{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T21:34:06Z","timestamp":1743111246653,"version":"3.40.3"},"publisher-location":"Cham","reference-count":23,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319784571"},{"type":"electronic","value":"9783319784588"}],"license":[{"start":{"date-parts":[[2018,3,23]],"date-time":"2018-03-23T00:00:00Z","timestamp":1521763200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019]]},"DOI":"10.1007\/978-3-319-78458-8_9","type":"book-chapter","created":{"date-parts":[[2018,3,21]],"date-time":"2018-03-21T22:40:29Z","timestamp":1521672029000},"page":"92-101","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Duality Properties of Variable-Type and -Order Differences"],"prefix":"10.1007","author":[{"given":"Wiktor","family":"Malesza","sequence":"first","affiliation":[]},{"given":"Dominik","family":"Sierociuk","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,3,23]]},"reference":[{"key":"9_CR1","doi-asserted-by":"crossref","unstructured":"Dzielinski, A., Sarwas, G., Sierociuk, D.: Time domain validation of ultracapacitor fractional order model. In: 2010 49th IEEE Conference on Decision and Control (CDC), pp. 3730\u20133735, December 2010","DOI":"10.1109\/CDC.2010.5717093"},{"key":"9_CR2","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1186\/1687-1847-2011-11","volume":"2011","author":"A Dzielinski","year":"2011","unstructured":"Dzielinski, A., Sarwas, G., Sierociuk, D.: Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. 2011, 11 (2011)","journal-title":"Adv. Differ. Equ."},{"issue":"1\u20134","key":"9_CR3","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1023\/A:1016586905654","volume":"29","author":"C Lorenzo","year":"2002","unstructured":"Lorenzo, C., Hartley, T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1\u20134), 57\u201398 (2002)","journal-title":"Nonlinear Dyn."},{"key":"9_CR4","doi-asserted-by":"crossref","unstructured":"Macias, M., Sierociuk, D.: An alternative recursive fractional variable-order derivative definition and its analog validation. In: Proceedings of International Conference on Fractional Differentiation and its Applications, Catania, Italy (2014)","DOI":"10.1109\/ICFDA.2014.6967452"},{"key":"9_CR5","doi-asserted-by":"crossref","unstructured":"Malesza, W., Macias, M., Sierociuk, D.: Matrix approach and analog modeling for solving fractional variable order differential equations. In: Latawiec, K.J., Lukaniszyn, M., Stanislawski, R. (eds.) Advances in Modelling and Control of Non-integer-Order Systems, Lecture Notes in Electrical Engineering, vol. 320, pp. 71\u201380. Springer (2015)","DOI":"10.1007\/978-3-319-09900-2_7"},{"key":"9_CR6","doi-asserted-by":"crossref","unstructured":"Malesza, W., Sierociuk, D.: Recursive variable type and order difference, its definition and basic properties. In: 2016 17th International Carpathian Control Conference (ICCC), pp. 473\u2013478, May 2016","DOI":"10.1109\/CarpathianCC.2016.7501144"},{"key":"9_CR7","volume-title":"An Introduction to the Fractional Calculus and Fractional Differenctial Equations","author":"K Miller","year":"1993","unstructured":"Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differenctial Equations. Wiley, New York (1993)"},{"key":"9_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84996-335-0","volume-title":"Fractional-Order Systems and Controls","author":"CA Monje","year":"2010","unstructured":"Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer, London (2010)"},{"key":"9_CR9","volume-title":"The Fractional Calculus","author":"KB Oldham","year":"1974","unstructured":"Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)"},{"key":"9_CR10","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)"},{"key":"9_CR11","volume-title":"Fractional Integrals and Derivative. Theory and Applications","author":"S Samko","year":"1987","unstructured":"Samko, S., Kilbas, A., Maritchev, O.: Fractional Integrals and Derivative. Theory and Applications. Gordon & Breach Science Publishers, New York (1987)"},{"key":"9_CR12","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Dzielinski, A., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A Mathe. Phys. Eng. Sci. 371(1990) (2013)","DOI":"10.1098\/rsta.2012.0146"},{"key":"9_CR13","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W.: On the differences of variable type and variable fractional order. In: 2016 European Control Conference (ECC), pp. 2191\u20132196, June 2016","DOI":"10.1109\/ECC.2016.7810616"},{"key":"9_CR14","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: Equivalent switching strategy and analog validation of the fractional variable order derivative definition. In: Proceedings of European Control Conference 2013, ECC 2013, pp. 3464\u20133469, Zurich, Switzerland (2013)","DOI":"10.23919\/ECC.2013.6669416"},{"key":"9_CR15","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: On a new definition of fractional variable-order derivative. In: Proceedings of the 14th International Carpathian Control Conference (ICCC), pp. 340\u2013345, Rytro, Poland (2013)","DOI":"10.1109\/CarpathianCC.2013.6560566"},{"key":"9_CR16","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: Switching scheme, equivalence, and analog validation of the alternative fractional variable-order derivative definition. In: Proceedings of the 52nd IEEE Conference on Decision and Control 10\u201313 December 2013, Florence, Italy (2013)","DOI":"10.1109\/CDC.2013.6760481"},{"issue":"4","key":"9_CR17","first-page":"809","volume":"62","author":"D Sierociuk","year":"2014","unstructured":"Sierociuk, D., Twardy, M.: Duality of variable fractional order difference operators and its application to identification. Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 809\u2013815 (2014)","journal-title":"Bull. Pol. Acad. Sci. Tech. Sci."},{"key":"9_CR18","unstructured":"Sierociuk, D.: Fractional variable order derivative simulink toolkit (2012). \n                    http:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/38801-fractional-variable-order-derivative-simulink-toolkit"},{"key":"9_CR19","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Macias, M., Malesza, W.: Analog modeling of fractional switched-order derivatives: experimental approach. In: Advances in the Theory and Applications of Non-Integer Order Systems, pp. 271\u2013280. Springer (2013)","DOI":"10.1007\/978-3-319-00933-9_25"},{"issue":"13","key":"9_CR20","doi-asserted-by":"publisher","first-page":"3876","DOI":"10.1016\/j.apm.2014.12.009","volume":"39","author":"D Sierociuk","year":"2015","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model. 39(13), 3876\u20133888 (2015). \n                    https:\/\/doi.org\/10.1016\/j.apm.2014.12.009","journal-title":"Appl. Math. Model."},{"issue":"4","key":"9_CR21","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1007\/s00034-014-9895-1","volume":"34","author":"D Sierociuk","year":"2015","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: On the recursive fractional variable-order derivative: Equivalent switching strategy, duality, and analog modeling. Circuits Syst. Sign. Proces. 34(4), 1077\u20131113 (2015)","journal-title":"Circuits Syst. Sign. Proces."},{"key":"9_CR22","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Malesza, W., Macias, M.: On the output-additive switching strategy for a new variable type and order difference, pp. 101\u2013111. Springer, Cham (2017)","DOI":"10.1007\/978-3-319-45474-0_10"},{"issue":"3, SI","key":"9_CR23","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1016\/j.sigpro.2010.04.006","volume":"91","author":"D Valerio","year":"2011","unstructured":"Valerio, D., da Costa, J.S.: Variable-order fractional derivatives and their numerical approximations. Sign. Proces. 91(3, SI), 470\u2013483 (2011)","journal-title":"Sign. Proces."}],"container-title":["Lecture Notes in Electrical Engineering","Non-Integer Order Calculus and its Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-78458-8_9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T22:20:29Z","timestamp":1558131629000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-78458-8_9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,23]]},"ISBN":["9783319784571","9783319784588"],"references-count":23,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-78458-8_9","relation":{},"ISSN":["1876-1100","1876-1119"],"issn-type":[{"type":"print","value":"1876-1100"},{"type":"electronic","value":"1876-1119"}],"subject":[],"published":{"date-parts":[[2018,3,23]]},"assertion":[{"value":"23 March 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}