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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Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.

The fractional derivatives are described in the Caputo sense. Such a step fractional Brownian motion can be obtained as a solution of the fractional Langevin equation with zero damping.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. We present the analytical technique for solving fractional -order, multi-term fractional differential equation.

Motivated by subdiffusive motion of biomolecules observed in living cells, we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. For the equation of water flux within a multi- fractional multidimensional confined aquifer, a dimensionally The equations have a sound physical basis and lead to physically correct coefficients in all flow situations.

For this purpose, a novel operational matrix of fractional -order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. Full Text Available We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The analytical solutions within the nondifferential terms are discussed.

### phosphate-water fractionation equation: Topics by

Fractional Klein-Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations.

Example demonstrates the generalized transfer function of an arbitrary viscoelastic system. Some properties of the fractional Schroedinger equation are studied. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. Integrable coupling gronwall-bellman-iinequality of fractional soliton equation hierarchy.

### Proof of Gronwall inequality – Mathematics Stack Exchange

Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics. Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations.

An example is constructed. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation.

There are two forms of the lemma, a differential form and an integral form. Full Text Available In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions.

## Grönwall’s inequality

Using the method of Lie point symmetry, we provide the associated vector fields, and derive the similarity reductions of the equationrespectively.

Full Text Available The fundamental solutions for linear fractional evolution equations are obtained.

The existence and uniqueness of the Mild solutions of the considered problem is also studied. The subordination principle, presented earlier in [J. For example, it can describe the process of tumor growth growth stimulation and growth inhibition in biomedical science.

We start with the standard Boltzmann equation ; averaging over photon frequencies leads to the appearance of a fractional derivative. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.

We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense.

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.

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.

Such a system appears in quantum mechanics. Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. The main advantage of this method is that it can reduce the error of the approximate solutions.