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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The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Inequalities for differential and integral equations.

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. Full Text Available Fractional calculus is a theory that studies the properties and application of arbitrary order differentiation and integration.

Being concise and straightforward, this method is applied to the space—time fractional Gardner equation with variable coefficients. Boundary value problems for multi-term fractional differential equations.

Finally, by using the nonlinear self-adjointness method and Riemann-Liouville time- fractional derivative operator as well as Euler-Lagrange operator, the conservation laws of the equation are obtained.

On matrix fractional differential equations. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. Fractional neutron point kinetics equations for nuclear reactor dynamics. Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus.

Grönwall’s inequality

These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio Integro-differential equations of fractional order with nonlocal fractional boundary conditions associated with financial asset model.

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. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. The obtained results are compared with the results obtained via other techniques. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi.


We investigate the oscillation of a class of fractional differential equations with damping term. The equations have a sound physical basis and lead to physically correct coefficients in all flow situations.

Several example equations are solved and the response of mechanical systems described by such equations is studied. An analytical method is used to solve the fractional model for the modified point kinetics equations.

The Klein—Gordon—Zakharov equations with the positive fractional It is shown that the considered method provides a very effective, convenient, and powe The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations.

The solutions are given in the form of series with easily computable terms. 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 consistent equation is also developed. Full Text Available In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equationshomogeneous or nonhomogeneous.

In order to have a better representation of these physical models, fractional calculus is used. Full Text Available In this article, the Sumudu transform series expansion method is used to handle the local fractional Laplace equation arising in the steady fractal heat-transfer problem via local fractional calculus.

Full Text Available We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The fractional Boltzmann equation for resonance radiation transport in plasma is proposed.

On the solution of the Schroedinger equation through continued fractions. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.


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

phosphate-water fractionation equation: Topics by

Boundary value problemfor multidimensional fractional advection-dispersion equation. 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. The method applies to both linear and nonlinear equations. Full Text Available The Fractional Complex Transform is extended to solve exactly time- fractional differential equations with the modified Riemann-Liouville derivative.

Using this operator it is possible to generate new ELDs that contain different fractional parts, in addition to the already known ELDs, which only differ by a sum of first-order partial derivatives of two arbitrary functions.

We establish the existence of a non-negative ground state solution by variational methods. Fractional derivatives have become important in physical and chemical phenomena as visco-elasticity and visco-plasticity, anomalous diffusion and electric circuits.

We represent explicit solution of formulated problem in particular case by Fourier series. Higher order multi-term time- fractional partial differential equations involving Caputo-Fabrizio derivative. Numerical simulation of fractional Cable equation of spiny neuronal dendrites. A survey of the fractional calculus; 3. The fraction -factor in this method gives it an edge over other existing analytical methods for non-linear differential equations.

Pramana — Journal of Physics News.

For gronwall-belman-inequality system of fractional evolution equationswe also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case. In this paper, nonlinear anomalous diffusion equations with time fractional derivatives Riemann-Liouville and Caputo of the order of are considered. Fractional differential equations prpof been discussed in this study. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces.

Then we give a judging theorem for this operator and with this judging theorem we prove that R—L, G—L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator.