318011616200_bib1","volume-title":"Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies","year":"2006"},{"key":"2019100318011616200_bib2","volume-title":"Fractional Differential Equations Mathematics in Science and Engineering","year":"1999"},{"key":"2019100318011616200_bib3","volume-title":"Fractional Integrals and Derivatives","year":"1993"},{"issue":"3","key":"2019100318011616200_bib4","doi-asserted-by":"publisher","first-page":"860","DOI":"10.1016\/j.amc.2011.03.062","article-title":"New Approach to a Generalized Fractional Integral","volume":"218","year":"2011","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"2019100318011616200_bib5","first-page":"1","article-title":"A New Approach to Generalized Fractional Derivatives","volume":"6","year":"2014","journal-title":"Bull. Math. Anal. Appl."},{"key":"2019100318011616200_bib6","unstructured":"Katugampola, U. N., 2016, \u201cExistence and Uniqueness Results for a Class of Generalized Fractional Differential Equations,\u201d arXiv:1411.5229"},{"issue":"10","key":"2019100318011616200_bib7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1687-1847-2014-10","article-title":"On Caputo Modification of the Hadamard Fractional Derivatives","volume":"2014","year":"2014","journal-title":"Adv. Differ. Equations"},{"issue":"142","key":"2019100318011616200_bib8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1687-1847-2012-142","article-title":"Caputo-Type Modification of the Hadamard Fractional Derivatives","volume":"2012","year":"2012","journal-title":"Adv. Differ. Equations"},{"issue":"3","key":"2019100318011616200_bib9","first-page":"297","article-title":"Cauchy Problem for Differential Equation With Caputo Derivative","volume":"7","year":"2004","journal-title":"Fractional Calculus Appl. Anal."},{"issue":"7","key":"2019100318011616200_bib10","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1016\/j.mechrescom.2008.05.003","article-title":"On a Numerical Scheme for Solving Differential Equations of Fractional Order","volume":"35","year":"2008","journal-title":"Mech. Res. Commun."},{"issue":"3","key":"2019100318011616200_bib11","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1002\/asjc.617","article-title":"Numerical Approximations of Fractional Derivatives With Applications","volume":"15","year":"2013","journal-title":"Asian J. Control"},{"issue":"3","key":"2019100318011616200_bib12","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1080\/01630563.2011.647197","article-title":"Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative","volume":"33","year":"2012","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"4","key":"2019100318011616200_bib13","doi-asserted-by":"publisher","first-page":"554","DOI":"10.2478\/s13540-011-0034-4","article-title":"Fractional Boundary Value Problems: Analysis and Numerical Methods","volume":"14","year":"2011","journal-title":"Fractional Calculus Appl. Anal."},{"issue":"10","key":"2019100318011616200_bib14","doi-asserted-by":"publisher","first-page":"2973","DOI":"10.1016\/j.camwa.2012.01.053","article-title":"Distributed Order Equations as Boundary Value Problems","volume":"64","year":"2012","journal-title":"Comput. Math. Appl."},{"issue":"C","key":"2019100318011616200_bib15","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2014.05.025","article-title":"Central Difference Approximation of Convection in Caputo Fractional Derivative Two-Point Boundary Value Problems","volume":"273","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"2019100318011616200_bib16","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1016\/j.jcp.2014.08.036","article-title":"An Explicit High Order Method for Fractional Advection Diffusion Equations","volume":"278","year":"2014","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2019100318011616200_bib17","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/s10543-013-0443-3","article-title":"Higher Order Numerical Methods for Solving Fractional Differential Equations","volume":"54","year":"2014","journal-title":"BIT Numer. Math."},{"issue":"1","key":"2019100318011616200_bib18","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/j.apm.2009.04.006","article-title":"Numerical Methods for Fractional Partial Differential Equations With Riesz Space Fractional Derivatives","volume":"34","year":"2010","journal-title":"Appl. Math. Modell."},{"key":"2019100318011616200_bib19","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1016\/j.amc.2014.06.047","article-title":"Convergence Analysis of a Numerical Scheme for Two Classes of Non-Linear Fractional Differential Equations","volume":"243","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"2019100318011616200_bib20","doi-asserted-by":"publisher","first-page":"725","DOI":"10.12988\/imf.2007.07064","article-title":"Some Formulas for Sums of Binomial Coefficients and Gamma Functions","volume":"2","year":"2007","journal-title":"Int. Math. Forum"},{"issue":"1","key":"2019100318011616200_bib21","doi-asserted-by":"publisher","first-page":"133","DOI":"10.2140\/pjm.1951.1.133","article-title":"The Asymptotic Expansion of a Ratio of Gamma Functions","volume":"1","year":"1951","journal-title":"Pac. J. Math."}],"container-title":["Journal of Computational and Nonlinear Dynamics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/asmedigitalcollection.asme.org\/computationalnonlinear\/article-pdf\/doi\/10.1115\/1.4034432\/6109595\/cnd_011_06_061017.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/asmedigitalcollection.asme.org\/computationalnonlinear\/article-pdf\/doi\/10.1115\/1.4034432\/6109595\/cnd_011_06_061017.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,3]],"date-time":"2019-10-03T22:01:25Z","timestamp":1570140085000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmedigitalcollection.asme.org\/computationalnonlinear\/article\/doi\/10.1115\/1.4034432\/384913\/Fractional-Differential-Equations-With-Dependence"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,16]]},"references-count":21,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2016,11,1]]}},"URL":"https:\/\/doi.org\/10.1115\/1.4034432","relation":{},"ISSN":["1555-1415","1555-1423"],"issn-type":[{"value":"1555-1415","type":"print"},{"value":"1555-1423","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,16]]},"article-number":"061017"}}