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Dumitru Baleanu

Publications and source records attributed to Dumitru Baleanu.

At least 37 records · Page 2Linked to original sources
arXiv2016

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the recently introduced nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the $Q-$operator to confirm our results. The corresponding Euler-Lagrange equations are obtained and one illustrative example is discussed.

math.CA↗
arXiv2016

On fractional derivatives with exponential kernel and their discrete versions

In this manuscript we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. Then, we provide the integration by parts formula and we use $Q-$operator to confirm our results. The related Euler-Lagrange equations were obtained and one example is analyzed. Moreover, we formulate and discus the discrete counterparts of our results.

math.DS↗
arXiv2016

Fractional derivative defined by non-singular kernels to capture anomalous relaxation and diffusion

Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that challenges numerical calculations. This study first explores physical properties of relaxation and diffusion models where the fractional derivative was defined recently using an exponential kernel. Analytical analysis shows that the fractional derivative model with an exponential kernel cannot characterize non-exponential dynamics well-documented in anomalous relaxation and diffusion. A legitimate extension of the previous fractional derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel. Numerical tests show that the fractional derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel, implying the potential applicability of the new fractional derivative in quantifying real-world, anomalous relaxation and diffusion processes.

cond-mat.stat-mech↗
arXiv2015

New Derivatives on Fractal Subset of Real-line

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.

math.CA↗
arXiv2013

A Kamenev-type oscillation result for a linear $(1+α)$--order fractional differential equation

We investigate the eventual sign changing for the solutions of the linear equation $\left(x^{(α)}\right)^{\prime}+q(t)x=0$, $t\geq0$, when the functional coefficient $q$ satisfies the Kamenev-type restriction $\limsup\limits_{t\rightarrow+\infty}\frac{1}{t^{\varepsilon}}\int_{t_0}^{t}(t-s)^{\varepsilon}q(s)ds=+\infty$ for some $\varepsilon>2$, $t_{0}>0$. The operator $x^{(α)}$ is the Caputo differential operator and $α\in(0,1)$.

math.CA↗
arXiv2013

Schrödinger Equation on Fractals Curves Imbedding in $R^3$

In this paper we have generalized the quantum mechanics on fractal time-space. The time is changing on Cantor-set like but space is considered as fractal curve like Von-Koch curve. The Feynman path method in quantum mechanics has been suggested on fractal curve. Using $F^α$-calculus and Feynman path method we found the Schrëdinger on fractal time-space. The Hamiltonian operator and momentum operator has been derived. More, the continuity equation and the probability density is given in generalized formulation.

math-ph↗
arXiv2012

Fractional differences and sums with binomial coefficients

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Litnikov fractional derivatives. In this article we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

math.DS↗
arXiv2011

Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and associated solitonic hierarchies. The constructions yield horizontal/vertical pairs of fractional vector sine-Gordon equations and fractional vector mKdV equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of Schrodinger maps.

math-ph↗
arXiv2011

Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.

math-ph↗
arXiv2010

Asymptotic integration of $(1+α)$-order fractional differential equations

\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+α)$--order fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x+a(t)x=0$, $t>0$, under some simple restrictions on the functional coefficient $a(t)$, where ${}_{0}^{\>i}{\cal O}_{t}^{1+α}$ is one of the fractional differential operators ${}_{0}D_{t}^α(x^{\prime})$, $({}_{0}D_{t}^αx)^{\prime}={}_{0}D_{t}^{1+α}x$ and ${}_{0}D_{t}^α(tx^{\prime}-x)$. Here, ${}_{0}D_{t}^α$ designates the Riemann-Liouville derivative of order $α\in(0,1)$. The asymptotic formula reads as $[a+O(1)]\cdot x_{\scriptstyle small}+b\cdot x_{\scriptstyle large}$ as $t\rightarrow+\infty$ for given $a$, $b\in\mathbb{R}$, where $x_{\scriptstyle small}$ and $x_{\scriptstyle large}$ represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x=0$, $t>0$.

math-ph↗
arXiv2010

Fractional Almost Kahler - Lagrange Geometry

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry and, in our approach, with fractional derivatives. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a "prime" Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.

math-ph↗
arXiv2010

Fractional Analogous Models in Mechanics and Gravity Theories

We briefly review our recent results on the geometry of nonholonomic manifolds and Lagrange--Finsler spaces and fractional calculus with Caputo derivatives. Such constructions are used for elaborating analogous models of fractional gravity and fractional Lagrange mechanics.

math-ph↗
arXiv2010

Fractional Exact Solutions and Solitons in Gravity

We survay our recent results on fractional gravity theory. It is also provided the Main Theorem on encoding of geometric data (metrics and connections in gravity and geometric mechanics) into solitonic hierarchies. Our approach is based on Caputo fractional derivative and nonlinear connection formalism.

math-ph↗
arXiv2010

Fedosov Quantization of Fractional Lagrange Spaces

The main goal of this work is to perform a nonolonomic deformation (Fedosov type) quantization of fractional Lagrange geometries. The constructions are provided for a (fractional) almost Kahler model encoding equivalently all data for fractional Euler-Lagrange equations with Caputo fractional derivative. For homogeneous generating Finsler functions, the geometric models contain quantum versions of fractional Finsler spaces. The scheme can be generalized for fractional Hamilton systems and various models of fractional classical and quantum gravity. We conclude that the approach with Caputo fractional derivative allows us to geometrize both classical and quantum (Fedosov type) fractional regular Lagrange interactions.

math-ph↗
arXiv2010

On the asymptotic integration of a class of sublinear fractional differential equations

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations $D_{0+}^α(x-x_0) =f(t,x)$ which includes $D_{0+}^α(x-x_0) =H(t)x^λ$ with $λ\in(0,1)$ for the case of slowly-decaying coefficients $H$. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as $x(t)=o(t^{aα})$ when $t\to+\infty$ for $1>α>1-a>λ>0$. Our result can be thought of as a non--integer counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal.

math.DS↗