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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Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi- fractional confined aquifer is developed. 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.

Reduced differential transform method for partial differential equations within local fractional derivative operators. The new exact solutions of nonlinear fractional partial differential equations FPDEs are established by adopting first integral method FIM.

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 origin of each mistaken statement has been clarified and corrected statements have been made. Then, we use the filstype of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions.

Fractional differential equation with the fuzzy initial condition.

Fractional diffusion equation for heterogeneous medium. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equationsthis self-contained text presents the possibility of gronwall-bellman-inequallty fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research.

The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc.

Intended for use gronwall-bellman-ibequality the field and designed for students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and gfonwall-bellman-inequality on time scales.

Full Text Available Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi- gronwalo-bellman-inequality confined aquifer is developed. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics. It has been shown that the fractional probability current equation is correct in the area of its applicability.


The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform gronwall-bellman-inequaliy Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function.

Sign up or log in Sign up using Google. This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The solutions are given in the form of series with easily computable terms. Sign up using Email and Password. In this paper we develop efficient, scalable techniques for solving fractional -in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector.

We consider the ‘ fractional ‘ continuous medium model for the fractal media and derive gronwall-bellmna-inequality fractional generalization of the equations of balance of mass fiketype, momentum density, and internal energy.

These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method. Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative.

For the second kind of equation with initial condition, the equivalent fractional sum form of the fractional difference equation are firstly proved. The fractional derivatives are described in the sense of conformable fractional derivatives. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform.

An example is provided to illustrate the theory. Full Text Available In this work we discuss higher order multi-term partial differential equation PDE with the Caputo-Fabrizio fractional derivative in time.

differential equations – Gronwall-Bellman inequality – Mathematics Stack Exchange

We used the standard and Krasnoselskii’s fixed point theorems. Lie group method provides an efficient tool to solve nonlinear partial differential equations. The fractional derivative is defined in the Caputo sense. Filehype improved fractional sub- equation method and its applications to the space—time fractional differential equations in fluid mechanics. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.


We begin by showing how our method applies to a simple class of problems and we give a convergence result. We determine the solutions of fractional nonlinear electrical transmission lines NETL and the perturbed nonlinear Schroedinger Gronwall-bellman-inequality equation with the Kerr law nonlinearity term.

Grönwall’s inequality

Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their growall-bellman-inequality and generalizations in one and more variables. Results for neutron dynamic behavior for both positive gronwall-bellman-unequality negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Inequalities for differential and integral equations.

However, the presence of a proof differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints. Examples illustrate the results obtained in this paper. In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian.

This paper presents a fractional Dirac equation and its solution. The Rayleigh differential equation has been generalized of fractional second order. Gronwall-Bellman inequality Ask Question.

Gronwall-bellman-inequqlity the method of Lie point symmetry, we provide the associated vector fields, and derive the similarity reductions of the equationrespectively. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

We firstly decompose gronwall-beklman-inequality multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Symmetry properties of fractional diffusion equations. Positive solutions of fractional differential equations with derivative terms.

Directory of Open Access Journals Sweden. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. The continuation of the solution of fileytpe fractional equation to the solution of the corresponding integer order equation is proved. We also give an application for stochastic integropartial differential equations of fractional order.

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