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Fractional Calculus is a study of an extension of derivatives and integrals to non-integer orders and also linking its origins with classical integral and di?erential calculus. The interesting part of this subject is that fractional derivatives and integrals are not a local or point property or quantity. In some few years considerable interest in fractional calculus has been seen by the applications it finds in various areas of engineering, science, applied mathematics, finance and bio-engineering as possibly it includes fractal phenomena too. This paper deals with the researchers of engineering and science who are learning about Fractional Calculus and its possible applications in their ?elds of study. The paper focuses on the review of new growth based on the fractional calculus in different fields both on theoretical and application facets. Savita Sharma"New Advancements in the Applications of Fractional Calculus in Science and Engineering" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-6 , October 2017, URL: http://www.ijtsrd.com/papers/ijtsrd3579.pdf Article URL: http://www.ijtsrd.com/mathemetics/applied-mathematics/3579/new-advancements-in-the-applications-of-fractional-calculus-in-science-and-engineering/savita-sharma

In this paper we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler-Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois-Reymond lemma and showing how Euler-Lagrange equations involving only Caputo derivatives can be obtained.

This is the first annual report to the U.S. Army Medical Research and Material Command for the three year project ''Advanced Soft Tissue Modeling for Telemedicine and Surgical Simulation'' supported by grant No. DAMD17-01-1-0673 to The Cleveland Clinic Foundation, to which the NASA Glenn Research Center is a subcontractor through Space Act Agreement SAA 3-445. The objective of this report is to extend popular one-dimensional (1D) fractional-order viscoelastic (FOV) materials models into their three-dimensional (3D) equivalents for finitely deforming continua, and to provide numerical algorithms for their solution.

Fractional Calculus is a study of an extension of derivatives and integrals to non-integer orders and also linking its origins with classical integral and di?erential calculus. The interesting part of this subject is that fractional derivatives and integrals are not a local or point property or quantity. In some few years considerable interest in fractional calculus has been seen by the applications it finds in various areas of engineering, science, applied mathematics, finance and bio-engineering as possibly it includes fractal phenomena too. This paper deals with the researchers of engineering and science who are learning about Fractional Calculus and its possible applications in their ?elds of study. The paper focuses on the review of new growth based on the fractional calculus in different fields both on theoretical and application facets. Savita Sharma"New Advancements in the Applications of Fractional Calculus in Science and Engineering" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-6 , October 2017, URL: http://www.ijtsrd.com/papers/ijtsrd3579.pdf Article URL: http://www.ijtsrd.com/mathemetics/applied-mathematics/3579/new-advancements-in-the-applications-of-fractional-calculus-in-science-and-engineering/savita-sharma

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha

In this paper, we resort to the Laplace transform method in order to show its efficiency when approaching some types of fractional differential equations. In particular, we present some applications of such methods when applied to possible generalizations of certain physical problems in linear viscoelasticity and harmonic oscillators, proving that fractional calculus is well suited for the modelling and solving of problems usually treated by ordinary integer calculus, with the promissing advantages of being able to provide more accurate theoretical predictions to fit with experimental data. OBS: Article in portuguese accepted for publication at RBEF (Revista Brasileira de Ensino de F\'isica).

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on time scales by proving two chain rules for the $\alpha$-fractional derivative and five inequalities for the $\alpha$-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order $\alpha = 1$ and the time scale coincides with the set of real numbers.

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on time scales by proving two chain rules for the $\alpha$-fractional derivative and five inequalities for the $\alpha$-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order $\alpha = 1$ and the time scale coincides with the set of real numbers.

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on time scales by proving two chain rules for the $\alpha$-fractional derivative and five inequalities for the $\alpha$-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order $\alpha = 1$ and the time scale coincides with the set of real numbers.

The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, e(t), and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, el, in terms of the R-function are developed. Also, some approximations for the R-function are developed.

A two-state master equation based decision making model has been shown to generate phase transitions, to be topologically complex and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $\nu \in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $\nu=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.

Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to analyze a wide class of physical systems in various fields of science and engineering. In this paper, we describe an ongoing project which aims at formalizing the basic theories of fractional calculus in the HOL Light theorem prover. Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones.

We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the calculus of variations with general fractional operators. Necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann-Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered.

We consider a partial differential equation associated with a mathematical model describing the concentration of nutrients in blood which interferes directly on the erythrocyte sedimentation rate in the case of an average fluid velocity equal to zero. Introducing the fractional derivative in the Caputo sense, we propose a time-fractional mathematical model which contains, as a particular case, the model proposed by Sharma et al. Our main purpose is to obtain an analytic solution of this time-fractional partial differential equation in terms of the Mittag-Leffler function and Wright function.

We consider a partial differential equation associated with a mathematical model describing the concentration of nutrients in blood which interferes directly on the erythrocyte sedimentation rate in the case of an average fluid velocity equal to zero. Introducing the fractional derivative in the Caputo sense, we propose a time-fractional mathematical model which contains, as a particular case, the model proposed by Sharma et al. Our main purpose is to obtain an analytic solution of this time-fractional partial differential equation in terms of the Mittag-Leffler function and Wright function.

The effect of Marichev-Saigo-Maeda (MSM) fractional operators involving third Appell function on the $I$ function is studied. It is shown that the order of the $I$-function increases on application of these operators to the power multiple of the $I$-function. The Caputo-type MSM fractional derivatives are introduced and studied for the $I$-function. As special cases, the corresponding assertions for Saigo and Erd\'elyi-Kober fractional operators are also presented. The results obtained in this paper generalize several known results obtained recently in the literature.

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of $\alpha$--stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of $\alpha$--stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.

We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms from fractional calculus. This affords us the ability to indefinitely sum more complicated functions than previously possible; such as holomorphic functions of order $n \in \mathbb{N}$ that have decay at plus or minus imaginary infinity. We then further investigate the indefinite summation operator by restricting ourselves to a space of functions of exponential type. We arrive at a second representation for the indefinite summation operator, equivalent to the first presented, and show we have defined a unique operator on this space. We develop a convolution using the indefinite sum that is commutative, associative, and distributive over addition. We arrive at a formula for the complex iterates of the indefinite sum (the differsum), using this convolution, that resembles the Riemann-Liouville differintegral. We close with a generalization of the Gamma function.

Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha

An analogy is drawn between the diffusion-wave equations derived from the fractional Kelvin-Voigt model and those obtained from Buckingham's grain-shearing (GS) model [J. Acoust. Soc. Am. 108, 2796-2815 (2000)] of wave propagation in saturated, unconsolidated granular materials. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and shear wave equation derived from the GS model turn out to be the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation respectively. Also, a physical interpretation of the characteristic fractional-order present in the Kelvin-Voigt fractional derivative wave equation and time-fractional diffusion-wave equation is inferred from the GS model. The shear wave equation from the GS model predicts both diffusion and wave propagation in the fractional framework. The overall goal is intended to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials, but rather it can be justified from real physical process of grain-shearing as well.

We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered.

Fractional Calculus is a study of an extension of derivatives and integrals to non-integer orders and also linking its origins with classical integral and di?erential calculus. The interesting part of this subject is that fractional derivatives and integrals are not a local or point property or quantity. In some few years considerable interest in fractional calculus has been seen by the applications it finds in various areas of engineering, science, applied mathematics, finance and bio-engineering as possibly it includes fractal phenomena too. This paper deals with the researchers of engineering and science who are learning about Fractional Calculus and its possible applications in their ?elds of study. The paper focuses on the review of new growth based on the fractional calculus in different fields both on theoretical and application facets. Savita Sharma"New Advancements in the Applications of Fractional Calculus in Science and Engineering" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-6 , October 2017, URL: http://www.ijtsrd.com/papers/ijtsrd3579.pdf     http://www.ijtsrd.com/mathemetics/applied-mathematics/3579/new-advancements-in-the-applications-of-fractional-calculus-in-science-and-engineering/savita-sharma

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental and isoperimetric problems, natural boundary conditions, and Noether type theorems.

We complement the theory of tick-by-tick dynamics of financial markets based on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et al., and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London.

We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of H\"{o}lder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $\alpha$. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is $\alpha>d-2$, which coincides with the condition obtained in Dalang (1999), when the noise is white in time. Under this condition, we obtain estimates for the $p$-th moments of the solution, we deduce its H\"older continuity, and we show that the solution is Malliavin differentiable of any order. When $d \leq 2$, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.

This paper begins by giving an historical context to fractional Brownian Motion and its development. Section 2 then introduces the fractional calculus, from the Riemann-Liouville perspective. In Section 3, we introduce Brownian motion and its properties, which is the framework for deriving the It\^o integral. In Section 4 we finally introduce the It\^o calculus and discuss the derivation of the It\^o integral. Section 4.1 continues the discussion about the It\^o calculus by introducing the It\^o formula, which is the analogue to the chain rule in classical calculus. In Section 5 we present our formal definition of fBm and derive some of its properties that give motivation for the development of a stochastic calculus with respect to fBm. Finally, in Section 6 we define and characterize a stochastic integral with respect to fBm from a pathwise perspective.

The recent theory of fractional $h$-difference equations introduced in [N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres: Discrete-time fractional variational problems, Signal Process. 91 (2011), no. 3, 513--524], is enriched with useful tools for the explicit solution of discrete equations involving left and right fractional difference operators. New results for the right fractional $h$ sum are proved. Illustrative examples show the effectiveness of the obtained results in solving fractional discrete Euler-Lagrange equations.

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

We developed an analytical approach, for a wave propagation in metal-dielectric nanostructures in the quasi-static limit. This consideration establishes a link between fractional geometry of the nanostructure and fractional integro-differentiation. The method is based on fractional calculus and permits to obtain analytical expressions for the electric field enhancement.

It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.

رريل

This is the first annual report to the U.S. Army Medical Research and Material Command for the three year project "Advanced Soft Tissue Modeling for Telemedicine and Surgical Simulation" supported by grant No. DAMD17-01-1-0673 to The Cleveland Clinic Foundation, to which the NASA Glenn Research Center is a subcontractor through Space Act Agreement SAA 3-445. The objective of this report is to extend popular one-dimensional (1D) fractional-order viscoelastic (FOV) materials models into their three-dimensional (3D) equivalents for finitely deforming continua, and to provide numerical algorithms for their solution.

The purpose of this study was to demonstrate the use of fractional derivatives to capture the frequency and temperature dependency of viscoelastic material behavior. To model the frequency dependency of viscoelastic material, fractional derivatives were included in the complex modulus. Solution techniques were performed in the Laplace domain to allow for easy manipulation of the fractional derivative terms. To incorporate the temperature dependency of viscoelastic material in the complex modulus model, the method of reduced variables was employed with the use of the WLF equation. With the frequency and temperature dependency built into complex modulus, a finite element formulation was devised that incorporated elastic and viscoelastic response of a truss structure. The formulation was limited to the use of the complex modulus in the transition region, the region where the damping ability of viscoelastic material is maximized. Quasi-static motion was also assumed, which limited the response to low frequencies. Theses.

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.