{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T15:42:10Z","timestamp":1774798930511,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,12,13]],"date-time":"2016-12-13T00:00:00Z","timestamp":1481587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general App\u00e8l polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schr\u00f6dinger and the Black\u2013Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations.<\/jats:p>","DOI":"10.3390\/axioms5040029","type":"journal-article","created":{"date-parts":[[2016,12,13]],"date-time":"2016-12-13T10:15:52Z","timestamp":1481624152000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations"],"prefix":"10.3390","volume":"5","author":[{"given":"Konstantin","family":"Zhukovsky","sequence":"first","affiliation":[{"name":"Faculty of Physics, Moscow State University, Leninskie Gory, Moscow 119991, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,13]]},"reference":[{"key":"ref_1","unstructured":"Mathai, M., Saxena, R.K., and Haubold, H.J. (2009). The H-Function: Theory and Applications, Springer Science & Business Media."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2008). Special Functions for Applied Scientists, Springer Science & Business Media.","DOI":"10.1007\/978-0-387-75894-7"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1515\/anly-2015-5002","article-title":"M-convolutions of products and ratios, statistical distributions and fractional calculus","volume":"36","author":"Mathai","year":"2016","journal-title":"Analysis"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cnsns.2015.02.021","article-title":"Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative","volume":"27","author":"Saxena","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.laa.2015.03.038","article-title":"Fractional differential operators in the complex matrix-variate case","volume":"478","author":"Mathai","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"103506","DOI":"10.1063\/1.3496829","article-title":"Solutions of certain fractional kinetic equations and a fractional diffusion equation","volume":"51","author":"Saxena","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"454865","DOI":"10.1155\/2014\/454865","article-title":"Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators","volume":"2014","author":"Zhukovsky","year":"2014","journal-title":"Sci. World J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.3103\/S0027134915020137","article-title":"A method of inverse differential operators using ortogonal polynomials and special functions for solving some types of differential equations and physical problems","volume":"70","author":"Zhukovsky","year":"2015","journal-title":"Moscow Univ. Phys. Bull."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1016\/j.jmaa.2016.08.054","article-title":"Operational solution for some types of second order differential equations and for relevant physical problems","volume":"446","author":"Zhukovsky","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1016\/j.amc.2006.07.001","article-title":"Operational Methods and Differential Equations with Applications to Initial-Value problems","volume":"184","author":"Dattoli","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1140\/epjc\/s10052-016-4464-3","article-title":"Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics","volume":"76","author":"Denisov","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"045021","DOI":"10.1103\/PhysRevD.94.045021","article-title":"Pulsar radiation in post-Maxwellian vacuum nonlinear electrodynamics","volume":"94","author":"Denisov","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7966","DOI":"10.1016\/j.amc.2011.02.088","article-title":"Evolution of non-spreading Airy wavepackets in time dependent linear potentials","volume":"217","author":"Zhukovsky","year":"2011","journal-title":"Appl. Math. Comp."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1869","DOI":"10.1080\/09205071.2014.945664","article-title":"Analytical account for a planar undulator performance in a constant magnetic field","volume":"28","author":"Zhukovsky","year":"2014","journal-title":"J. Electromagn. Waves Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"232","DOI":"10.3103\/S0027134915040177","article-title":"Harmonic Radiation in a Double-Frequency Undulator with Account for Broadening","volume":"70","author":"Zhukovsky","year":"2015","journal-title":"Moscow Univ. Phys. Bull."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.nimb.2015.10.041","article-title":"High harmonic generation in undulators for FEL","volume":"369","author":"Zhukovsky","year":"2015","journal-title":"Nuclear Instrum. Methods B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1080\/09205071.2014.985854","article-title":"Harmonic generation by ultrarelativistic electrons in a planar undulator and the emission-line broadening","volume":"29","author":"Zhukovsky","year":"2015","journal-title":"J. Electromagn. Waves Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.optcom.2015.04.079","article-title":"High harmonic generation in the undulators for free electron lasers","volume":"353","author":"Zhukovsky","year":"2015","journal-title":"Opt. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1017\/S0263034616000264","article-title":"Emission and tuning of harmonics in a planar two-frequency undulator with account for broadening","volume":"34","author":"Zhukovsky","year":"2016","journal-title":"Laser Part Beams"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.amc.2016.08.038","article-title":"Analytical solutions for heat diffusion beyond Fourier law","volume":"293","author":"Zhukovsky","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"237","DOI":"10.3103\/S0027134916030164","article-title":"Operational solution of differential equations with derivatives of non-integer order, Black\u2013Scholes type and heat conduction","volume":"71","author":"Zhukovsky","year":"2016","journal-title":"Moscow Univ. Phys. Bull."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/j.ijheatmasstransfer.2016.03.021","article-title":"Violation of the maximum principle and negative solutions with pulse propagation in Guyer-Krumhansl model","volume":"98","author":"Zhukovsky","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0022-247X(92)90191-F","article-title":"A representation theory for solutions of a higher-order heat equation. I","volume":"168","author":"Haimo","year":"1992","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0022-247X(92)90158-A","article-title":"A representation theory for solutions of a higher-order heat equation. II","volume":"168","author":"Haimo","year":"1992","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ijheatmasstransfer.2016.01.005","article-title":"Exact solution of Guyer-Krumhansl type heat equation by operational method","volume":"96","author":"Zhukovsky","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_26","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. Higher Transcendental Functions, McGraw-Hill Book Company."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1140\/epjc\/s10052-007-0263-1","article-title":"Quark Flavour Mixing and the Exponential Form of the Kobayashi-Maskawa Matrix","volume":"50","author":"Dattoli","year":"2007","journal-title":"Eur. Phys. J. C"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1140\/epjc\/s10052-007-0399-z","article-title":"Quark Mixing in the Standard Model and the Space Rotations","volume":"52","author":"Dattoli","year":"2007","journal-title":"Eur. Phys. J. C"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1807","DOI":"10.1134\/S1063778808100153","article-title":"Quark Mixing and Exponential Form of the Cabibbo-Kobayashi-Maskawa Matrix","volume":"71","author":"Zhukovsky","year":"2008","journal-title":"Phys. At. Nucl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1140\/epjc\/s10052-008-0620-8","article-title":"Neutrino Mixing and the exponential form of the Pontecorvo-Maki-Nakagawa-Sakata matrix","volume":"55","author":"Dattoli","year":"2008","journal-title":"Eur. Phys. J. C"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1140\/epjc\/s10052-016-4487-9","article-title":"Exponential parameterization of the neutrino mixing matrix: Comparative analysis with different data sets and CP violation","volume":"76","author":"Zhukovsky","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1140\/epjc\/s10052-016-4310-7","article-title":"Exponential parameterization of neutrino mixing matrix with account of CP-violation data","volume":"76","author":"Zhukovsky","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.3390\/e16031414","article-title":"Analysis of solar neutrino data from Super-Kamiokande I and II","volume":"16","author":"Haubold","year":"2014","journal-title":"Entropy"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4011","DOI":"10.3390\/e15104011","article-title":"On a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data","volume":"15","author":"Mathai","year":"2013","journal-title":"Entropy"},{"key":"ref_35","first-page":"112","article-title":"Back to the Solar Neutrino Problem","volume":"185","author":"Mathai","year":"2012","journal-title":"Space Res. Today"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0377-0427(00)00283-1","article-title":"Generalized polynomials, operational identities and their applications","volume":"118","author":"Dattoli","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.cam.2004.10.021","article-title":"\u041erthogonality properties of the Hermite and related polynomials","volume":"182","author":"Dattoli","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Abd-Elhameed, W.M., and Youssri, Y.H. (2016). A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations. Entropy, 18.","DOI":"10.3390\/e18100345"},{"key":"ref_39","first-page":"795","article-title":"Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives","volume":"61","author":"Youssri","year":"2016","journal-title":"Romanian J. Phys."},{"key":"ref_40","first-page":"375","article-title":"New spectral solutions of multi-term fractional order initial value problems with error analysis","volume":"105","author":"Youssri","year":"2015","journal-title":"Comp. Model. Eng. Sci."},{"key":"ref_41","first-page":"626275","article-title":"New ultraspherical wavelets spectral solutions for fractional Riccati differential equations","volume":"2014","author":"Youssri","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1660","DOI":"10.1016\/j.optmat.2013.12.039","article-title":"Energy transfer in solid solutions ZnxMg1\u2212xWO4","volume":"36","author":"Spassky","year":"2014","journal-title":"Opt. Mater."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"20578","DOI":"10.1021\/acs.jpcc.5b05627","article-title":"Kinetic Model of Energy Relaxation in Csl:A (A = Tl and In) Scintillators","volume":"119","author":"Gridin","year":"2015","journal-title":"J. Phys. Chem. C"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.jlumin.2013.06.039","article-title":"The features of energy transfer to the emission centers in ZnWO4 and ZnWO4:Mo","volume":"144","author":"Krutyak","year":"2013","journal-title":"J. Lumin."},{"key":"ref_45","unstructured":"Hermite, C. (1864). Sur un Nouveau D\u00e9veloppement en S\u00e9rie de Fonctions, Gauthier-Villar."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1215\/S0012-7094-62-02907-1","article-title":"Operational formulas connected with two generalizations of Hermite polynomials","volume":"29","author":"Gould","year":"1962","journal-title":"Duke Math. J."},{"key":"ref_47","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited)."},{"key":"ref_48","unstructured":"Appell, P., and Kamp\u00e9 de F\u00e9riet, J. (1926). Fonctions Hyperg\u00e9om\u00e9triques et Hypersph\u00e9riques Polyn\u00f4mes, d\u2019Hermite, Gauthier-Villars."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1016\/j.jmaa.2008.11.002","article-title":"Hermite-based Appell polynomials: Properties and applications","volume":"351","author":"Khan","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.mcm.2010.02.022","article-title":"Laguerre-based Appell polynomials: Properties and applications","volume":"52","author":"Khan","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0034-4877(14)00008-1","article-title":"Hermite-Laguerre matrix polynomials and generating relations","volume":"73","author":"Khan","year":"2014","journal-title":"Rep. Math. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel. [Revised English ed.].","DOI":"10.1007\/978-94-010-2196-8"},{"key":"ref_53","unstructured":"Roman, S. (1984). The Umbral Calculus, Academic Press."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Wolf, K.B. (1979). Integral Transforms in Science and Engineering, Plenum Press.","DOI":"10.1007\/978-1-4757-0872-1"},{"key":"ref_55","first-page":"150","article-title":"Theories and heat pulse experiments of non-Fourier heat conduction","volume":"7","year":"2016","journal-title":"Commun. Appl. Ind. Math."},{"key":"ref_56","unstructured":"Watson, G.N. (1944). A Treatise on the Theory of Bessel Functions, Cambridge University Press. [2nd ed.]."},{"key":"ref_57","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Publishers. North-Holland Mathematical Studies."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/5\/4\/29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:28:27Z","timestamp":1760210907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/5\/4\/29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,13]]},"references-count":57,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,12]]}},"alternative-id":["axioms5040029"],"URL":"https:\/\/doi.org\/10.3390\/axioms5040029","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,13]]}}}