# Fractional Integrals And Derivatives Theory And Applications Pdf

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- Fractional Integrals and Derivatives: Theory and Applications
- Fractional Calculus via Functional Calculus: Theory and Applications
- FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications

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## Fractional Integrals and Derivatives: Theory and Applications

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D.

More generally, one can look at the question of defining a linear operator. Fractional differential equations, also known as extraordinary differential equations, [1] are a generalization of differential equations through the application of fractional calculus.

In applied mathematics and mathematical analysis , a fractional derivative is a derivative of any arbitrary order, real or complex. Therefore, it is expected that the fractional derivative operation involves some sort of boundary conditions , involving information on the function further out. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.

A fairly natural question to ask is whether there exists a linear operator H , or half-derivative, such that. Form the definite integral from 0 to x. Call this. The Cauchy formula for repeated integration , namely. Using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.

It is straightforward to show that the J operator satisfies. The inner integral is the beta function which satisfies the following property:. This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general. For negative integer power k, the gamma function is undefined and we have to use the following relation: [10].

This extension of the above differential operator need not be constrained only to real powers. Also setting negative values for a yields integrals. For example,.

We can also come at the question via the Laplace transform. Knowing that. Indeed, given the convolution rule. Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations. The classical form of fractional calculus is given by the Riemann—Liouville integral , which is essentially what has been described above. The theory for periodic functions therefore including the "boundary condition" of repeating after a period is the Weyl integral.

It is defined on Fourier series , and requires the constant Fourier coefficient to vanish thus, it applies to functions on the unit circle whose integrals evaluate to zero. The Riemann-Liouville integral exists in two forms, upper and lower. Considering the interval [ a , b ] , the integrals are defined as. The Hadamard fractional integral is introduced by Jacques Hadamard [12] and is given by the following formula,. Recently, using the generalized Mittag-Leffler function, Atangana and Baleanu suggested a new formulation of the fractional derivative with a nonlocal and nonsingular kernel.

The integral is defined as:. Unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral. The corresponding derivative is calculated using Lagrange's rule for differential operators.

Similar to the definitions for the Riemann-Liouville integral, the derivative has upper and lower variants.

Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his paper. Moreover, there is the Caputo fractional derivative of distributed order defined as.

In a paper of , M. Caputo and M. Like the integral, there is also a fractional derivative using the general Mittag-Leffler function as a kernel. The theory of pseudo-differential operators also allows one to consider powers of D. The operators arising are examples of singular integral operators ; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials.

So there are a number of contemporary theories available, within which fractional calculus can be discussed. As described by Wheatcraft and Meerschaert , [31] a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear.

In the referenced paper, the fractional conservation of mass equation for fluid flow is:. When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by Fick's Law.

Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation here in dimensionless form :. Taking the derivative of C x,s and then the inverse Laplace transform yields the following relationship:.

For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction. In — Atangana et al. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. This equation [ clarification needed ] has been shown useful for modeling contaminant flow in heterogenous porous media. Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a variational order derivative.

The modified equation was numerically solved via the Crank—Nicolson method. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives [39].

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. The time-space fractional diffusion governing equation can be written as. Its applications in anomalous diffusion modeling can be found in reference. Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.

Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable u t in terms of a measured error value e t can be written as. The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law.

This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This book on power-law attenuation also covers the topic in more detail. Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.

The potential energy function V r , t depends on the system. From Wikipedia, the free encyclopedia. Limits of functions Continuity. Mean value theorem Rolle's theorem. Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations. Not to be confused with Fractal derivative.

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Acoustic attenuation Autoregressive fractionally integrated moving average Differintegral Differential equation Erdelyi—Kober operator Neopolarogram Nonlocal operator Riemann—Liouville integral Weyl integral. Handbook of Differential Equations. Elsevier Science. Bulletin of Mathematical Analysis and Applications.

Bibcode : arXiv Magazin for Naturvidenskaberne. Kristiania Oslo : 55— Magin, and Irina Trymorush Fractional Calculus and Applied Analysis. Historia Mathematica. Retrieved New Jersey: World Scientific Publishing. Bibcode : fcip. Fractional Calculus. Geophysical Journal International. Bibcode : GeoJ Progress in Fractional Differentiation and Applications. Retrieved 7 August

## Fractional Calculus via Functional Calculus: Theory and Applications

Preliminaries The spaces H x and H x p , The spaces L p and L p p Some special functions Integral transforms Riemann-Liouville Fractional Integrals and Derivatives The Abel integral equation On the solvability of the Abel equation in the space of integrable functions Definition of fractional integrals and derivatives and their simplest properties Fractional integrals and derivatives of complex order Fractional integrals of some elementary functions Fractional integration and differentiation as reciprocal operations Composition formulae. Connection with semigroups of operators Lizorkin's space of test functions. Schwartz's approach The case of the half-axis. The approach via the adjoint operator McBride's spaces The case of an interval Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results relating to Tables of fractional integrals and derivatives Chapter 3 Further Properties of Fractional Integrals and Derivatives Compositions of Fractional Integrals and Derivatives with Weights Compositions of two one-sided integrals with power weights Compositions of two-sided integrals with power weights Compositions of several integrals with power weights Compositions with exponential and power-exponential weights Connection between Fractional Integrals and the Singular Operator The singular operator S The case of the whole line The case of an interval and a half-axis Some other composition relations Fractional Integrals of the Potential Type The real axis. The Riesz and Feller potentials On the "truncation" of the Riesz potential to the half-axis The case of the half-axis The case of a finite interval Functions Representable by Fractional Integrals on an Interval The Marchaud fractional derivative on an interval Characterization of fractional integrals of functions in L p Continuation, restriction and "sewing" of fractional integrals Characterization of fractional integrals of Holderian functions Connections with Fourier series Elementary properties of Weyl fractional integrals Other forms of fractional integration of periodic functions The coincidence of Weyl and Marchaud fractional derivatives The representability of periodic functions by the Weyl fractional integral Weyl fractional integration and differentiation in the space of Holderian functions Weyl fractional integrals and derivatives of periodic functions in H x The Bernstein inequality for fractional integrals of trigonometric polynomials An Approach to Fractional Integro-differentiation via Fractional Differences The Grunwald-Letnikov Approach Differences of a fractional order and their properties Coincidence of the Grunwald-Letnikov derivative with the Marchaud derivative.

Metrics details. In this article, we present some new properties of the fractional proportional derivatives of a function with respect to a certain function. We use a modified Laplace transform to find the relation between the derivatives in the Riemann—Liouville setting and the one in Caputo. In addition, we provide an integration by parts formulas related to the considered operators. For the last 30 years or more, some scientists have shown a great deal of interest in the field of fractional calculus which addresses the derivatives and integrals with any order. As a matter of fact, this interest has sprung out by the dint of the substantial results obtained when these scientists used the tools in this calculus in order to study some models from the real world [ 1 — 6 ].

## FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications

We hope this content on epidemiology, disease modeling, pandemics and vaccines will help in the rapid fight against this global problem. Click on title above or here to access this collection. Due to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. This monograph introduces fundamental information on fractional calculus and provides a detailed treatment of existing numerical approximations.

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Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D.

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