RIEMANN LIOUVILLE FRACTIONAL DERIVATIVE PDF



Riemann Liouville Fractional Derivative Pdf

QUASILINEARIZATION AND BOUNDARY VALUE PROBLEMS FOR. Tables of Fractional Derivatives and q-Derivatives In this appendix, we collect the Riemann–Liouvillefractional derivative and Caputo fractional of some q-analogues of the celebrated special functions and we also include a table of Riemann–Liouville fractional derivative for comparison. A.1 Table of Riemann–Liouville Fractional Derivatives, Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems by Piyachat Borisut 1,2 , Poom Kumam 1,2,* , Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4,*.

(PDF) On the k-Riemann-Liouville fractional derivative

Appendix A Tables of Fractional Derivatives and q-Derivatives. It worth noting that there exists only two main definition of fractional derivative, the first was proposed by Riemann and Liouville and is the derivative of the convolution of a given function and a power law kernel, the second one was suggested by Caputo and it is the convolution of the local derivative of a given function with power law, 21/06/2019 · In this paper, we introduce and study a new kind of coupled fractional differential system involving right Caputo and left Riemann–Liouville fractional derivatives, supplemented with nonlocal three-point coupled boundary conditions. Existence and uniqueness results for the given problem are derived with the aid of modern techniques of.

Modified Riemann-Liouville Derivative 1369 DEFINITION. RIEMANN-LIouVILLE. Let f : 91 ~ 9l, x --~ f(x) denote a continuous function. Its fractional derivative of order a is defined by the expressions (2.4), (2.5), and (2.6) for every f(.) But this definition gives rise to the following problem. PDF The main object of this paper is to introduce a new fractional operator called k-Riemann-Liouville fractional derivative defined by using the k-gamma function, which is a generalization of the classical gamma function. We also investigate relationships with the...

These operators are called the Riemann-Liouville fractional integral op-erators, or simply R-L operators. The special case of the fractional deriva-tive when KW! is called the semi-derivative. The connection between the Riemann-Liouville fractional integral and derivative can, as Riemann real- These operators are called the Riemann-Liouville fractional integral op-erators, or simply R-L operators. The special case of the fractional deriva-tive when KW! is called the semi-derivative. The connection between the Riemann-Liouville fractional integral and derivative can, as Riemann real-

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second These operators are called the Riemann-Liouville fractional integral op-erators, or simply R-L operators. The special case of the fractional deriva-tive when KW! is called the semi-derivative. The connection between the Riemann-Liouville fractional integral and derivative can, as Riemann real-

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A is a square matrix. At the same time, power-type estimate for them has been given. Extended Riemann-Liouville fractional derivative operator and its applications Praveen Agarwala, Junesang Choib,∗, R. B. Parisc aDepartment of Mathematics, Anand International College of Engineering, Jaipur-303012, India. bDepartment of Mathematics, Dongguk University, Gyeongju 780 …

This derivative was induced by the Riemann-Liouville derivative and is useful in physics. Definition 1.6. The Caputo derivative of fractional order αof function x t is defined as C Dα 0,t x t D − m−α 0,t dm dtm x t 1 Γ m−α t 0 t−τ m−α−1xm τ dτ, 1.8 in which m−1 <α

However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann-Liouville definition, from power-law kernel to generalized Mittag-Leffler-law via exponential-decay-law. On the k-Riemann-Liouville fractional derivative 43 And let F[ϕ] be the Fourier Transform of the function ϕ belonging to S(R) the Schwartzian space of functions that decay rapidly at infinity together with

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second These operators are called the Riemann-Liouville fractional integral op-erators, or simply R-L operators. The special case of the fractional deriva-tive when KW! is called the semi-derivative. The connection between the Riemann-Liouville fractional integral and derivative can, as Riemann real-

On the k-Riemann-Liouville Fractional Derivative

riemann liouville fractional derivative pdf

Fractional Noether’s Theorem with Classical and Riemann. Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems Piyachat Borisut 1,2, Poom Kumam 1,2,, Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4, 1 KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory, 29/12/2017 · The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5..

Merdan On the Solutions Fractional Riccati Differential. In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some …, 29/12/2017 · The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5..

On Riemann-Liouville and Caputo Impulsive Fractional Calculus

riemann liouville fractional derivative pdf

Fractional diffusion equation with a generalized Riemann. Fractional variational iteration method via modified Riemann–Liouville derivative 417 Yildirim, Ahmet, 2010. Analytical approach to Fokker–Planck equa- Yildirim, A., Momani, S., 2010. Series solution of a fractional tion with space- and time-fractional derivatives by means of the oscillator by means of the homotopy perturbation method. https://pt.wikipedia.org/wiki/Derivada_Fracion%C3%A1ria_de_Gr%C3%BCnwald-Letnikov In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems..

riemann liouville fractional derivative pdf


Riemann-Liouville derivative is the most used generalization of the derivative. It is based on Cauchy's formula for calculating iterated integrals. If the first integral of a function, which must equal to deriving it to -1, is as follows In this work, we study existence theorem of the initial value problem for the system of fractional differential equations 1 0, x. Dxt Axt t xt b , where . D denotes standard Riemann-Liouville fractional derivative, 01 , 12,, T T. x txtxt xt b b 12,,, , , n. bbn. and . A. is a square matrix.

Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper. In contrast to the Riemann-Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. In this article, a new application of fractional variational iteration method (FVIM) has been used for solution of nonlinear fractional reaction-diffusion with modified Riemann-Liouville derivative. A new application of FVIM was extended to derive analytical solutions in …

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems Piyachat Borisut 1,2, Poom Kumam 1,2,, Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4, 1 KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper. In contrast to the Riemann-Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions.

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations 1 0, x. Dxt Axt t xt b , where . D denotes standard Riemann-Liouville fractional derivative, 01 , 12,, T T. x txtxt xt b b 12,,, , , n. bbn. and . A. is a square matrix. 21/06/2019 · In this paper, we introduce and study a new kind of coupled fractional differential system involving right Caputo and left Riemann–Liouville fractional derivatives, supplemented with nonlocal three-point coupled boundary conditions. Existence and uniqueness results for the given problem are derived with the aid of modern techniques of

PDF The main object of this paper is to introduce a new fractional operator called k-Riemann-Liouville fractional derivative defined by using the k-gamma function, which is a generalization of the classical gamma function. We also investigate relationships with the... Riemann-Liouville derivative is the most used generalization of the derivative. It is based on Cauchy's formula for calculating iterated integrals. If the first integral of a function, which must equal to deriving it to -1, is as follows

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems by Piyachat Borisut 1,2 , Poom Kumam 1,2,* , Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4,* Fractional variational iteration method via modified Riemann–Liouville derivative 417 Yildirim, Ahmet, 2010. Analytical approach to Fokker–Planck equa- Yildirim, A., Momani, S., 2010. Series solution of a fractional tion with space- and time-fractional derivatives by means of the oscillator by means of the homotopy perturbation method.

Tables of Fractional Derivatives and q-Derivatives In this appendix, we collect the Riemann–Liouvillefractional derivative and Caputo fractional of some q-analogues of the celebrated special functions and we also include a table of Riemann–Liouville fractional derivative for comparison. A.1 Table of Riemann–Liouville Fractional Derivatives Fractional variational iteration method via modified Riemann–Liouville derivative 417 Yildirim, Ahmet, 2010. Analytical approach to Fokker–Planck equa- Yildirim, A., Momani, S., 2010. Series solution of a fractional tion with space- and time-fractional derivatives by means of the oscillator by means of the homotopy perturbation method.

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems by Piyachat Borisut 1,2 , Poom Kumam 1,2,* , Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4,* Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order О± are indicated graphically.

Study for System of Nonlinear Differential Equations with

riemann liouville fractional derivative pdf

A new Riemann–Liouville type fractional derivative. Modified Riemann-Liouville Derivative 1369 DEFINITION. RIEMANN-LIouVILLE. Let f : 91 ~ 9l, x --~ f(x) denote a continuous function. Its fractional derivative of order a is defined by the expressions (2.4), (2.5), and (2.6) for every f(.) But this definition gives rise to the following problem., Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order α are indicated graphically..

(PDF) On the k-Riemann-Liouville fractional derivative

ANALYSIS AND NUMERICAL SOLUTION OF A RIEMANN. Fractional variational iteration method via modified Riemann–Liouville derivative 417 Yildirim, Ahmet, 2010. Analytical approach to Fokker–Planck equa- Yildirim, A., Momani, S., 2010. Series solution of a fractional tion with space- and time-fractional derivatives by means of the oscillator by means of the homotopy perturbation method., In this article, a new application of fractional variational iteration method (FVIM) has been used for solution of nonlinear fractional reaction-diffusion with modified Riemann-Liouville derivative. A new application of FVIM was extended to derive analytical solutions in ….

Composition of Mixed Riemann-Liouville Fractional Integral and Mixed Fractional Derivative . By T. Mamatov . Bukhara Technological Institute of Engineering. Abstract- We study the question of the composition of a mixed fractional integral and a mixed fractional … 29/12/2017 · The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5.

Fractional Noether’s Theorem with Classical and Riemann–Liouville Derivatives Gasta˜o S. F. Frederico1,2 and Delfim F. M. Torres 2 Abstract—We prove a Noether type symmetry theorem to fractional problems of the calculus of variations with classical and Riemann–Liouville derivatives. As result, we obtain con- The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second

However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann-Liouville definition, from power-law kernel to generalized Mittag-Leffler-law via exponential-decay-law. The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions.

Composition of Mixed Riemann-Liouville Fractional Integral and Mixed Fractional Derivative . By T. Mamatov . Bukhara Technological Institute of Engineering. Abstract- We study the question of the composition of a mixed fractional integral and a mixed fractional … On the k-Riemann-Liouville fractional derivative 43 And let F[ϕ] be the Fourier Transform of the function ϕ belonging to S(R) the Schwartzian space of functions that decay rapidly at infinity together with

the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be … 26/02/2015 · We present an analysis of projectile motion in view of fractional calculus. We obtain the solution for the problem using the Riemann-Liouville derivative, and then we compute some features of projectile motion in the framework of Riemann-Liouville fractional calculus. We compare the solutions using Caputo derivatives and Riemann-Liouville

With the Riemann–Liouville fractional derivative, an arbitrary function needs not be continuous at the origin and it needs not be differentiable. 2. One of the great advantages of the Caputo fractional derivative is that it allows traditional initial and boundary conditions to … The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions.

Tables of Fractional Derivatives and q-Derivatives In this appendix, we collect the Riemann–Liouvillefractional derivative and Caputo fractional of some q-analogues of the celebrated special functions and we also include a table of Riemann–Liouville fractional derivative for comparison. A.1 Table of Riemann–Liouville Fractional Derivatives Fractional Noether’s Theorem with Classical and Riemann–Liouville Derivatives Gasta˜o S. F. Frederico1,2 and Delfim F. M. Torres 2 Abstract—We prove a Noether type symmetry theorem to fractional problems of the calculus of variations with classical and Riemann–Liouville derivatives. As result, we obtain con-

for the integral fractional calculus, the most popular ones being the Riemann-Liouville fractional integral. There is also a fractional Riemann- Liouville derivative. However, the well- known Caputo fractional derivative are less involved since the associated integral operator manipulates the derivatives of the primitive function under the integral In this paper using topological degree we study the existence of nontrivial solutions for a higher order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives. Here, the nonlinearity can be sign-changing and can also depend on the derivatives of unknown functions.

On the k -Riemann-Liouvillefractional integral and applications Int. J. Stat. Math. 034 k -Riemann-Liouville fractional integral Here we want to present the fractional integration which generalizes all of the above Rimann-Liouville fractional integrals as follows (see Romero (2013)): Let be a real non negative number. 08/05/2018 · Here, the concept of a new and interesting Riemann–Liouville type fractional derivative operator is exploited. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two variables and Lauricella hypergeometric function of three variables. With a view on analytic properties

On the k -Riemann-Liouvillefractional integral and applications Int. J. Stat. Math. 034 k -Riemann-Liouville fractional integral Here we want to present the fractional integration which generalizes all of the above Rimann-Liouville fractional integrals as follows (see Romero (2013)): Let be a real non negative number. In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where DО± denotes standard Riemann-Liouville fractional derivative, 0 and A is a square matrix. At the same time, power-type estimate for them has been given.

for the integral fractional calculus, the most popular ones being the Riemann-Liouville fractional integral. There is also a fractional Riemann- Liouville derivative. However, the well- known Caputo fractional derivative are less involved since the associated integral operator manipulates the derivatives of the primitive function under the integral Fractional variational iteration method via modified Riemann–Liouville derivative 417 Yildirim, Ahmet, 2010. Analytical approach to Fokker–Planck equa- Yildirim, A., Momani, S., 2010. Series solution of a fractional tion with space- and time-fractional derivatives by means of the oscillator by means of the homotopy perturbation method.

1.2 Fractional Derivatives of Exponential Functions Joseph Liouville wrote about the theory of fractional calculus in his 1832 mem-oirs. According to [6] he fimade the –rst major study of fractional calculus.fl The derivative of an exponential function, where n2 Z + and bis a constant, is given by the following formula: Dne bx= bne (5) When Benoit Mandelbrot discussed the problem of fractional Brownian motion in his classic book The Fractal Geometry of Nature, he already pointed out some strong relations to the Riemann-Liouville fractional integral and differential calculus.

QUASILINEARIZATION AND BOUNDARY VALUE PROBLEMS FOR

riemann liouville fractional derivative pdf

Stability of the nonlinear fractional differential. The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions., lems for fractional di erential equations of Riemann-Liouville type. In this article, we consider two boundary value problems for fractional di erential equations of Riemann-Liouville type and apply the method of quasilinearization. Speci cally, we 2010 Mathematics Subject Classi cation. 26A33, 34K10, 34A45, 47H05. Key words and phrases..

ANALYSIS AND NUMERICAL SOLUTION OF A RIEMANN

riemann liouville fractional derivative pdf

Study for System of Nonlinear Differential Equations with. In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A is a square matrix. At the same time, power-type estimate for them has been given. https://en.wikipedia.org/wiki/Conformal_map Extended Riemann-Liouville fractional derivative operator and its applications Praveen Agarwala, Junesang Choib,∗, R. B. Parisc aDepartment of Mathematics, Anand International College of Engineering, Jaipur-303012, India. bDepartment of Mathematics, Dongguk University, Gyeongju 780 ….

riemann liouville fractional derivative pdf

  • Study for System of Nonlinear Differential Equations with
  • Analysis and numerical solution of a Riemann-Liouville
  • On the k-Riemann-Liouville fractional integral and
  • QUASILINEARIZATION AND BOUNDARY VALUE PROBLEMS FOR

  • Boundary Value Problems Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations Jessada Tariboon 0 2 Sotiris K Ntouyas 1 Weerawat Sudsutad 0 2 0 Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology Riemann–Liouville Derivative and their main properties. In section 3, we study some properties of linear fractional differential operators with constant coefficients. In section 4, we adopt the method of inverse fractional differential operators to find the particular

    leading term in the di erential operator is a Riemann-Liouville fractional derivative of order 2 with 0 < <1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind. In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some …

    PDF The main object of this paper is to introduce a new fractional operator called k-Riemann-Liouville fractional derivative defined by using the k-gamma function, which is a generalization of the classical gamma function. We also investigate relationships with the... Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems by Piyachat Borisut 1,2 , Poom Kumam 1,2,* , Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4,*

    In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems. The fractional integral (or the Riemann-Liouville integral) with fractional order of function is defined as has an important convolution property (or semigroup property), that is, for arbitrary and . This implies that . Definition 1.3. The GrГјnwald-Letnikov fractional derivative with fractional order is defined by, if, where .

    Riemann–Liouville Derivative and their main properties. In section 3, we study some properties of linear fractional differential operators with constant coefficients. In section 4, we adopt the method of inverse fractional differential operators to find the particular - Fractional integral according to Riemann-Liouville - Caputo fractional derivative - Riesz-Feller fractional derivative - Grünwal-Letnikov - Integral equations - Relaxation and oscillation equations - Fractional di⁄usion equation - A nonlinear fractional di⁄erential equation. Stochastic solution - Geometrical interpretation of fractional

    - Fractional integral according to Riemann-Liouville - Caputo fractional derivative - Riesz-Feller fractional derivative - GrГјnwal-Letnikov - Integral equations - Relaxation and oscillation equations - Fractional diвЃ„usion equation - A nonlinear fractional diвЃ„erential equation. Stochastic solution - Geometrical interpretation of fractional Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems Piyachat Borisut 1,2, Poom Kumam 1,2,, Idris Ahmed 1,2,3 and Kanokwan Sitthithakerngkiet 4, 1 KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory

    equation with generalized Riemann-Liouville fractional derivatives and con-stant coefficients by using operational calculus of Mikusin ski’s type. We prove that the initial value problem has the solution of if and only if some initial values should be zero. Keywords: generalized Riemann … In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems.

    Tables of Fractional Derivatives and q-Derivatives In this appendix, we collect the Riemann–Liouvillefractional derivative and Caputo fractional of some q-analogues of the celebrated special functions and we also include a table of Riemann–Liouville fractional derivative for comparison. A.1 Table of Riemann–Liouville Fractional Derivatives 08/05/2018 · Here, the concept of a new and interesting Riemann–Liouville type fractional derivative operator is exploited. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two variables and Lauricella hypergeometric function of three variables. With a view on analytic properties

    The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. In this article, a new application of fractional variational iteration method (FVIM) has been used for solution of nonlinear fractional reaction-diffusion with modified Riemann-Liouville derivative. A new application of FVIM was extended to derive analytical solutions in …

    a fractional derivative was an ongoing topic in the last 300 years. Several mathematicians contributed to this subject over the years. People like Liouville, Riemann, and Weyl made major contributions to the theory of fractional calculus. The story of the fractional calculus for the integral fractional calculus, the most popular ones being the Riemann-Liouville fractional integral. There is also a fractional Riemann- Liouville derivative. However, the well- known Caputo fractional derivative are less involved since the associated integral operator manipulates the derivatives of the primitive function under the integral

    In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems. leading term in the di erential operator is a Riemann-Liouville fractional derivative of order 2 with 0 < <1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind.

    9.1. FRACTIONAL CALCULUS: RIEMANN-LIOUVILLE 323 then the latter expression is a first-order integral. If g(t) = (t) is the Dirac delta-function, then transformation represented by the former integral reproduces the in- where αis fractional derivative order, nis an integer, A x ,B x,andC x are known real functions, and d k is a constant. The goal of this paper is to extend the application of the variational iteration method to solve fractional nonlinear Riccati differential equations with modified Riemann-Liouville derivative.

    In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some … lems for fractional di erential equations of Riemann-Liouville type. In this article, we consider two boundary value problems for fractional di erential equations of Riemann-Liouville type and apply the method of quasilinearization. Speci cally, we 2010 Mathematics Subject Classi cation. 26A33, 34K10, 34A45, 47H05. Key words and phrases.