In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional ...calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.
Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under ...stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.
We consider an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. We prove existence and uniqueness of global solutions in the ...space of weighted continuous functions. The stability of the solution for a weighted Cauchy-type problem is also analyzed.
Small-molecule natural products (NPs) have a long and successful track record of providing first-in-class drugs and pharmacophore (scaffolds) in all therapeutic areas, serving as a bridge between ...modern and traditional medicine. This trajectory has been remarkably successful in three key areas of modern therapeutics: cancers, infections, and CNS diseases. Beginning with the discovery of morphine 200 years ago, natural products have remained the primary source of new drugs/scaffolds for CNS diseases. In this perspective, we address the question: why are the majority of active compounds in the CNS domain natural products? Our analysis indicates that ∼84% approved drugs for CNS diseases are NPs or NP-inspired, and interestingly, 20 natural products provided more than 400 clinically approved CNS drugs. We have discussed unique physicochemical properties of NPs and NP-inspired vis-à-vis synthetic drugs, isoform selectivity, and evolutionary relationship, providing a rationale for increasing focus on natural product driven discovery for next-generation drugs for neurodegenerative diseases.
This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional ...calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention.
We use a generalization of the Hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval-valued functions. The properties of ...Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative and Caputo fractional derivative for interval-valued functions are investigated. Several examples are presented to illustrate the concepts and results.
Cellulose ethers have proven to be highly useful natural-based polymers, finding application in areas including food, personal care products, oil field chemicals, construction, paper, adhesives, and ...textiles. They have particular value in pharmaceutical applications due to characteristics including high glass transition temperatures, high chemical and photochemical stability, solubility, limited crystallinity, hydrogen bonding capability, and low toxicity. With regard to toxicity, cellulose ethers have essentially no ability to permeate through gastrointestinal enterocytes and many are already in formulations approved by the U.S. Food and Drug Administration. We review pharmaceutical applications of these valuable polymers from a structure–property–function perspective, discussing each important commercial cellulose ether class; carboxymethyl cellulose, methyl cellulose, hydroxypropylcellulose, hydroxypropyl methyl cellulose, and ethyl cellulose, and cellulose ether esters including hydroxypropyl methyl cellulose acetate succinate and carboxymethyl cellulose acetate butyrate. We also summarize their syntheses, basic material properties, and key pharmaceutical applications.
The paper presents a new fractional integration, which generalizes the Riemann–Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration ...operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.
Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative ...(GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term
t
1
-
α
in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.
PDF
