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

Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative.

Numerical study of fractional nonlinear Schrodinger equations. The method applies to both linear pdoof nonlinear equations. Inserting the assumed integral inequality for the function u into the remainder gives.

Examples are given to illustrate the obtained results. We consider the ‘ fractional ‘ continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. Exact solutions to the time- fractional differential equations via local fractional derivatives. On some impulsive fractional differential equations in Banach spaces.

Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.

Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation ,which extend some results of the previous papers. Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems. Finite element method for gronwall-bellkan-inequality fractional Schrodinger equation. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed.

We convert each term of the problem to the matrix form by means of fractional Bernstein matrices. Full Text Available In this paper, we study Lie symmetry analysis and conservation laws for the time fractional nonlinear fourth-order evolution equation.

### phosphate-water fractionation equation: Topics by

Sign up using Facebook. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. 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.

In this study, we implement a well known transformation technique, Differential Transform Method DTMto the area of fractional differential equations. Full Text Available In the present paper, we study the integro-differential equations which are combination of differential and Fredholm—Volterra equations that have the fractional order with constant coefficients by the homotopy perturbation and the variational iteration.

The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Fractional corresponding operator in quantum mechanics and applications: Then the condition of the function is used to satisfy the contradiction, that is, the assumption is false, which verifies the oscillation of the solution. The analytical solutions within the nondifferential terms are discussed.

The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets. The solutions are obtained for the parameters in the range 0 Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation.

We use the fractional integrals in order to describe dynamical processes in the fractal medium. For the equation of water flux within a multi- fractional multidimensional confined aquifer, a dimensionally Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.

This paper suggests Lie group method for fractional partial differential equations. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view.

The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. Under this circumstance, it can be said that the NFDE has the space history. The method used to solve the problem is Homotopy Analysis Method. The results show that conformable fractional derivative definition is usable and convenient As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.

Many problems of filtration of liquids in fractal high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order. Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0 equations.

Full Text Available We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion.

We also discuss the relationships between the fractional and standard Schroedinger equations. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. Thomas Kirven 96 7. 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. A survey of the fractional calculus; 3. The approach is a generalization to our recent work for single fractional differential equations.

The fractional Hamiltonian equations corresponding to the Lagrangians of constrained systems within Caputo derivatives are investigated. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.

Periodicity and positivity of a class of fractional differential equations. The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The improved fractional sub- equation method and its applications to the space—time fractional differential equations in fluid mechanics.

Example of the fractional Hamiltonian system of the C-KdV soliton equation hierarchy is constructed, which is a new Hamiltonian structure.

## Grönwall’s inequality

The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. The explanation of this behaviour is given in terms of the mathematical approximations involved and its relationship to physically interesting quantities.

Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations.