Živorad Tomovski

Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative

In this paper, the solution of a fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time fractional derivative is obtained in terms of Mittag–Leffler-type functions and Foxs H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be expressed via Foxs H-function. It is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time fractional derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The fractional moments of the fundamental solution of the fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.

The general time fractional wave equation for a vibrating string

The solution of a general time fractional wave equation for a vibrating string is obtained in terms of the Mittag–Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. The time fractional derivative used is taken in the Caputo sense, and the method of separation of variables and the Laplace transform method are used to solve the equation. Some results for special cases of the initial and boundary conditions are obtained and it is shown that the corresponding solutions of the integer order equations are special cases of those of time fractional equations. The proposed general equation may be used for modeling different processes in complex or viscoelastic media, disordered materials, etc.Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.

Velocity and displacement correlation functions for fractional generalized Langevin equations

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise

We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.

Fractional wave equation with a frictional memory kernel of Mittag-Leffler type

Abstract In this paper we give an analytical treatment of a fractional wave equation with Caputo time fractional derivative and frictional memory kernel of Mittag-Leffler type. This problem generalizes a recently solved problem [62] of a wave equation for a vibrating string in presence of a fractional friction with power-law memory kernel. Such equations can be used in the context of modeling processes in complex and viscoelastic media.

On multiple generalized Mathieu series

Closed integral form expressions and different kinds of bounding inequalities are obtained for the m-fold generalized Mathieu series where s, q are positive m-tuples that ensure the convergence of Mathieu series. Connections to the first kind Bessel functions are given.

Laplace type integral expressions for a certain three-parameter family of generalized Mittag–Leffler functions with applications involving complete monotonicity

Abstract In this paper, we derive a Laplace type integral expression for the function e α , β γ ( t ; λ ) defined by e α , β γ ( t ; λ ) ≔ t β − 1 E α , β γ ( − λ t α ) , where E α , β γ ( z ) stands for the generalized three-parameter Mittag–Leffler function occurring in many interesting applied problems involving fractional differential equations. Our result is shown to enable us to extend certain findings by Mainardi (2010) [21] and others. As an application of the obtained Laplace type integral representation, we prove the complete monotonicity of the function e α , β γ ( t ; λ ) . We also establish several related positivity results and some uniform upper bounds for the function e α , β γ ( t ; λ ) .

Asymptotic behavior of a harmonic oscillator driven by a generalized Mittag-Leffler noise

The asymptotic behavior of a harmonic oscillator driven by a generalized Mittag-Leffler noise was studied by analyzing the generalized Langevin equation. The mean square displacement (MSD) and the velocity autocorrelation function (VACF) of a diffusing particle were obtained by using the Laplace transform method and Tauberian theorem. It was found that the MSD and VACF for various values of the parameters show a power-law decay, i.e. an anomalous diffusive behavior of the oscillator.

Integral representations and integral transforms of some families of Mathieu type series

By using some integral representations for several Mathieu type series (see P.L. Butzer, T.K. Pogány, and H.M. Srivastava, A linear ODE for the Omega function associated with the Euler function E α(z) and the Bernoulli function B α(z), Appl. Math. Lett. 19 (2006), pp. 1073–1077; P. Cerone and C.T. Lenard, On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003), Article 100, pp. 1–11 (electronic), T.K. Pogány; H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a-series and alternating Mathieu a-series, Appl. Math. Comput. 173 (2006), pp. 69–108; H.M. Srivastava and Ž. Tomovski, Some problems and solutions involving Mathieus series and its generalizations, J. Inequal. Pure Appl. Math. 5 (2) (2004), Article 45, pp. 1–13 (electronic); Ž. Tomovski, Integral representations of generalized Mathieu series via Mittag-Leffler type functions, Fract. Calc. Appl. Anal. 10 (2007), pp. 127–138.) via the Bessel function J ν of the first kind, the Gauss hypergeometric function 2 F 1, the generalized hypergeometric function p F q and the Fox–Wright generalization p Ψ q of the hypergeometric function p F q , a number of integral representations of the Laplace, Fourier, and Mellin types are derived here for certain general families of Mathieu type series. Some interesting corollaries and consequences of these integral representations are also considered.

Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity

By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with some generalizations of the Mathieu series. In particular, the mathematical expectation, variance and the characteristic functions, related to the probability density functions of the considered probability distributions are derived. As a consequence, some interesting inequalities involving complete monotonicity and log-convexity are derived.