Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations

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 Appel 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 Schrodinger and the Black–Scholes-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.

[1]  K. V. Zhukovsky,et al.  High harmonic generation in the undulators for free electron lasers , 2015 .

[2]  Subuhi Khan,et al.  Hermite-based Appell polynomials: Properties and applications , 2009 .

[3]  K. V. Zhukovsky,et al.  Harmonic radiation in a double-frequency undulator with account for broadening , 2015 .

[4]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[5]  Arak M. Mathai,et al.  Special Functions for Applied Scientists , 2008 .

[6]  Hari M. Srivastava,et al.  Analytical solutions for heat diffusion beyond Fourier law , 2017, Appl. Math. Comput..

[7]  Clemens Markett,et al.  A representation theory for solutions of a higher order heat equation, II , 1992 .

[8]  Christophe Dujardin,et al.  Kinetic Model of Energy Relaxation in CsI:A (A = Tl and In) Scintillators , 2015 .

[9]  K. V. Zhukovsky,et al.  Operational solution for some types of second order differential equations and for relevant physical problems , 2017 .

[10]  T. Neumann.  Advanced Combinatorics The Art Of Finite And Infinite Expansions , 2016 .

[11]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[12]  Arak M. Mathai,et al.  Analysis of Solar Neutrino Data from Super-Kamiokande I and II , 2014, Entropy.

[13]  Giuseppe Dattoli,et al.  QUARK MIXING IN THE STANDARD MODEL AND THE SPACE ROTATIONS , 2009 .

[14]  G. Dattoli,et al.  Generalized polynomials, operational identities and their applications , 2000 .

[15]  D. A. Spassky,et al.  The features of energy transfer to the emission centers in ZnWO4 and ZnWO4:Mo , 2013 .

[16]  Arak M. Mathai,et al.  On a Generalized Entropy Measure Leading to the Pathway Model with a Preliminary Application to Solar Neutrino Data , 2013, Entropy.

[17]  Arakaparampil M. Mathai.  M-convolutions of products and ratios, statistical distributions and fractional calculus , 2015 .

[18]  Arak M. Mathai.  Fractional differential operators in the complex matrix-variate case , 2015 .

[19]  P. Appell,et al.  Fonctions hypergéométriques et hypersphériques : polynomes d'Hermite , 1926 .

[20]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[21]  A. M. Mathai,et al.  Analysis of Solar Neutrino Data from SuperKamiokande I and II: Back to the Solar Neutrino Problem , 2012 .

[22]  Anatoly Borisov,et al.  Exponential parameterization of the neutrino mixing matrix: comparative analysis with different data sets and CP violation , 2016 .

[23]  A. M. Mathai,et al.  Solutions of certain fractional kinetic equations and a fractional diffusion equation , 2007, 0704.1916.

[24]  Youssri Hassan Youssri,et al.  SPECTRAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS VIA A NOVEL LUCAS OPERATIONAL MATRIX OF FRACTIONAL DERIVATIVES , 2016 .

[25]  Victor I. Denisov,et al.  Pulsar radiation in post-Maxwellian vacuum nonlinear electrodynamics , 2016 .

[26]  Giuseppe Dattoli,et al.  Quark flavour mixing and the exponential form of the Kobayashi–Maskawa matrix , 2007 .

[27]  Hari M. Srivastava,et al.  Orthogonality properties of the Hermite and related polynomials , 2005 .

[28]  K. V. Zhukovsky,et al.  High harmonic generation in undulators for FEL , 2016 .

[29]  Kurt Bernardo Wolf,et al.  Integral transforms in science and engineering , 1979 .

[30]  K. V. Zhukovsky,et al.  A method of inverse differential operators using ortogonal polynomials and special functions for solving some types of differential equations and physical problems , 2015 .

[31]  K. V. Zhukovsky,et al.  Analytical account for a planar undulator performance in a constant magnetic field , 2014 .

[32]  K. V. Zhukovsky,et al.  The operational solution of fractional-order differential equations, as well as Black–Scholes and heat-conduction equations , 2016 .

[33]  Youssri H. Youssri,et al.  A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations , 2016, Entropy.

[34]  K. V. Zhukovsky,et al.  Violation of the maximum principle and negative solutions for pulse propagation in Guyer–Krumhansl model , 2016 .

[35]  Giuseppe Dattoli,et al.  Evolution of non-spreading Airy wavepackets in time dependent linear potentials , 2011, Appl. Math. Comput..

[36]  H. W. Gould,et al.  Operational formulas connected with two generalizations of Hermite polynomials , 1962 .

[37]  A. W. Kemp,et al.  A treatise on generating functions , 1984 .

[38]  D. A. Spassky,et al.  Energy transfer in solid solutions ZnxMg1−xWO4 , 2014 .

[39]  P. Ván.  Theories and heat pulse experiments of non-Fourier heat conduction , 2015, 1501.04234.

[40]  K. V. Zhukovsky,et al.  Harmonic generation by ultrarelativistic electrons in a planar undulator and the emission-line broadening , 2015 .

[41]  K. V. Zhukovsky,et al.  Exponential parameterization of neutrino mixing matrix with account of CP-violation data , 2016 .

[42]  K. V. Zhukovsky,et al.  Quark mixing and exponential form of the Cabibbo-Kobayashi-Maskawa matrix , 2008 .

[43]  K. V. Zhukovsky,et al.  Emission and tuning of harmonics in a planar two-frequency undulator with account for broadening , 2016 .

[44]  R. K. Saxena,et al.  Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative , 2012, Commun. Nonlinear Sci. Numer. Simul..

[45]  Steven Roman.  The Umbral Calculus , 1984 .

[46]  K. Zhukovsky.  Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators , 2014, TheScientificWorldJournal.

[47]  K. V. Zhukovsky,et al.  Exact solution of Guyer–Krumhansl type heat equation by operational method , 2016 .

[48]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[49]  H. M. Srivastava,et al.  Operational methods and differential equations with applications to initial-value problems , 2007, Appl. Math. Comput..

[50]  E. Savaş.  On generalized A , 2010 .

[51]  Y. H. Youssri,et al.  New Spectral Solutions of Multi-Term Fractional-OrderInitial Value ProblemsWith Error Analysis , 2015 .

[52]  Subuhi Khan,et al.  Laguerre-based Appell polynomials: Properties and applications , 2010, Math. Comput. Model..

[53]  Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics , 2016, 1610.06017.

[54]  Waleed M. Abd-Elhameed,et al.  New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations , 2014 .

[55]  Giuseppe Dattoli,et al.  Neutrino mixing and the exponential form of the Pontecorvo–Maki–Nakagawa–Sakata matrix , 2008 .

[56]  Subuhi Khan,et al.  Hermite-Laguerre Matrix Polynomials and Generating Relations , 2014 .