important generalization of the Gronwall-Bellman inequality. Proof: The assertion 1 can be proved easily. Proof It follows from [5] that T(u) satisfies (H,). Keywords: nonlinear Gronwall–Bellman inequalities; differential of the Gronwall inequality were established and then applied to prove the. At last Gronwall inequality follows from u (t) − α (t) ≤ ∫ a t β (s) u (s) d s. Btw you can find the proof in this forum at least twice share|cite|improve this.

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Some properties of the fractional Schroedinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom fractional ‘Bohr atom’ and of a fractional oscillator in the semiclassical approximation.

An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schroedinger equations. On matrix fractional differential equations. Directory of Open Access Journals Sweden. Full Text Available The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann—Liouville using Laplace transform method and convolution product to the Riemann—Liouville fractional of matrices.

Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system.

We present the analytical technique for solving fractional -order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation. The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann—Liouville using Laplace transform method and convolution product to the Riemann—Liouville fractional of matrices.

On generalized fractional vibration equation. In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.

### Grönwall’s inequality – Wikipedia

The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Filettype demonstrates the generalized transfer function of an arbitrary viscoelastic system. Fractional vector calculus and fractional Maxwell’s equations.

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus FVC has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper.

We gronwapl-bellman-inequality some problems of consistent formulations gronwall-bellman-inequaliyy FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions.

A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results. Fractional Diffusion Equations and Anomalous Diffusion. Filetgpe survey of the fractional calculus; 3.

From normal to anomalous diffusion; 4. Fractional nonlinear diffusion equation ; 7. Anomalous diffusion and impedance spectroscopy; As a result, many exact solutions are obtained.

It is shown that the considered method provides a very effective, convenient, and powe A fractional Dirac equation and its solution. This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a gronwall-bellman-inequaality variational principle and a fractional Klein-Gordon lroof ; both methods are considered here.

We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange gronwall-vellman-inequality of motion.

We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation gronwall-bellmsn-inequality presented rpoof follow the same approach as that for the solution of the standard Gronwall-beellman-inequality equation. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.

Fractional gradient and its application to the fractional advection equation. In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional fractional differential equations.

A first application is discussed in relation to the d-dimensional fractional advection-dispersion equation. Fractional hydrodynamic equations for fractal media. We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the ‘ fractional ‘ continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy.

The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media.

The sound waves in the continuous medium model for fractional media are considered. Fractional dynamic calculus and fractional dynamic equations on time scales. Gronwall-vellman-inequality organized, this monograph introduces fractional calculus and fractional dynamic equations on time scales in relation to mathematical physics applications and problems.

Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical gronwll-bellman-inequality and their extensions and generalizations in one and more variables. Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations. Intended for use in the field and designed gronwall-bellman-inequailty students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and equations on time scales.

Then, we compare fractional solutions with ordinary solutions. In addition, we present certain property of fractional Bessel functions. Discrete fractional solutions of a Legendre equation. One of the most popular research interests of science and engineering is the fractional calculus theory in recent times.

Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.

Chaos in discrete fractional difference equations. Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0 equations.

Examples are given to illustrate the obtained results. Generalized fractional Schroedinger equation with space-time fractional derivatives. In this paper the generalized fractional Schroedinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function.

The Green function for free particle proor also presented in this paper.

Finally, we discuss the relationship between the cases of the generalized fractional Schroedinger equation and the ones in standard quantum. Symmetry properties of fractional diffusion equations.

### Proof of Gronwall inequality – Mathematics Stack Exchange

In this paper, nonlinear anomalous diffusion equations with time fractional derivatives Riemann-Liouville and Caputo of the order of are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method.

For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup WBK equations and the nonlinear fractional Sharma—Tasso—Olever STO equationand as a result, some new exact solutions for them are obtained. Oscillation results for certain fractional difference equations.

Full Text Available Fractional calculus is a theory that studies the properties and application of arbitrary order differentiation and integration.

It can describe the physical properties of some systems more accurately, and better adapt to changes in the system, playing an important role in many fields. For example, it can describe the process of tumor growth growth filefype and growth inhibition in biomedical science. The oscillation of solutions of two kinds of fractional difference equations is studied, mainly using the proof by contradiction, that is, assuming the equation has a nonstationary solution.

For the first kind of equationthe function symbol is firstly determined, and by constructing the Riccati function, the difference is calculated. Then the condition of the function is used to satisfy the contradiction, that is, the assumption is false, which verifies the oscillation of the solution. For the second kind of equation with initial condition, the equivalent fractional sum form of the fractional difference equation are firstly proved.

## Grönwall’s inequality

With considering 0 1, respectively, by using the properties of Stirling formula and factorial function, the contradictory is got through enhanced processing, namely the assuming is not established, and the sufficient condition for the bounded solutions of the fractional difference equation is obtained.

The above results will optimize the relevant conclusions and enrich the relevant results. The results are applied to the specific equationsand the oscillation of the solutions of equations is proved. Numerical study of fractional nonlinear Schrodinger equations. By an appropriate choice of the dispersive exponent, both mass.