Class Of Derivatives Research Articles

The mathematical description of the bulk fluid flow and that of the content impurity dispersion, obtained by replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective

In the field of fractional calculus applications, there is a tendency to admit that “integer-order derivatives cannot simply be replaced by fractional-order derivatives to develop fractional-order theories”. There are different arguments for that: initialization problem, inconsistency, use of nonsingular or singular kernels, loss of objectivity. In this paper it is shown that the mathematical description of the bulk fluid flow and that of the content impurity spread replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. More precisely, it is proven that, the mathematical description of the bulk fluid 2D flow and that of the content impurity spread, in a horizontal unconfined aquifer, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. It is also proven that, the mathematical description of a Newtonian, incompressible, viscous bulk fluid 3D flow and that of the contained impurity dispersion, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. The obtained results show the compatibility of the general temporal Caputo and general temporal Riemann-Liouville fractional order derivatives with the understanding of the “measured time” evolution. In the same time these results reveal that, the objectivity violation, when integer order temporal derivatives are replaced by classic temporal Caputo or classic temporal Riemann-Liouville fractional order derivatives, is originated in the incompatibility of the classic fractional order derivatives, with the understanding of the “measured time” evolution.

Introduction to weather derivatives

AbstractThe weather is one of the factors that may have an impact on the countries' economies. There are two main hedging ways against unexpected weather conditions: weather derivatives and weather insurances. During the last two decades, companies started to use weather derivatives against weather issues, especially in the energy and agriculture sectors. Starting from weather derivatives' first launch, their transaction volumes at the exchange and over‐the‐counter markets have increased. In addition to the increasing dependency of the economies on the weather, providing the weather derivative contracts with a reasonable premium amount is another reason which helps to have this positive trend. Since weather derivatives have similar parameters and rules with classical financial derivatives, it is possible to use the same pricing approaches for financial and weather derivatives. Monte–Carlo simulation, based on random number generation, is one of the existing methods of pricing derivative contracts. A difference between simulated values and really occurred data is the base point of the expected payoff or price of the contract. The current article introduces weather derivatives and shows two different approaches to their pricing, where one of them requires deeper statistical analysis. Adding the statistical analysis into the consideration, defining the relation between each data value, helps to provide better estimation and less volatility. Having less volatility can provide more accurate estimations and reasonable prices that are affordable and desired by the companies.This article is categorized under: Energy Systems Economics > Economics and Policy Energy Systems Economics > Systems and Infrastructure

Dynamics of Different Nonlinearities to the Perturbed Nonlinear Schrödinger Equation via Solitary Wave Solutions with Numerical Simulation

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned nonlinearities are known as the Kerr law, power, dual power law, triple power law, quadratic–cubic law and anti-cubic law. The dark, bright, singular and combinations of these solutions are retrieved using an efficient, simple integration scheme. These solutions suggest that this method is more simple, straightforward and reliable compared to existing methods in the literature. The novelty of this paper is that the perturbed nonlinear Schrödinger equation is investigated in different nonlinear media using a novel derivative operator. Furthermore, the numerical simulation for certain solutions is also presented.

Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

In this paper, we study the existence of solutions to initial value problems for a nonlinear generalized Caputo fractional differential inclusion with Lipschitz set-valued functions. The applied fractional operator is given by the kernel k ρ , s = ξ ρ − ξ s and the derivative operator 1 / ξ ′ ρ d / d ρ . The existence result is obtained via fixed point theorems due to Covitz and Nadler. Moreover, we also characterize the topological properties of the set of solutions for such inclusions. The obtained results generalize previous works in the literature, where the classical Caputo fractional derivative is considered. In the end, an example demonstrating the effectiveness of the theoretical results is presented.

Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

Dynamics of new optical solitons for the Triki–Biswas model using beta-time derivative

This paper comprises the different types of optical soliton solutions of an important Triki–Biswas model equation with beta-time derivative. The beta derivative is considered as a generalized version of the classical derivative. The aforesaid model equation is the generalization of the derivative nonlinear Schrödinger equation that describes the ultrashort pulse propagation with non-Kerr dispersion. The study is carried out by means of a novel beta derivative operator and three efficient integration schemes. During this work, a sequence of new optical solitons is produced that may have an importance in optical fiber systems. These solutions are verified and numerically simulated through soft computation.

Semi-supervised extensions of multi-task tree ensembles

Scale inconsistency is a widely encountered issue in multi-output learning problems. Specifically, target sets with multiple real valued or a mixture of categorical and real valued variables require addressing the scale differences to obtain predictive models with sufficiently good performance. Data transformation techniques are often employed to solve that problem. However, these operations are susceptible to different shortcomings such as changing the statistical properties of the data and increase the computational burden. Scale differences also pose problem in semi-supervised learning (SSL) models as they require processing of unsupervised information where distance measures are commonly employed. Classical distance metrics can be criticized as they lose efficiency when variables exhibit type or scale differences, too. Besides, in higher dimensions distance metrics cause problems due to loss of discriminative power. This paper introduces alternative semi-supervised tree-based strategies that are robust to scale differences both in terms of feature and target variables. We propose use of a scale-invariant proximity measure by means of tree-based ensembles to preserve the original characteristics of the data. We update classical tree derivation procedure to a multi-criteria form to resolve scale inconsistencies. We define proximity based clustering indicators and extend the supervised model with unsupervised criteria. Our experiments show that proposed method significantly outperforms its benchmark learning model that is predictive clustering trees.

A study on fractional HIV‐AIDs transmission model with awareness effect

The main aim of this work is to numerically investigate a mathematical model of the HIV/AIDS disease with awareness effect, as the fractional derivatives capture more the genetic and memory properties of the biological models in comparison to the classical derivative. Therefore, in this article, we utilize the generalized Caputo fractional derivative to extend the integer‐order model of HIV/AIDS transmission to fractional order. A novel fixed‐point approach is used to identify the existence and uniqueness of the FO model. We use a new numerical technique based on the Adams‐Bashforth predictor‐corrector method to solve the fractional‐order HIV/AIDS transmission model. We illustrated the obtained numerical results through graphs.

Radiative contributions to gravitational scattering

The linear-order effects of radiation-reaction on the classical scattering of two point masses, in General Relativity, are derived by a variation-of-constants method. Explicit expressions for the radiation-reaction contributions to the changes of 4-momentum during scattering are given to linear order in the radiative losses of energy, linear-momentum and angular momentum. The polynomial dependence on the masses of the 4-momentum changes is shown to lead to non-trivial identities relating the various radiative losses. At order $G^3$ our results lead to a streamlined classical derivation of results recently derived within a quantum approach. At order $G^4$ we compute the needed radiative losses to next-to-next-to-leading-order in the post-Newtonian expansion, thereby reaching the absolute fourth and a half post-Newtonian level of accuracy in the 4-momentum changes. We also provide explicit expressions, at the absolute sixth post-Newtonian accuracy, for the radiation-graviton contribution to {\it conservative} $O(G^4)$ scattering. At orders $G^5$ and $G^6$ we derive explicit theoretical expressions for the last two hitherto undetermined parameters describing the fifth-post-Newtonian dynamics. Our results at the fifth-post-Newtonian level confirm results of [Nucl. Phys. B \textbf{965}, 115352 (2021)] but exhibit some disagreements with results of [Phys. Rev. D \textbf{101}, 064033 (2020)].

A numerical approach for a class of nonlinear optimal control problems with piecewise fractional derivative

In this study, a kind of piecewise fractional derivatives based on the Caputo fractional derivative is used to define a novel category of fractional optimal control problems. The piecewise Chebyshev cardinal functions as an appropriate family of basis functions are considered to construct a numerical method for solving such problems. The classical and piecewise fractional derivative matrices of these basis functions are derived and used in constructing the proposed technique. The established scheme transforms obtaining the solution of such problems into finding the solution of algebraic systems of equations by approximating the state and control variables using the mentioned basis functions. The accuracy of the expressed approach is investigated by solving some examples.

Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

A nonlocal generalization of the Exner law

Here we consider a new, more general mathematical formulation of the nonlocal Exner law, widely used in morphodynamics and sediment transport models. In some recent literature the role of nonlocal effects in the sediment flux was analyzed, by using a fractional calculus approach, based on the application of Caputo fractional derivatives. In this paper we apply fractional derivatives of a function with respect to another function in order to have a more general and realistic picture of nonlocal effects in sediment transport models. We then discuss the meaning of this approach, considering a logarithmic profile emerging from the generalized formulation of the nonlocal steady Exner law. We finally suggest that this can be the way to overcome the theoretical shortcomings of the fractional Exner equation based on the application of classical Caputo derivatives.

A new method for obtaining model-free viscoelastic material properties from atomic force microscopy experiments using discrete integral transform techniques.

Viscoelastic characterization of materials at the micro- and the nanoscale is commonly performed with the aid of force–distance relationships acquired using atomic force microscopy (AFM). The general strategy for existing methods is to fit the observed material behavior to specific viscoelastic models, such as generalized viscoelastic models or power-law rheology models, among others. Here we propose a new method to invert and obtain the viscoelastic properties of a material through the use of the Z-transform, without using a model. We present the rheological viscoelastic relations in their classical derivation and their z-domain correspondence. We illustrate the proposed technique on a model experiment involving a traditional ramp-shaped force–distance AFM curve, demonstrating good agreement between the viscoelastic characteristics extracted from the simulated experiment and the theoretical expectations. We also provide a path for calculating standard viscoelastic responses from the extracted material characteristics. The new technique based on the Z-transform is complementary to previous model-based viscoelastic analyses and can be advantageous with respect to Fourier techniques due to its generality. Additionally, it can handle the unbounded inputs traditionally used to acquire force–distance relationships in AFM, such as ramp functions, in which the cantilever position is displaced linearly with time for a finite period of time.

Mathematical Calculation of the COVID-19 Disease in Pakistan by Emergency Response Modeling Based on Intuitionistic Fuzzy Decision Process

The objective of this paper is to investigate three new methods to regulate and analyze the spreading of coronavirus disease (COVID-19). COVID-19 started in the city of Wuhan, China at the end of 2019 and spread to the whole world in a short time. This infection caused millions of infected cases globally and still poses a disturbing situation for the people. Recently, some mathematical models have been constructed for better understanding of the coronavirus infection. Mostly, these models are based on classical integer-order derivative using real numbers which cannot capture the fading memory. So, at the current position, it is a challenge for the world to control the spreading of COVID-19. Therefore, the aim of this paper is to utilize fuzzy logic to control the transmission and spreading of COVID-19. Here, we develop three new methods such as the generalized interval-valued intuitionistic fuzzy Einstein weighted geometric (GIVIFEWG) operator, the generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (GIVIFEOWG) operator, and the generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric (GIVIFEHG) operator. Intuitionistic fuzzy information for interval-valued is the moral and decent method to precise the fuzzy information for judgment or decision and Einstein operations are the best approximations, and the generalized aggregation operators are a generalization of most aggregation operators so, in these notes, we can spread the Einstein operations to aggregate the interval-valued intuitionistic fuzzy information based on the generalized aggregation operators. At the end of the paper, an illustration of the emergency situation of COVID-19 is given for demonstrating the effectiveness of the suggested approach, showing the feasibility and reliability of the new methods.

Stochastic covid-19 model with fractional global and classical piecewise derivative

Several methodologies have been advocated in the last decades with the aim to better understand behaviours displayed by some real-world problems. Among which, stochastics modelling and fractional modelling, fuzzy and others. These methodologies have been suggested to threat specific problems; however, It have been noticed that some problems exhibit different patterns as time passes by. Randomness and nonlocality can be combined to depict complex real-world behaviours. It has been observed that, covid-19 virus spread does not follow a single pattern; sometimes we obtained stochastic behaviours, another nonlocal behaviour and others. In this paper, we shall consider a covid-19 model with fractional stochastic behaviours. More precisely a covid-19 model, where the model considers nonlocalities and randomness is suggested. Then a comprehensive analysis of the model is conducted. Numerical simulations and illustrations are done to show the efficiency of the model.

A Multiple Excited-State Engineering of Boron-Functionalized Diazapentacene Via a Tuning of the Molecular Orbital Coupling.

Harvesting high-energy excited-state energy is still challenging in organic chromophores. An introduction of boron atoms along the short axis of the diazapentacene backbone induces multiple emission characteristics. Our studies reveal that the weak molecular orbital (MO) coupling of the S3-S1 transition is responsible for the slow internal conversion rates. Such MO coupling-regulated anti-Kasha emission is different from the large band gap-induced anti-Kasha emission character of classical azulene derivatives. Theoretical studies reveal that a strong MO coupling of the S3-S0 transition is responsible for the higher photoluminescence quantum yield of the anti-Kasha emission in a more polar solution (tetrahydrofuran: 11%; cyclohexane: 0%). Such an MO coupling factor is generally overlooked in anti-Kasha emitters reported previously. Furthermore, the multiple emission can be regulated by solvent polarity, solvent temperature, and fluoride anion binding. As a proof of concept of harvesting high-energy emission, the multiple emission character has allowed us to design single-molecule white-light-emitting materials.

Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

On inverse problem for linear and semilinear diffusion equation with Caputo–Fabrizio derivative

We investigate a fractional time diffusion equation with Caputo–Fabrizio derivative instead of classical derivative. We set up the existence, uniqueness, and regularity of the mild solution and then consider the above problem in some various cases including linear and nonlinear source function. Thanks to applying some Sobolev embeddings, we derive the regularity of the mild solution for linear case. For nonlinear case, we apply the fixed point theory to obtain the existence of the mild solution.

CHAOTIC BEHAVIOR OF BHALEKAR–GEJJI DYNAMICAL SYSTEM UNDER ATANGANA–BALEANU FRACTAL FRACTIONAL OPERATOR

In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the new fractal–fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.

Pre-shape calculus and its application to mesh quality optimization

Abstract Deformations of the computational mesh, arising from optimization routines, usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize the classical shape optimization and calculus. We define pre-shape derivatives and derive corresponding structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives, featuring only normal directions. Techniques from classical shape optimization and calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable with the use of pre-shape derivatives. This class allows for simultaneous optimization of the classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.