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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It this respect, this retardationless equation is similar to the fractional KFP by Barkai and Silbey. Some numerical examples are provided to confirm the accuracy of the proposed method. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered. Regarding on the exact solutions for the nonlinear fractional differential equations. This method can be seen as the fractional version of the known projective Riccati equation method.

Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are considered in the Caputo sense. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.

## Grönwall’s inequality

The specified idea is very effective, expedient for fractional Peoof, and could be extended to other physical problems. The method is given with some priori error estimate. Full Text Available In this paper the more general comparison theorems for conformable fractional differential equations is proposed and tested. 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 gronwall-gellman-inequality.

By an appropriate choice of the dispersive exponent, both mass. Oscillation results for certain fractional difference equations.

This subject has at least three hundred years of history. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. We develop and extend some recent works.

We further discuss possible filetypr to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of filteype time- fractional differential equations FDEs.

It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations. The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. Lie symmetry analysis and conservation gronwall-belkman-inequality for the time fractional fourth-order evolution equation.

One of the most popular research interests of science gronwall-bellman-inequa,ity engineering is the fractional calculus theory in recent times. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. Gronwall-Bellman inequality Ask Question. This book systematically presents solutions to the linear time- fractional diffusion-wave equation.

It has been shown that the fractional probability current equation is correct in the area of its applicability. In addition, several other sufficient conditions are established for the existence of at least triple, n or 2n-1 positive solutions. Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equationwe analyze a possible interpolation between the Dirac and the poof equations in one space dimension.

The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained.

### differential equations – Gronwall-Bellman inequality – Mathematics Stack Exchange

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

Hence, the integrability assumption on u implies that. Using the definition of v t for the first step, and then this inequality and the gronwall-bellman-ineqyality equation of the exponential function, we obtain. Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations.

Fractional Klein-Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter. Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. The solution proof shown to be unique.

The coefficients of these equations are a family of linear closed operators in the Banach space.

As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids.

This page was last edited on 5 Septemberat In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided.

The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP. Full Text Available In this paper, we investigate the local fractional Laplace equation in the steady heat-conduction problem.

A procedure to construct exact solutions of nonlinear fractional differential equations. Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their extensions and generalizations in one and more variables.

Then new existence results for smooth solutions gronwall-bellmwn-inequality established by using the Schauder fixed point theorem. Furthermore, using Caputo fractional derivative, the matrix form of the fractional derivative is constructed for the fractional Bernstein matrices. Fractional vector calculus and fractional Maxwell’s equations.

Then, various ansatz method are implemented to construct the solutions for both equations.