Reflections on Two Parallel Ways in the Progress of Fractional Calculus in Mechanics of Solids
Annotatsiya
Dedicated to Professor Stanislav Meshkov on the occasion of his 75th birthdayInterest in fractional calculus has quickened profoundly in the past few decades, resulting in a large body of articles devoted to this challenge, and sometimes researchers, especially young ones, who have tried to or will attempt the fractional calculus in problems of mechanics, may be hard pressed to orient themselves in such information explosion. Thus, as a result, certain findings are rediscovered, references are cited improperly or even incorrectly, priorities are placed erroneously, and some results remain concealed. In this connection, a story about two papers dealing with fractional calculus application in mechanics is rather instructive.The first paper to be under consideration is “A New Dissipation Model Based on Memory Mechanism” by the Italian researchers Caputo and Mainardi (1), which was submitted in March 5, 1971 to Pure and Applied Geophysics and was published in its December 1971 issue, while the second article “Integral Representation of ∍γ-functions and Their Application to Problems in Linear Viscoelasticity” by the Russian scholars Meshkov et al. (2) was submitted to International Journal of Engineering Science in June 5, 1970 and was published in its April 1971 issue.The story began from a letter (3) I received by e-mail in the summer of 1997 from Prof. Mainardi, wherein he complained that his paper (1), which he considered as a classical contribution to linear viscoelasticity, had been unfairly forgotten, and that is why he called Ref. 1 as a “phantom-paper.” I offered to join our forces in order to improve a historical unfairness in priorities ordering, since I was also one of the co-authors of the other phantom-paper, i.e., Ref. 2. The years of creating both papers, i.e., 1969–1970, belong to the period of so-called “Cold War,” so in those times there was no opportunity for us to know about each other. But Prof. Mainardi declined my offer.A decade has since gone by. Meanwhile, many publishing houses such as Elsevier, Springer, and Wiley have launched tremendous projects for publishing online technical articles using their hard copies published in peer-reviewed journals, including Russian academic journals, beginning from their first issues, in so doing covering the 1950s through the 1970s, and even earlier. The digital object identifier (DOI), as an identification system for intellectual property in the digital environment, has been developed by the International DOI Foundation on behalf of the publishing industry with the goals to provide a framework for managing intellectual content, link customers with publishers, facilitate electronic commerce, and enable automated copyright management.Thus, Ref. 2 has been available online since February 27, 2003 via ScienceDirect website, while Ref. 1 was published online on December 29, 2004 by SpringerLink website. Nevertheless, the situation with the citation of Refs. 12 was not dramatically changed till 2007. In January 2007 I received a new e-mail message from Prof. Mainardi (4) notifying me and all the research community dealing with fractional calculus applications that Ref. 1 has been published online, and from now on it is available via SpringerLink website for anyone who is interested. At the same time I was aware that Prof. Virginia Kiryakova, Managing Editor of Fractional Calculus and Applied Analysis (FCAA), had a project to reproduce Ref. 1 in the 10th Jubilee volume of FCAA.Since I saw little reason for reprinting the paper (1), which is available for the research audience both in hard and digital versions, I sent to Prof. Kiryakova a copy of Ref. 2 and, as an alternative, made a suggestion to write a tutorial survey (5) with a comparative analysis of Refs. 12 and their contribution to the linear viscoelasticity theory. But it went unnoticed.By the end of 2007, Prof. Kiryakova has formalized her plans. First, Ref. 1 was reprinted (6) with the Editor’s comments (7) involving the following:“Some other authors worked also on similar problems either simultaneously but independently, or years afterwards. Although they recognized the Caputo–Mainardi achievements and priorities in private discussions, correspondence and public lectures, the Caputo-Mainardi PAGEOPH-1971 paper (our Ref. 1) was often forgotten to be (or not properly) referred to…”It remains a surprising thing that the editor of the journal “closely specialized on Applied Fractional Calculus and Special Functions” (7) published in Bulgaria, has been unaware of, or has forgotten, a major contribution of Russian researchers to this field of knowledge.The next FCAA issue devoted to the 80th anniversary of Prof. Caputo and the 40th anniversary of the so-called “Caputo derivative” involves the reprint of Caputo’s paper (8) originally published in 1967 in Geophysical Journal of the Royal Astronomical Society (9) and recently online (on April 2, 2007 in Wiley InterScience website). As this takes place, the historical facts concerning the of this fractional order have been In her Prof. Kiryakova about the Caputo the of the for of fractional as applications in to linear of viscoelasticity, first in Caputo’s paper of i.e., in Ref. and the same in his paper he an for of order of the referred to as the Caputo 5, the research community in fractional calculus its since this in his first paper on the in As for the first application of this in mechanics, it was by in his “A of Linear of and Application to the Problems of made in the of Russian of Science on 29, which was published as a technical paper in In Ref. 12 the of the order for the the linear the and the for the of and the and the for the 2 was in Ref. 12 for the of a the of its two one of which is but the other to the first one to the and (2) by the this paper by was not since the Russian academic journal Applied and was since the that some authors cited Ref. 12 that the of has and this in one with the papers by as the first who the fractional 2 in the is that the fractional 1 in viscoelasticity problems years Caputo fractional was also by in while the for the to which is years Ref. was was with the by few years the same fractional order was also by with to in the published in the Caputo’s paper (9) which was in and has available for since as the volume of the in Applied and was by that 1 is the of the fractional order to and the was in mechanics of by But this researchers and was of no and this for problems of of fractional of the be to Prof. Caputo for his in this via the and its in mechanics and but not for the of Caputo of is by other researchers as be also that this of the fractional has a large this I have on to the and of all co-authors of the both papers since Ref. 2 remains in the of is a major contribution of Prof. Meshkov to the of applications of fractional calculus in problems of mechanics and the papers that are cited who was a of my and I worked with from 1967 to my a in the by was the first to write the fractional linear in 1967 in i.e., years Caputo and Mainardi which is a little for our as the article with Prof. which applications of fractional calculus to problems of linear and mechanics of it has been that of the by Russian and researchers in the field are not by their since they published in Russian journals, and But now this situation has been since of by both and are available is references DOI of papers will be as as the of their i.e., of online of of and technical research have i.e., and Engineering covering all of and from of and the that paper in was by reason of its be Nevertheless, a tremendous contribution of Russian researchers in fractional calculus viscoelasticity is the papers which in the are cited and all of have been unfairly in the historical of the recently published and technical papers but not a that has me to write this was Professor as a to the FCAA issue to the years of Caputo and to the 80th anniversary of Prof. wherein he his of on of those But I that the research community in viscoelasticity aware of the and now it is my to some both papers in 1971 the same time and simultaneously had gone in the i.e., they have been unfairly The the findings of articles have been many times by researchers, and few know or have been in the Refs. 12 are devoted to the same it is rather to the comparative analysis of their results and applications it be that one and the same in both papers in the two in Ref. in the of the of the linear by the with of order one those Caputo and Mainardi unaware that Meshkov had this years but from Refs. it is that this has not been the while in Ref. 2, in the of the with the fractional as a of The was by in and in who had the as a and its which was by as the fractional the same as the of the is the fractional and is the or the was the first to in about the of two of the for the fractional linear he considered fractional as some and and to the mechanics of the of fractional in the his the via fractional in the fractional Caputo’s paper (9) and the fractional and in Meshkov the fractional and fractional linear be that was the first to in a of fractional the of a by its on with the and Meshkov the of the fractional while Caputo and the Caputo it is that the two of a linear and as Mainardi as from the past a of authors have or the fractional calculus as an of the of is to the of mechanics developed by that fractional my the of the two or application of fractional calculus in viscoelasticity is not and the of the is that they are and the same for all of the under consideration will be in by the results of Refs. Thus, Meshkov the of the of a linear with fractional and the with the fractional as a of The in the two is in in the article of the first researchers who recognized that of mechanics is to that the be to the fractional of the was in he with his via of second in and from the fractional 2 in Refs. and about Refs. the fractional in In so authors the the a fractional and the on the of linear viscoelasticity via the was in Refs. for the fractional of two as the or of fractional in viscoelasticity, it is to the of the via with of and in of fractional as the first and second the the was developed and in problems of mechanics in while the was in and the of in those and of one and the two papers under consideration i.e., Refs. are the Caputo and Mainardi (1), of the second from the Meshkov et al. a of the first and is the is the and are the and and are the of or the and of or the of the and are the and of the is the time of the fractional order and is the fractional which is to the for in Refs. 12 the authors the for the same and for the of the are for of from both in Ref. the and the in of the which not in Ref. in Ref. 2, the and the of the and the are as the the the and to and and the of the of the of which have the and not in Ref. but the which is to the the of to the fractional was in Ref. 2. that the for the fractional linear was by Meshkov in of the and time the was in both their for of the fractional 2 in Ref. 1 is with 1 in Ref. 2, it is that they little from each the of of the which has no has been in Ref. 2 the of the findings of the two papers the the two developed in Refs. and (2) that the in the both articles is to that in his had the in the involving the fractional as is the of the fractional order is to that the be also in the are in and have the the which is by the involving and the of the fractional linear by in in the and are the in Refs. and Ref. the some applications in both Thus, in Ref. the of the was for the and the with the available for and the and a was In Ref. 2, of an and in a using the will not on their in a was and by Meshkov and in with the of the of the the of the fractional for the is to the is the of the and is the has been that the of the as the to and to a some our paper on this published in is cited sometimes but not Ref. which is available in the of our the fractional calculus linear has been in the of in Ref. and in Ref. that and in considered the for the is the fractional it is to that from the of not belong to the of the it is the linear the But rather of the fractional order in the by in Caputo (9) in and are some using the fractional and of and be in Refs. us the research on the the of which are by the fractional linear In it be that the on the fractional was for the first time in Ref. in wherein the fractional was but this remains in the of the of the from the that the results in Ref. 2 years by and and are referred to authors for in Ref. as as Ref. has been the in Ref. of all the of of a was in the is the is the and is the was in the the via the in was As a result, the was in the in Ref. the be all for the the of the has been that two and the in Ref. has been for their The of the in the as of the for of the fractional are in in Ref. the of as the was in the with and in Ref. and are by a of of the the of the and of the involving the of the referred the of the of the system as is one to the that not in by which are to the fractional calculus linear it was a to in in Ref. that results in by and the authors of Ref. our on the that they considered a is from and the of two for this have to since the similar is the about the of on the fractional which was in Ref. in As this takes place, the was in the of the the two in the as of the is in for of 2 in Ref. may in the authors of Ref. know about Refs. but a few years the of Ref. in wherein Refs. and in et al. their for the of a as it was in Ref. that the authors of Refs. that the of the which in Ref. years the first in Ref. the historical of the about 12 it may be that the paper by Caputo and Mainardi has received recently its as Ref. with the that it is in Ref. contribution of this paper is of one of the first a of the those times to of and this was and Prof. Mainardi is of our paper (2) has not been reprinted our it is since it is for an to it via the ScienceDirect but there are to be of the that it was by Prof. one of the of the the the my co-authors of Ref. 2 in those I had the of the with fractional and the with by its in I not to a is or of is rather a of But one is and the two in of the each other and in the historical be considered as classical papers in that field of which is with the application of fractional calculus in linear viscoelasticity Refs. and the of the I to research in this field two and the of the of a a was in my research to in linear and as as and to of major results have been in a of papers in the in there was an of in fractional calculus viscoelasticity and it the of in of on the dealing with fractional received a by a and Russian academic such papers as while it was in to a paper to journal to an article was those is one In Prof. Mainardi has in Ref. about the same and he had in those times in which to to other of research as and in fractional calculus was a the in Ref. was in that I went to the applications of Fractional Calculus I aware of the or on the application of fractional order by the of similar has to a of years my in with fractional was to rather In January for the first the on Applied in to the from the International Science The period of the had but it was to all the for such a from to from a Russian a But all problems I had to as with the of an of this I a of from the and I had by their by one of the by Prof. who the of Applied two years I had the by the scholars from concerning of a fractional with so-called i.e., fractional of the order of its it that and about the of Russian in the field they even had no of the the of fractional and of in the and and with they me that they had made a for and aware of all papers in the field their I that a had for a to be with of Russian in the applications of fractional calculus in problems of the mechanics of and, in with the contribution of the of by the which was published in his two as as in technical papers of his not the articles published in Russian as they be in Ref. was a for the of our first article including references devoted to applications of fractional calculus to problems of linear and mechanics of the of references in Ref. is from and some papers in the field published by have been be consideration in Ref. was under had no to journal via which is a and in Russian the in that there was a for journal to I to the opportunity to the of Ref. who sent us some papers and some I will this some through for papers on the application of fractional order in mechanics in order to has been in this field in the past I with for the of the in this in there was the of in the research of fractional of viscoelasticity the second and especially in their application for problems of In the in the of and et al. and et al. the of has been using and but from the references cited in Refs. it was that their authors had about the contribution of Russian researchers in the The papers of and and in the have the of the of the of fractional the a fractional calculus application in mechanics began to in and my the Prof. in and Prof. in As for the on the application of fractional in mechanics, and other of and it that there is not a a research dealing with this in one or the of Ref. in which was our second paper in this journal the first one was published in on Prof. me to as an Editor of which was my and that my with this journal and it has been to the June of one of my International for a on and Fractional Calculus in which was by and Mainardi, he made a of our results it was Prof. Mainardi aware of Russian in the field through our paper first with the contribution of Russian authors and in he was one of the of the on In Ref. Prof. Mainardi on his this a I had the by the of in the of I aware that the by the Russian in from our of with the Fractional Linear the Prof. Mainardi a correspondence with me by e-mail that there was a of in our each of us has two of research in fractional in viscoelasticity, as years of and little resulting in the of Refs. by Mainardi with and Refs. by and other and (2) the period in and for Mainardi and the first from the comparative analysis of Refs. 12 made in it is that worked on for the our have Mainardi has been the fractional by the second in the Refs. as as the references on his while the major of our is devoted to and both a both of us in of But even in that period there was one of the application of the to the the paper our results in this and also the of and his co-authors is reason to for a this is a or from my research has been in of linear and and and other fractional and other fractional the of linear our was with the analysis of the of the of the with fractional their with to the of the fractional their and so As for the the of received the of the of the fractional and the on the as as the of the and so that the two in the analysis of the problems in the mechanics of i.e., the on the with of and (2) the on the fractional and each other. Their for of some and to using the correspondence to the of a the of which is by the linear fractional it is to the by the with the fractional as the there is a to the in they be by of the of the fractional in the and of the the with the In this it is not to the with an the one the of the involving fractional or fractional The of is that the is its the of while for the there is a each time to their as a results in some for the and fractional that Thus, for and the fractional linear with fractional of the for the and in of its with In on the of the to that it a of and a it has been that the of the fractional on the and be to each other. The for involving fractional of have been in the article our papers, it has been that the application of the involving fractional to the problems of of the for the of the the one the as the in the to the fractional from both the fractional and fractional the on such a the under some of its and fractional In other the fractional of other of the fractional or may to new years have since the of our paper body of has in this period of which the fractional calculus in of are many and involves fractional calculus its of theory. fractional in the of published in that it to such by with for and applications in many to of with their be in Ref. recently the of research in the fractional calculus application to problems to the audience of researchers and in the field of and has a article papers, has been recently with Prof. and submitted for in Applied in December of and results in the field by the of both which are in the the of the and of fractional calculus in problems and of this I to the of the made to by and Russian researchers in the field of fractional calculus applications in linear viscoelasticity, which are in the of was for the of the of fractional calculus The fractional calculus the of using the two i.e., via the with and via fractional or fractional as as other of fractional are in the first while the Russian and authors who for the first time our in problems of viscoelasticity are cited in the second and The the papers wherein for the first time for problems of mechanics of and of the is from the fractional by and using the second and first fractional calculus was by as the with the it has the property that its has from the to the was by Caputo to of an fractional calculus linear by and was with in the by Russian scholars for a of problems Thus, in and the first to the of an the order of the fractional as the and are which results in the of the is now by the authors dealing with in with fractional The in the of the in 1970 by et al. for the fractional linear and in Refs. for the in in by Meshkov and and the in a made of the fractional linear was by and this paper was originally published in a Russian the results of Ref. in of Ref. The of a with the of a was by et al. in The of the in a by the application to its end the time was in by and years the in the of the in a similar for a made of the fractional was by and Mainardi using the second for was a in the of the fractional calculus in in Ref. the which the with times on the fractional such in Ref. for an of the and for The of a large of one to the of the the and involving and in of as a which was in the of a fractional from the fractional or fractional of was by in In Meshkov the for the while it for in with a of years Meshkov and to the as the in the and this to the of a and the in Ref. for the of a in a and its fractional and linear first by and Meshkov in and and but a time by Caputo (9) and Caputo and Mainardi (1), The applications of made by Caputo (9) in 1967 and Caputo and Mainardi in 1971 for the problems dealing with In the fractional was by Caputo to of an the 1970s, an on fractional and their application to the problems of was in the by and Their first paper in the field to was devoted to an as a fractional the next authors to the fractional linear the for from the papers cited in the of 1 it is a tremendous was in the and by Russian and Italian researchers in the application of fractional calculus or in the same period using the two as in the first of 1) for problems in the mechanics of and to first of all Professor Stanislav the of my who first me in the of the fractional calculus viscoelasticity, years all my to I have the to on his 75th I Professor Meshkov and mechanics research community Professor who was the of Applied for for his to a new in the volume of from has been publishing by authors on their and of my there is no other research journal to have such a I to for this and opportunity to my on fractional calculus viscoelasticity application in of and the contribution of all my co-authors in the papers in the I to Professor for her in of this paper and the of this