Operational solution for some types of second order differential equations and for relevant physical problems

2016

Axioms

Abstract:We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power-exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

“…The application of the operational method for solution of the second order PDE was given in [72]. The following PDE with initial conditions:…”

Section: Operational Solution Of the Hyperbolic Heat Conduction Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

2016

Axioms

“…giving a link to the operational approach for DE [41][42][43][44][45][46][47]. So expressed as a matrix polynomial, the group element depends on the group rotation angle θ and on the sole invariant det(H), which is encoded cyclometrically as another angle (see [40]):…”

Section: Exponential Parameterization and The Matrix Logarithmmentioning

confidence: 99%

Exponential parameterization of the neutrino mixing matrix: comparative analysis with different data sets and CP violation

Borisov

2016

Eur. Phys. J. C

The exponential parameterization of the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix for neutrino is used for a comparative analysis of different neutrino mixing data. The U PMNS matrix is considered as the element of the SU(3) group and the second-order matrix polynomial is constructed for it. The inverse problem of constructing the logarithm of the mixing matrix is addressed. In this way the standard parameterization is exactly related to the exponential parameterization. The exponential form allows easy factorization and separate analysis of the rotation and the CP violation. With the most recent experimental data on neutrino mixing (May 2016), we calculate the values of the exponential parameterization matrix for neutrinos with account for the CP violation. The complementarity hypothesis for quarks and neutrinos is demonstrated to hold, despite a significant change in the neutrino mixing data. The values of the entries of the exponential mixing matrix are evaluated with account for the actual degree of the CP violation in neutrino mixing and without it. Various factorizations of the CP-violating term are investigated in the framework of the exponential parameterization.

“…Analytical study gives deeper insight in the problem; operational analytical approach and solutions to HHE were developed in [45][46][47][48][49][50]. This method easily handles also other linear DE of high order and fractional DE [51][52][53][54][55][56][57][58]. Use of the exponential differential operators, such as the heat operator S = e ∂ 2 x [59] allows operational solution of GK-type Equation (4) as demonstrated in [47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Oskolkov

Gubina

2018

Axioms

Abstract:One-dimensional equations of telegrapher's-type (TE) and Guyer-Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated-fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.