Class Of Derivatives Research Articles

A Novel Approach to Solving Fractional Diffusion Equations Using Fractional Beta Derivative

This paper introduces a novel application of fractional beta derivatives in solving fractional diffusion equations with an emphasis on systems exhibiting anomalous diffusion and memory effects. The work explores the fractional beta derivative as an extension of classical fractional derivatives by incorporating a parameter β (beta) that controls the system’s memory behavior. We investigate both the analytical and numerical solutions of these equations, demonstrating the superior flexibility of fractional beta derivatives in modeling complex diffusion processes. Additionally, we provide a comparison between classical fractional derivatives and the new fractional beta approach to highlight the advantages in terms of accuracy and computational efficiency. We begin by reviewing the theoretical background of fractional derivatives and proceed to introduce the beta derivative as a modification that provides additional flexibility in modeling complex systems. Applications in fields such as control theory, signal processing, and bioengineering are highlighted. Furthermore, numerical methods for solving fractional beta differential equations are discussed, along with potential areas for future research.

Solution of Well-known Physical Problems with Fractional Conformable Derivative

This paper presents a new definition of fractional derivatives and integrals through the conformable derivative approach. This innovative framework offers a closer alignment with classical derivative concepts while providing a more practical and intuitive basis for fractional calculus. The new definition is applicable in two primary ranges: 0 ≤ α < 1 and n − 1 ≤ α < n, where n is a positive integer. It is shown that when α = 1, this definition corresponds precisely to the classical first-order derivative. Key benefits of this approach include its improved consistency with traditional calculus and greater computational ease, making it a useful tool for both theoretical research and practical applications. By integrating fractional calculus with conventional derivative ideas, this definition simplifies the analysis and interpretation of fractional differential equations and their solutions. Additionally, we examine the definition’s effects on stability and convergence in numerical methods and provide examples demonstrating its effectiveness and applicability.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Cobalt Tetracationic 3,4-Pyridinoporphyrazine for Direct CO2 to Methanol Conversion Escaping the CO Intermediate Pathway.

Molecular catalysts offer a unique opportunity to implement different chemical functionalities to steer the efficiency and selectivity for the CO2 reduction for instance. Metalloporphyrins and metallophthalocyanines are under high scrutiny since their most classic derivatives the tetraphenylporphyrin (TPP) and parent phthalocyanine (Pc), have been used as the molecular platform to install, hydrogen bonds donors, proton relays, cationic fragments, incorporation in MOFs and COFs, to enhance the catalytic power of these catalysts. Herein, we examine the electrocatalytic properties of the tetramethyl cobalt (II) tetrapyridinoporphyrazine (CoTmTPyPz) for the reduction of CO2 in heterogeneous medium when adsorbed on carbon nanotubes (CNT) at a carbon paper (CP) electrode. Unlike reported electrocatalysis with cobalt based phthalocyanine where CO was advocated as the two electron and two protons reduced intermediate on the way to the formation of methanol, we found here that CoTmTPyPz does not reduce CO to methanol. Henceforth, ruling out a mechanistic pathway where CO is a reaction intermediate.

Cobalt Tetracationic 3,4‐Pyridinoporphyrazine for Direct CO2 to Methanol Conversion Escaping the CO Intermediate Pathway

AbstractMolecular catalysts offer a unique opportunity to implement different chemical functionalities to steer the efficiency and selectivity for the CO2 reduction for instance. Metalloporphyrins and metallophthalocyanines are under high scrutiny since their most classic derivatives the tetraphenylporphyrin (TPP) and parent phthalocyanine (Pc), have been used as the molecular platform to install, hydrogen bonds donors, proton relays, cationic fragments, incorporation in MOFs and COFs, to enhance the catalytic power of these catalysts. Herein, we examine the electrocatalytic properties of the tetramethyl cobalt (II) tetrapyridinoporphyrazine (CoTmTPyPz) for the reduction of CO2 in heterogeneous medium when adsorbed on carbon nanotubes (CNT) at a carbon paper (CP) electrode. Unlike reported electrocatalysis with cobalt based phthalocyanine where CO was advocated as the two electron and two protons reduced intermediate on the way to the formation of methanol, we found here that CoTmTPyPz does not reduce CO to methanol. Henceforth, ruling out a mechanistic pathway where CO is a reaction intermediate.

Characterization of solutions in Besov spaces for fractional Rayleigh–Stokes equations

This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some Lp−Lq estimates in the framework of Lp spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.

On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.

Dynamic Modeling and Control Strategy Optimization of a Volkswagen Crafter Hybrid Electrified Powertrain

This article studies the transformation and assembly process of the Volkswagen (VW) Crafter from conventional to hybrid vehicle of the department of vehicles engineering, University of Debrecen, and uses a computer-aided simulation (CAS) to design the vehicle based on the real measurement data (hardware-in-the-loop, HIL method) obtained from an online CAN bus data measurement platform using MATLAB/Simulink/Simscape and LabVIEW software. The conventional vehicle powered by a 6-speed manual transmission and a 4-stroke, 2.0 Turbocharged Direct Injection Common Rail (TDI CR) Diesel engine and the transformed hybrid electrified powertrain are designed to compare performance. A novel methodology is introduced using Netcan plus 110 devices for the CAN bus analysis of the vehicle’s hybrid version. The acquired raw CAN data is analyzed using LabVIEW and decoded with the help of the database (DBC) file into physical values. A classical proportional integral derivative (PID) controller is utilized in the hybrid powertrain system to manage the vehicle consumption and CO2 emissions. However, the intricate nonlinearities and other external environments could make its performance unsatisfactory. This study develops the energy management strategies (EMSs) on the basis of enhanced proportional integral derivative-based genetic algorithm (GA-PID), and compares with proportional integral-based particle swarm optimization (PSO-PI) and fractional order proportional integral derivative (FOPID) controllers, regulating the vehicle speed, allocating optimal torque and speed to the motor and engine and reducing the fuel and energy consumption and the CO2 emissions. The integral time absolute error (ITAE) is proposed as a fitness function for the optimization. The GA-PID demonstrates superior performance, achieving energy efficiency of 90%, extending the battery pack range from 128.75 km to 185.3281 km and reducing the emissions to 74.79 gCO2/km. It outperforms the PSO-PI and FOPID strategies by consuming less battery and motor energy and achieving higher system efficiency.

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through modified ABC operator

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Cα-CURVES AND THEIR Cα-FRAME IN CONFORMABLE DIFFERENTIAL GEOMETRY

The aim of this study is to redesign the space curve and its Frenet framework, which are extremely important in terms of differential geometry, by using conformable derivative arguments. In this context, conformable counterparts of basic geometric concepts such as angle, vector, line, plane and sphere have been obtained. The advantages of the conformable derivative over the classical (Newton) derivative are mentioned. Finally, new concepts produced by conformable derivative are supported with the help of examples and figures.

Active vibration control effect of 3D printed cruciform honeycomb laminates based on fiber-reinforced shape memory polymer

Honeycomb structures offer the benefits of being lightweight and having high out-of-face compressive stiffness, making them a popular choice for aerostructures. The present study delineates the design and active vibration control effect for fiber-reinforced cruciform honeycomb laminates (CHLs) characterized by variable parameters. Employing a continuous fiber 3D printer, carbon fiber filament and shape memory polymers (SMPs) were utilized in the fabrication of the specimens. Differential equations of motion for CHLs were deduced from Kirchhoff plate theory. To ascertain their frequency response, these laminates were subjected to a sinusoidal swept frequency excitation. Subsequently, the classical proportional integral derivative (PID) program was employed to scrutinize the efficacy of closed-loop active vibration control on CHLs. The investigation extended to analyzing the impact of active vibration control across laminates with disparate parameters, including an examination of the convergence speed during vibration control procedures. A comparative analysis of simulated versus experimental data revealed a substantial consonance between the two, thereby corroborating the accuracy of the experimental findings. This study offers crucial insights and reference value for the utilization of CHLs in the aerospace industry.

Parametrized predictor–corrector method for initial value problems with classical and Caputo–Fabrizio derivatives

Ordinary nonlinear differential equations with classical and fractional derivatives are used to simulate several real-world problems. Nonetheless, numerical approaches are used to acquire their solutions. While various have been proposed, they are susceptible to both disadvantages and advantages. In this paper, we propose a more accurate numerical system for solving nonlinear differential equations with classical and Caputo–Fabrizio derivatives by combining two concepts: the parametrized method and the predictor–corrector method. We gave theoretical analyses to demonstrate the method’s correctness, as well as several illustrated examples for both scenarios.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

Exact Solutions for the Sharma–Tasso–Olver Equation via the Sardar Subequation Method with a Comparison between Atangana Space–Time Beta-Derivatives and Classical Derivatives

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type.

Exploring the dynamics of nonlocal coupled systems of fractional [formula omitted]-integro-differential equations with infinite delay

In this study, we explore a coupled system of fractional integro-differential equations with infinite delay and nonlocal conditions. This system encompasses classical derivatives of different orders and the fractional derivative of Caputo-Fabrizio type, as well as the fractional integral of the q-Riemann-Liouville operator. We introduce a novel definition of the Caputo and Fabrizio differential operators, enhancing the mathematical formulation. Our main focus is to investigate the system's fundamental properties, including existence, uniqueness, and continuous dependence. Through rigorous mathematical analysis, we establish the existence and uniqueness of solutions and examine how small perturbations in initial conditions or parameters impact the solutions. For the numerical aspect, we use the finite-trapezoidal approach, a reliable method for solving fractional integro-differential equations. We provide a concise explanation of the approach and demonstrate its effectiveness through two numerical examples. Overall, this comprehensive study contributes to the understanding of coupled systems with fractional derivatives and infinite delays, with implications for various scientific and engineering fields.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

An approach to metric space-valued Sobolev maps via weak* derivatives

Abstract We give a characterization of metric space-valued Sobolev maps in terms of weak* derivatives. More precisely, we show that Sobolev maps with values in dual-to-separable Banach spaces can be defined in terms of classical weak derivatives in a weak* sense. Since every separable metric space X X embeds isometrically into ℓ ∞ {\ell }^{\infty } , we conclude that Sobolev maps with values in X X can be characterized by postcomposition with such embedding and the mentioned weak gradients. A slight variation on our definition was proposed previously by Hajłasz and Tyson. However, we show that their definition does not work in the sense that for technical reasons the arising Sobolev space is essentially empty.

An operational matrix strategy for time fractional Fokker-Planck equation in an unbounded space domain

In this study, the Caputo fractional derivative is used to define time fractional Fokker-Planck equation in an unbounded domain. To solve this equation, the Jacobi polynomials together with the tanh-Jacobi functions are employed. The operational matrices of the classical and fractional derivatives of these basis functions are obtained to use them in constructing a numerical method for the expressed equation. In the proposed method, the introduced basis functions are used simultaneously to approximate the equation’s unknown solution. More precisely, the shifted Jacobi polynomials are applied to approximate the solution in the temporal direction and the tanh-Jacobi functions are utilized to approximate the solution in the spatial direction. By substitute the expressed approximation into the equation and employing the introduced operational matrix, solving the problem under consideration transforms into solving an algebraic system of equations, which can be solved easily. The accuracy and efficiency of the presented method are investigated numerically by solving some numerical examples. The reported results confirms the high accuracy of the established method.