Operational methods and differential equations with applications to initial-value problems

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

“…The operational identity (98) and the function (128) together yield, in accordance with the previously computed in [10], the result:…”

“…Equation (118) is in fact the general form of the equation, which unifies the Laguerre heat equation and the matter diffusion equation with Laguerre derivative, as previously explored in [10,37]. In order to solve Equation (118), we employ the operational method.…”

“…With the help of the operational definition of Laguerre polynomials (30) and of the Gleisher operational rule [10],…”

“…The method of operational solution of DE demonstrated in [7][8][9][10] is applicable to a wide spectrum of physical problems, described by linear partial differential equations (PDE), such as propagation and radiation from charged particles [11][12][13][14][15][16][17][18][19], heat diffusion [20][21][22], including processes not described by Fourier law, and others [23][24][25]. In the context of the operational approach, the operational definitions for the polynomials through the operational exponent are very useful [26].The operational exponent is also applied when describing the fundamentals of structures in nature, including elementary particles and quarks [27][28][29]; such modern mathematical instruments are also used for the theoretical study of…”

Some properties of Hermite based Appell matrix polynomials

“…Recently Srivastava et al [5,7,8,15,17,18] has established some results for the sheffer and Appell polynomials and also for the new classes of mixed special polynomials related to these polynomials by employing certain operational methods. This paper is an attempt to further stress the importance of operational methods in introducing new families of special matrix polynomials.…”