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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A nonlocal Cauchy problem is discussed for the evolution equations. A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using of differential forms and exterior derivatives of fractional orders.

phosphate-water fractionation equation: Topics by

Views Read Edit View history. Asymptotic behavior of solutions of forced fractional differential equations.

Some illustrative examples are presented. An example is constructed. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. Full Text Available In this paper we study a class of one-dimensional Dirichlet boundary value problems involving the Caputo fractional derivatives.

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.

Grönwall’s inequality

As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution. Bifurcation dynamics of the tempered fractional Langevin equation. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method.

The method yields a powerful flietype algorithm for fractional order derivative to implement. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions.

Fractional diffusion equations and anomalous diffusion. The oscillation of solutions of two kinds of fractional difference equations is studied, mainly using the proof by contradiction, that is, gronwall-bellman-inequalify the equation has a nonstationary solution.


The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. Stationarity-conservation laws for fractional differential equations with variable coefficients. The main property of the suggested fractional maps is a long-term memory. By means of the Mittag-Leffler function gornwall-bellman-inequality the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.

We also discuss Caputo impulsive fractional differential equations with finite delay. We use the fractional integrals in order to describe dynamical processes in the fractal medium. Several example equations are solved and the response prooc mechanical systems described by such equations is studied.

A map between standard 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. Fractional hydrodynamic equations for fractal media.

The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral propf. The proof is divided into three steps. In this paper, fractional differential transform method DTM is implemented on the Bagley Torvik equation.

The new exact solutions of nonlinear fractional partial differential equations FPDEs are established by adopting first integral method FIM.

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. On the solutions of fractional reaction-diffusion equations. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters.

For this system of fractional evolution equationswe also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case.

The time- fractional derivative is described in Caputo’s sense and the space- fractional derivative in Riesz’s sense. This new relationship is described in terms of a new fractional operator that includes both left- and right-sided fractional derivatives.

In this work we construct a closed-form solution for the multidimensional transport equation rewritten in integral form which is expressed in terms of a fractional derivative of the angular flux.


The presented approach is more straightforward and allows some simplification in application of the variational iteration method to fractional differential equationsthus improving the convergence of the successive iterations. Subordination principle for fractional evolution equations. The fractional derivatives are described in the Caputo sense. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.

However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework.

Equations are developed for calculating interfacial drag and shear coefficients for dispersed vapour flows from void fraction correlations. Fractional differential equation with the fuzzy initial condition. Klein-Gordon equation is one of the basic steps towards relativistic quantum mechanics. Full Text Available In the present paper, we apply the Bezier curves method for solving fractional optimal control problems OCPs and fractional Riccati differential equations.

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity.

Lie group method provides an efficient tool to solve nonlinear partial differential equations. Lie point symmetries of these equations are investigated and compared. Using the fractional Klein-Gordon equationwe can overcome the problem.

As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom fractional ‘Bohr atom’ and of a fractional oscillator in the semiclassical approximation.

The homotopy analysis transform method HATM is applied in this work in order to find the analytical solution of fractional diffusion equations FDE. In the NFDE model, neutron flux in each zone depends directly on the all previous zones not only on the nearest neighbors.