Sandeep P Bhairat
Aim of this paper is to study the non-existence of global solutions of the fractional differential problem involving generalized Katugampola derivative. We utilize the test function method and fractional integration by parts formula to obtain the result. An illustrative example is also given.
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In this work, we give the power series solutions around an ordinary point, in the case of variable coefficients, homogeneous sequential linear conformable fractional differential equations of order 2α. Further, we introduce the conformable fractional Hermite differential equations, conformable fractional Hermite polynomials and basic properties of these polynomials.
Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions is developed in [2]. In this paper, we give conformable fractional differential transform method and its application to conformable fractional differential equations.
In this paper, we obtain new results related to Minkowski fractional integral inequality using generalized k-fractional integral operator which is in terms of the Gauss hypergeometric function.
We establish partial semigroup property of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems, and prove existence and uniqueness of solution to multi-term Caputo fractional differential systems
This paper deals with the local existence and uniqueness results for the solution of fractional differential equations with Hilfer-Hadamrd fractional derivative. Using Picard's approximations and generalizing the restrictive conditions imposed on nonlinear function, the iterative scheme for uniformly approximating the solution is constructed.
We consider a Hilfer fractional differential equation with nonlocal Erdélyi-Kober fractional integral boundary conditions. The existence, uniqueness and Ulam-Hyers stability results are investigated by means of the Krasnoselskii's fixed point theorem and Banach's fixed point theorem. An example is given to illustrate the main results.
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of continuous functions, in particular the Comparison's principle, and others that allow, the study of the limitation of fractional nonlinear differential equations.
In this paper, we show some results about the existence and the uniqueness of the positive solution for a $p$-Laplacian fractional differential equations with fractional derivative boundary condition. Our results are based on Krasnosel'skii's fixed point theorem, the nonlinear alternative of Leray-Schauder type and contraction mapping principle. Three examples are given to illustrate the applicability of our main results.
In this paper, we investigate the existence and uniqueness of solutions for a fractional boundary value problem supplemented with nonlocal Riemann-Liouville fractional integral and Caputo fractional derivative boundary conditions. Our results are based on some known tools of fixed point theory. Finally, some illustrative examples are included to verify the validity of our results.
In this paper, Lyapunov type inequality is establish for fractional boundary value problem involving the k-Prabhakar fractional derivative.
In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations.
In this work, we give the general solution sequential linear conformable fractional differential equations in the case of constant coefficients for α(\in)(0,1]. In homogeneous case, we use a fractional exponential function which generalizes the corresponding ordinary function. In non-homogeneous case, we present to fractional the method of variation of parameters for a particular solution of sequential linear conformable fractional differential equations.
In this paper Lyapunov type inequality is developed for hybrid fractional boundary value problem involving the prabhakar fractional derivative.
In this paper, using generalized k-fractional integral operator (in terms of the Gauss hypergeometric function), we establish new results on generalized k-fractional integral inequalities by considering the extended Chebyshev functional in case of synchronous function and some other inequalities.
We investigate fractional Cauchy type problem. By using Schauder fixed point theorem we obtain sufficient conditions for the global attractivity of solutions for nonlinear fractional differential equations in weighted spaces.
In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.
Recently, a new fractional derivative called the conformable fractional derivative is given on based basic limit definition derivative in [4]. Then, the fractional versions of chain rules, exponential functions, Gronwalls inequality, integration by parts, Taylor power series expansions is developed in [5]. In this paper, we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.
Using the $ψ-$Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
There is no unified method to solve the fractional differential equation. The type of derivative here used in this paper is of Jumarie formulation, for the several differential equations studied. Here we develop an algorithm to solve the linear fractional differential equation composed via Jumarie fractional derivative in terms of Mittag-Leffler function; and show its conjugation with ordinary calculus. In these fractional differential equations the one parameter Mittag-Leffler function plays the role similar as exponential function used in ordinary differential equations.
Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional differential equations in-terms of Mittag-Leffler function and generalized Sine and Cosine functions, where the fractional derivative operator is of Jumarie type. The use of Jumarie type fractional derivative, which is modified Rieman-Liouvellie fractional derivative, eases the solution to such fractional order systems. The use of this type of Jumarie fractional deriv…
In present paper, Hartman-Wintner-type inequality is established for a nonlocal fractional boundary value problem involving k-Prabhakar fractional derivative.
Using Gronwall inequality we will investigate the Ulam-Hyers and generalized Ulam-Hyers-Rassias stabilities for the solution of a fractional order pseudoparabolic partial differential equation.
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions for various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the generalized modified $k$-Bessel function of the first kind. It is also pointed out that the main results presented here are general enough to be able to be specialized to yield many known…
In this paper, we will develop a definition of mild solution for impulsive fractional differential equation of order $α\in (1,2)$ with the help of solution operator and study the existence results of mild solution for impulsive fractional differential equations with state dependent delay by using fixed point theorems. Finally, we present an example of partial fractional derivative to illustrate the existence and uniqueness result.
In this paper, we present a study on the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solution of the fractional functional differential equation using the Banach fixed point theorem.
The existence and stability results for a class of fractional differential equations involving generalized Katugampola derivative are presented herein. Some fixed point theorems are used and enlightening examples of obtained result are also given.
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2α.
In this paper, using the Riemann-Liouville fractional integral with respect to another function and the $ψ-$Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the $α$-resolvent operator in the Sobolev space $W^{1,1}(\ma…
In this paper we study the existence and continuation of solution to general fractional differential equation with Hilfer fractional derivative. First we establish new local existence theorems. Then we derive the continuation theorems. With the help of continuation theorems derived in this paper, several global existence results are constructed.
In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.
In this paper, we investigate the sufficient conditions for existence and uniqueness of solutions and δ-Ulam-Hyers-Rassias stability of an impulsive fractional differential equation involving $ψ$-Hilfer fractional derivative. Fixed point approach is used to obtain our main results.
In this article we study the asymptotic behavior of solutions of some fractional differential equations. We prove convergence to power type functions under some assumptions on the nonlinearities. Our results extend and generalize some existing well-known results on solutions of ordinary differential equations. Appropriate estimations and lemmas such as a fractional version of L'Hopital's rule are used.
In this paper, we consider a nonlocal fractional boundary value problem with Prabhakar derivative and obtained a Hartman-Wintner type inequality for it.
In this work we wish to highlight some consequences of a recent result proved in [N. D. Cong and H. T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations, J. Integral Equations Appl. 29 (2017), no. 4, 585-608]. Particular emphasis will be given to its application on fractional variational problems of Herglotz type.
The Cauchy-type problem for a nonlinear differential equation involving Hilfer fractional derivative is considered. We prove existence, uniqueness and continuous dependence of a solution for Cauchy-type problem using successive approximations and generalized Gronwall inequality.
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function due to Marichev-Saigo-Maeda, to the generalized Struve function. The results are expressed in terms of generalized Wright function. The results obtained here are general in nature and can easily obtain various known results.
This article investigates a class of discrete nabla fractional operators by using the discrete nabla convolution theorem. Inspired by this, we define the discrete generalized nabla fractional sum and differences of Riemann-Liouville and Caputo types. In the process, we give a relationship between the generalized discrete delta fractional operators introduced by Ferreira \cite{Ferreira} and the proposed discrete generalized nabla fractional operators via the dual identities. Also, we present some test examples to justify the relationship. Moreover, we prove the fundamental theorem of calculus f…
By means of the recent $ψ$-Hilfer fractional derivative and of the Banach fixed-point theorem, we investigate stabilities of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias on closed intervals $[a,b]$ and $[a,\infty)$ for a particular class of fractional integro-differential equations.
In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with fe…
This paper gives the existence and uniqueness results for solution of fractional differential equations with Hilfer derivative. Using some new techniques and generalizing the restrictive conditions imposed on considered function, the iterative scheme for uniformly approximating the solution is established.
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm-Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal and impulsive conditions. Furthermore, our study includes Riemann-Liouville, Caputo, Hadamard, Prabhakar, Hilfer and conformable type fractional derivatives. Results for boundary value problems involving fractional $p$-Laplacian, fractional operators with nonsingular Mittag-Leffler kernels, $q$-difference, di…
We present a new smoothness result for Caputo-type fractional ordinary differential equations, which reveals that, subtracting a non-smooth function that can be obtained by the information available, a non-smooth solution belongs to $ C^m $ for some positive integer $ m $.
We use Sadavoskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order αbetween 0 and 1 with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and complement the existing one.
In this paper, we generalized the known Laplace-transform final-value theorem. From our conclusion, one can deduce the existing results in [1, 3, 12]. By using final value theorem, we give a new proof that Caputo fractional differential equations have no nonconstant periodic solution.
Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions.
In present paper, we establish sufficient conditions for existence and stability of solutions for system of nonlinear implicit fractional differential equations. The main techniques are based on method of successive approximations. Finally, an illustrative example is given to show the applicability of our theoretical results.
In this paper, we are concerned with a fractional differential inequality containing a lower order fractional derivative and a polynomial source term in the right hand side. A non-existence of non-trivial global solutions result is proved in an appropriate space by means of the test-function method. The range of blow up is found to depend only on the lower order derivative. This is in line with the well-known fact for an internally weakly damped wave equation that solutions will converge to solutions of the parabolic part.
In this paper, by means of the Gronwall inequality, the ψ-Riemann-Liouville fractional partial integral and the ψ-Hilfer fractional partial derivative are introduced and some of its particular cases are recovered. Using these results, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solutions of a fractional partial differential equation of hyperbolic type in a Banach space (B, |.|), real or complex.
Discrete mathematics, the study of finite structures, is one of the fastest growing areas in mathematics and optimization. Discrete fractional calculus (DFC) theory that is an important subject of the fractional calculus includes the difference of fractional order. In present paper, we mention the radial Schr{ö}dinger equation which is a physical and singular differential equation. And, we can obtain the particular solutions of this equation by applying nabla ($ \nabla $) discrete fractional operator. This operator gives successful results for the singular equations, and solutions have fractio…
In this work, we study the regularity of positive solutions for nonlinear fractional differential equation with a singular weight. We define the new Banach space and use this space to show the regularity. We also give an example with a singular weight which may not be in $L^1.$
In this paper, we study the existence and uniqueness of pseudo $S$-asymptotically $ω$-periodic mild solutions of class $r$ for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula to a general clas…
In this paper, the generalized fractional operators involving Appell's function $F_3$ in the kernel due to Marichev-Saigo-Maeda are applied to the generalized $K$-Wright function. These fractional operators when applied to power multipliers of the generalized $K$-Wright function ${}_{p}Ψ^k_q$ yields a higher ordered generalized $K$-Wright function, namely, ${}_{p+3}Ψ^k_{q+3}$. The Caputo-type modification of Marichev-Saigo-Maeda fractional differentiation is introduced and the corresponding assertions for Saigo and Erdélyi-Kober fractional operators are also presented. The results derived in t…
In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of Fractional Calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann-Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikus…
This paper presents some sufficient conditions for the existence of solutions of fractional differential equation with nonlocal multi-point boundary conditions involving Caputo fractional derivative and integral boundary conditions. Our analysis relies on the Banach contraction principle, Boyd and Wong fixed point theorem, Leray-Schauder nonlinear alternative. Finally, examples are provided to illustrate our main results.
It is shown that the attractor of an autonomous Caputo fractional differential equation of order $α\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of "N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dyn…
Fractional integral and fractional maximal operators on generalized Fofana spaces
本文在广义Fofana空间上建立了分数次积分算子、分数次极大算子和广义分数次积分算子的有界性，并推导了涉及Riesz势的广义Olsen型不等式。
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new …
In this paper, we focus on the existence and uniqueness of solutions of boundary value problems for a coupled system of fractional differential equations with four-point boundary conditions involving $ψ$-Caputo fractional derivatives. Our investigation is based on Leray-Schauder alternative and Banach's fixed point theorem. Two examples are presented to illustrate the applicability of the results developed.
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