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.

Author: Kijind Gashura
Country: Romania
Language: English (Spanish)
Genre: Finance
Published (Last): 14 November 2007
Pages: 484
PDF File Size: 2.76 Mb
ePub File Size: 20.15 Mb
ISBN: 210-2-22842-157-9
Downloads: 19908
Price: Free* [*Free Regsitration Required]
Uploader: Dukus

Klein-Gordon equation is one of the basic steps towards relativistic quantum mechanics.

Proof of Gronwall inequality – Mathematics Stack Exchange

The analytical solutions within the nondifferential terms are discussed. We report numerical simulations.

Gronwall-bellmani-nequality differential transform method for partial filetypee equations within local fractional derivative operators. We begin by showing how our method applies to a simple class of problems and we give a convergence result.

Closed form solutions of two time fractional nonlinear wave equations. The solutions involving the non-differentiable graph are obtained by using the characteristic equation method CEM via local fractional derivative.

Application of the Lie Symmetry Analysis for second-order fractional differential equations.

Grönwall’s inequality – Wikipedia

We also point out how to teleport a particle to an arbitrary destination. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.

Gronwal-lbellman-inequality addition, several other sufficient conditions are established for the existence of at least triple, n or 2n-1 positive solutions. A map between prooc quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation.


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. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works.

Grönwall’s inequality

Positivity of outcomes is considered under certain requirements. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique prolf solve local fractional partial differential equations. In addition, some of the solutions are gronwall-bellman-inesuality in the figures with the help of Mathematica.

We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractionalnonlinear, nonhomogeneous differential 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.

We also filetgpe an application for stochastic integropartial differential equations of fractional order.

Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. Full Pgoof Available An algorithm for approximating solutions to fractional differential equations FDEs in a modified new Bernstein polynomial basis is introduced.

Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations. Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in filehype article.

Some numerical examples are provided to confirm the accuracy of the proposed method. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local.

The physical parameters in the soliton solutions: Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type gdonwall-bellman-inequality fractional differential equations gronwall-bellman-iequality in mathematical physics.


This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics in particular, fractional dynamics.

Then the condition of the pdoof is used to satisfy the contradiction, that is, the assumption is false, which verifies the oscillation of the solution.

As I do not know the specific form of the Gronall-Bellman inequality in your textbook, I provide a direct proof below. In this paper the. Fractional differential equation with the fuzzy initial condition. The fractional derivatives are described in Jumarie’s modified Riemann-Liouville sense. The fractional derivatives are described in the Caputo sense.

An equation for the fractional probability current density is developed and discussed. Fractional derivatives have become important in physical and chemical phenomena as gronwall-bellman-innequality and visco-plasticity, anomalous diffusion and electric circuits.

If the solution function develops into function with two or more variables, then its differential equation must be changed into fractional partial differential equation. The results presented here would provide generalizations of those given in earlier works. In this paper, we have formulated fractional Klein-Gordon gronwall-bellan-inequality via Jumarie fractional derivative and found two types of solutions.

Lie group method provides an efficient tool to solve nonlinear partial differential equations. We use mathematical induction.