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

Zhukovsky, K. V.; Srivástava, H. M.

doi:10.3390/axioms5040029

Cited by 7 publications

(5 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More recently, much attention has been paid to the solutions of FDEs using various methods, such as the Adomian decomposition method (2005) [8], the first integral method (2014) [9], the Lie group theory method (2012,2015) [10,11], the homotopy analysis method (2016) [12], the inverse differential operational method (2016) [13][14][15], the F-expansion method (2017) [16], M-Wright transforms (2017) [17], exponential differential operators (2017,2018) [18,19], and so on. In reality, the finding of exact solutions of the FDEs is hard work and remains a problem.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

Yong¹,

Li²

2018

Axioms

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

Yong¹,

Li²

2018

Axioms

View full text Add to dashboard Cite

show abstract

“…Analytical solutions are highly appreciated, but only a few types of DE allow explicit, if any, exact analytical solutions. Recently some fractional ordinary DE and partial differential equations (PDE) were analyzed and analytically solved in [24][25][26][27][28][29][30][31][32][33][34][35][36][37]. They benefit from the use of special functions [38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Zhukovsky

2016

Axioms

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The physics and the approach with regard to the betatron oscillations remain the same for any undulator. For the bi-harmonic planar undulator d 1 = d 2 = 0, and the result (31) reduces to that in [44] in different notations. For the common planar undulator with single field harmonic…”

Section: Spontaneous Ur Intensity and Spectrum Distortionsmentioning

confidence: 99%

“…where for practical evaluations we can use kN 0.934L[cm]H 0 [Tesla]. The special function S can be expressed as the action of the operational differential operators, also employed for the studies of Hermite and Laguerre families of orthogonal polynomials in [31][32][33]. The generalized multivariable Hermite polynomials:…”

Section: Spontaneous Ur Intensity and Spectrum Distortionsmentioning

confidence: 99%

Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments

Zhukovsky

2020

Symmetry

Self Cite

View full text Add to dashboard Cite

A theoretical study of the synchrotron radiation (SR) from electrons in periodic magnetic fields with non-periodic magnetic components is presented. It is applied to several free electron lasers (FELs) accounting for the real characteristics of their electron beams: finite sizes, energy spread, divergence etc. All the losses and off-axis effects are accounted analytically. Exact expressions for the harmonic radiation in multiperiodic magnetic fields with non-periodic components and off-axis effects are given in terms of the generalized Bessel and Airy-type functions. Their analytical forms clearly distinguish all contributions in each polarization of the undulator radiation (UR). The application to FELs is demonstrated with the help of the analytical model for FEL harmonic power evolution, which accounts for all major losses and has been verified with the results of well documented FEL experiments. The analysis of the off-axis effects for the odd and even harmonics is performed for SPRING8 Angstrom Compact free-electron LAser (SACLA) and Pohang Accelerator Laboratory (PAL-XFEL). The modelling describes theoretically the power levels of odd and even harmonics and the spectral line width and shape. The obtained theoretical results agree well with the available data for FEL experiments; where no data exist, we predict and explain the FEL radiation properties. The proposed theoretical approach is applicable to practically any FEL.

show abstract

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

Cited by 7 publications

References 50 publications

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments

Contact Info

Product

Resources

About