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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In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. By an appropriate choice of the dispersive exponent, both mass.

In addition, we present certain property of fractional Bessel functions. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator.

Integro-differential equations of fractional order with nonlocal fractional boundary conditions associated with financial asset model. Full Text Available Similarity method is employed to solve multiterm time- fractional diffusion equation. The improved fractional sub- equation method and its applications to the space—time fractional differential equations in gronwall–bellman-inequality mechanics.

The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.

differential equations – Gronwall-Bellman inequality – Mathematics Stack Exchange

The Caputo type fractional derivative is employed. Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems. Applying the finite-dimensional reduction method and the penalization method, we obtain the high-energy solutions for this equation. Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative.


The fractional derivatives are described in Jumarie’s modified Riemann-Liouville sense. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field.

We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller.

Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations. We also extend the method to the two-dimensional time-space- fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted.

Grönwall’s inequality

In addition, several other sufficient conditions are established for the existence of at least triple, n or 2n-1 positive solutions.

The other three methods follow the particle strength exchange PSE methodology and are by construction conservative, in the sense that the total particle strength is time invariant. Fractional nonlinear diffusion equation ; 7. A nonlocal Cauchy problem is discussed for filetjpe evolution equations. The convergence and the accuracy gronwall-bellmn-inequality the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers-Fokker-Planck KFP equation by Barkai and Silbey and its generalization by Friedrich et al. The fraction -factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. The fractional Boltzmann equation for resonance radiation transport in plasma is proposed.

We used the standard and Krasnoselskii’s fixed point theorems. For the equation of water flux within a multi- fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The method is based on analytic technique of the fixed point theory.


We study a fuzzy fractional differential equation FFDE and present its solution using Zadeh’s extension principle. The fundamental solution of these problems is established and its moments are calculated. New Hamiltonian structure of the fractional C-KdV soliton equation hierarchy.

phosphate-water fractionation equation: Topics by

This page was last edited on 5 Septemberat The technique used in obtaining their results will apply to related fractional differential equations with Caputo derivatives of any order. In this paper, fractional differential transform method DTM is implemented on the Bagley Torvik equation.

This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis.

Full Text Available We discuss new approaches to handling Fokker Planck gronwall-bellman-inequakity 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. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed.

Full Text Available In recent time there is a very great interest in the study fietype differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. Existence of solutions of abstract fractional impulsive semilinear evolution equations.