Fractional Operator Research Articles

Efficient pure qP-wave modeling and reverse time migration in tilted transversely isotropic media calculated by a finite-difference approach

Subsurface anisotropy is commonly induced by shale layers, aligned cracks, and fine bedding and has a significant impact on seismic wave propagation. Ignoring anisotropic effects during seismic migration degrades image quality. Therefore, we derive a pure qP-wave equation with high accuracy for modeling seismic wave propagation in tilted transversely isotropic (TTI) media. However, the derived pure qP-wave equation requires a computationally expensive spectral-based method for performing numerical simulations. This is unsuitable for large-scale industrial applications, particularly 3D applications. For numerical efficiency, we first decompose the newly derived wave equation into some fractional differential operators and spatial derivatives. The spatial derivatives can be directly solved by conventional finite-difference (FD) approaches. Then, we use an asymptotic approximation to find an equivalent form of fractional differential operators, obtaining scalar operators that we can discretize with the FD method. Numerical tests show that our TTI pure qP-wave equation with an FD discretization can produce accurate and highly efficient wavefield simulations in TTI media. We also use our TTI pure qP-wave equation with an FD discretization as a forward engine to implement TTI reverse time migration (RTM). Synthetic examples and a field data test demonstrate that our TTI RTM can effectively correct the anisotropic effects, providing high-quality imaging results while maintaining good computational efficiency.

Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (H-H)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of I.V-(⋏,ℏ) generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and I.V-(⋏,ℏ) convexity to derive new improvements of the H-H- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for I.V R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results.

Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ−fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators.

A new approach to the directional derivative of fractional order

The fractional derivative approximations offer many approaches to understanding real‐world problems. The conformable fractional derivative operator that is one of the fractional derivative operators has recently attracted a lot of interesting. In this paper, a new approach to the directional derivative obtained with the help of the conformal fractional derivative is presented. Considering this approach, the definition of the fractional partial derivative is reformulated. In addition, a new definition of fractional gradient, fractional curl, and fractional divergence is given, and the properties of these new concepts are examined.

(L p,λ,μ,L q,λ,μ) boundedness of the G-fractional integral operator on G-Morrey spaces

Abstract In this paper we find conditions for the boundedness of the strong and weak fractional integral and fractional maximal operator generated by Gegenbauer differential operator on G-Morrey spaces.

Sinusoidal heating on convective heat transfer of nanofluid under differential technique

AbstractSinusoidal convection describes a smooth and repetitive heat transfer oscillation that reasonably calculates the optimal heat transfer for nanoparticle's intermolecular interactions. This study aims to investigate the heat transfer effects of Stokes second problem on the performance of novel fractional differential operators based on rheological characteristics of nanofluid. For the sake of augmentation in thermal characteristics of the nanofluid, the fractionalized governing equations based on the suspension of nanoparticles namely copper and silver in ethylene‐glycol have been developed. The fractional Stokes second problem is simulated with the help of transformation techniques and statistical data analysis. The analytic optimization of mathematical solutions of velocity and temperature have been established in terms of newly defined different Mittag‐Leffler functions. The statistical data for velocity field is generated by varying Hartman number, Prandtl number and Eckert number. The comparative analysis for two types of fractional differential operators of temperature distribution is perceived through bar column label, three‐dimensional color pie chart, three‐dimensional color map surface with projection, three‐dimensional bars and linear fitting of data. The comparative results revealed that temperature distribution has a significant effect and great influence on the stability of nanofluid by the embedded parameters.

Heat transfer analysis of Walter’s-B flow over a vertical plate by using fractional operator

The present study deals with the unsteady Walter’s-B fluid with hybrid fractional derivative namely constant proportional Caputo type with singular kernel. In this paper, we find new analytical solutions of a well-known problem of the fluid dynamics known as Stokes’ first problem. Using dimensional variables governing equations convert into dimensionless form. To solve the model analytically, one uses the Laplace transform approach. Analytical and numerical evaluations of the inverse Laplace transform have been conducted. The influence of several embedded flow properties, including the magnetic parameter, Grashof number, dimensionless time, Prandtl number, Schmidt number, and fractional parameter is analyzed graphically. More specifically, compared to the classical model, the fractional model provides a wider range of integral curves and better represents the flow behavior. The temperature and velocity of the fluid decrease with increasing fractional parameters during short intervals of time, but exhibit the reverse pattern over longer durations. Skin friction, the Sherwood number, and the Nusselt number are numerical quantities linked to engineering that are statistically computed and provided in tabular form.

Bilinear Θ-type generalized fractional integral and its commutator on product of generalized fractional weighted Morrey spaces

The aim of this paper is to establish the boundedness of a bilinear θ-type generalized fractional integral operator T˜α and its commutator T˜α,b1,b2 on generalized fractional weighted Morrey spaces Lωp,η,φ(Rn). Under assumption that the function φ satisfies doubling conditions, the authors prove that the T˜α is bounded from product of generalized fractional weighted Morrey spaces Lω1p1p1,η1,φ(Rn)×Lω2p2p2,η2,φ(Rn) into generalized fractional weighted Morrey spaces Lνω→qq,η−α,φ(Rn), and it is also bounded from product of spaces Lω1p1p1,η1,φ(Rn)×Lω2p2p2,η2,φ(Rn) into generalized fractional weighted weak Morrey spaces WLνω→qq,η−α,φ(Rn), where η=η1+η2, ω→=(ω1,ω2)∈A(P→,q), P→=(p1, p2), 1q=1p−αn, 1p=1p1+1p2 for 1<p<nα and 0<α<η. Furthermore, via the known result of fractional integral operators, the boundedness of the commutator T˜α,b1,b2 which is generated by b1,b2∈BMO(Rn) and the T˜α on spaces Lωp,η,φ(Rn) and on spaces WLωp,η,φ(Rn) is also obtained.

Fractional-order Sliding Mode Control of Manipulator Combined with Disturbance and State Observer

A fractional-order sliding mode control (FSMC) method for a manipulator based on disturbance and state observers is proposed. First, a state estimator is designed that can estimate the velocity and acceleration, and only joint position feedback and the mathematical model of the manipulator are needed. The state estimator converges in finite time. Then, the disturbance observer is designed. By designing the nominal system model of the manipulator, a disturbance observation error is introduced into the closed-loop control so that the performance of the manipulator can track the nominal system. Finally, a sliding mode controller (SMC) based on the fractional differential operator theory is also designed. The value of the sliding mode variable in the derivation of the controller is composed of the fractional derivative of the trajectory tracking error of the manipulator, whereas the fractional differentiation operation uses integration in its realization, and the integration is a low-pass filter, thus, high-frequency noise is suppressed. In the experimental section, the method designed is compared with the conventional sliding mode, which further reveals the rapidity and control accuracy of FSMC.

Fractal-fractional estimations of Bullen-type inequalities with applications

The study of inequalities inside fractal domains has been stimulated by the growing interest in fractional calculus for the applied and mathematical sciences. This work uses extended fractional integrals in fractal domains to prove new Bullen-type inequalities for differentiable convex functions. By showing how these operators and inequalities can convert classical inequalities into fractal sets, it fills a vacuum in the literature on fractal-fractional integral inequalities. Using extended fractional integral operators, we derive new fractal-fractional estimates and Bullen-type inequalities, accompanied by thorough mathematical derivations and graphical validations. We show optimal results derived from new Bullen-type inequalities for fractal domains. We confirm our results with mathematical examples and graphs, improving models for complicated problems in fractal contexts. In particular, our findings have important ramifications for special means on fractal sets, probability density functions, and quadrature formulas, as well as for future study and applications.

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The Mittag-Leffler and confluent hypergeometric functions were originally developed to extend the exponential function and its area of applications. This study aims to examine some operators involving generalized Mittag-Leffler-type functions in the kernels, employing the generalized Fox-Wright function in specific circumstances. Furthermore, we investigate some of the commonly utilized generalized fractional integral operators in fractional calculus. Moreover, a numerical technique is developed to solve fractional differential equations of both kinds, linear and nonlinear. The graphic results of the examples show how effective this method is at solving fractional differential equations. Lastly, various effects and implications of these results are thoroughly examined.

Spectral theory for fractal pseudodifferential operators

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator Tτμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T^\\mu _\ au $$\\end{document}, (Tτμf)(x)=∫Rne-ixξτ(x,ξ)(fμ)∨(ξ)dξ,x∈Rn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\big ( T^\\mu _\ au f\\big )(x) = \\int _{{{\\mathbb {R}}}^n} e^{-ix\\xi } \\, \ au (x,\\xi ) \\, \\big ( f\\mu \\big )^\\vee (\\xi ) \\, {\\mathrm d}\\xi , \\qquad x\\in {{\\mathbb {R}}}^n, \\end{aligned}$$\\end{document}in suitable special Besov spaces Bps(Rn)=Bp,ps(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^s_p ({{\\mathbb {R}}}^n) = B^s_{p,p} ({{\\mathbb {R}}}^n)$$\\end{document}, s>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s>0$$\\end{document}, 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}. Here τ(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au (x,\\xi )$$\\end{document} are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes Ψ1,δσ(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi ^\\sigma _{1, \\delta } ({{\\mathbb {R}}}^n)$$\\end{document}, σ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma <0$$\\end{document}, 0≤δ≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le \\delta \\le 1$$\\end{document} (including the exotic case δ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta =1$$\\end{document}) whereas μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} is the Hausdorff measure of a compact d–set Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^n$$\\end{document}, 0<d<n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<d<n$$\\end{document}. This extends previous assertions for the positive-definite selfadjoint fractal differential operator (id-Δ)σ/2μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extrm{id}- \\Delta )^{\\sigma /2} \\mu $$\\end{document} based on Hilbert space arguments in the context of suitable Sobolev spaces Hs(Rn)=B2s(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s ({{\\mathbb {R}}}^n) = B^s_2 ({{\\mathbb {R}}}^n)$$\\end{document}. We collect the outcome in the Main Theorem below. Proofs are based on estimates for the entropy numbers of the compact trace operator trμ:Bps(Rn)↪Lp(Γ,μ),s>0,1<p<∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{tr}\\,_\\mu : \\quad B^s_p ({{\\mathbb {R}}}^n) \\hookrightarrow L_p (\\Gamma , \\mu ), \\quad s>0, \\quad 1<p<\\infty . \\end{aligned}$$\\end{document}We add at the end of the paper a few personal reminiscences illuminating the role of Pietsch in connection with the creation of approximation numbers and entropy numbers.

A generalization of the Exner law for sediment nonlocal transport at bedform scale

Recent researches have highlighted the significance of nonlocal processes in understanding the morphodynamics and sediment transport across landscapes. Nonlocal processes in sediment transport refer to the influence of landscape properties beyond the immediate vicinity of a given point on the sediment flux. Existing nonlocal models, employing fractional operators, aim to capture global correlated nonlocality using a global convolution operator. Nevertheless, such models tend to disregard nonlocal phenomena occurring at regional scales, potentially resulting in substantial inaccuracies due to an inadequate representation of sediment transport processes at these scales. This study presents a novel and more comprehensive mathematical formulation of the nonlocal Exner law, leveraging the peridynamic differential operator (PDDO). The proposed regional nonlocal model incorporates nonlocal sediment transport processes by utilizing a pre-defined weight function and interaction domain within the framework of the PDDO. The novel regional nonlocal model effectively bridges the gap between local and global models by integrating both short-range and long-range interactions in sediment transport. Application reveals that that the regional nonlocal model provides a significantly enhanced accuracy in depicting profiles at the bedform scale compared to the local and the global models.

The Integrals of Fractional Operators with Non-Singular Kernels: A Conceptual Approach, Formulations, and Normalization Functions

The Integrals of Fractional Operators with Non-Singular Kernels: A Conceptual Approach, Formulations, and Normalization Functions

3rd-Order Differential Subordination and Superordination Results for Univalent Analytic Functions Associated with Zeta-Riemann Fractional Differential Operator

Using the results of 3rd order differential subordination, we introduce certain families of admissible functions and discuss some applications of 3rd order differential subordination for normalized analytic functions associated with novel fractional operator namely Zeta-Riemann Fractional differential operator. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted.

On Extended Class of Totally Ordered Interval-Valued Convex Stochastic Processes and Applications

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping γ, and center-radius ordering relations to establish a class of extended cr-interval-valued convex stochastic processes. This class yields a combination of innovative convex and non-convex stochastic processes. We characterize our class by illustrating its relationships with other classes as well as certain key attributes and sufficient conditions for this class of processes. Additionally, leveraging Riemann–Liouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensen’s inequality, Hermite–Hadamard’s inequality, Fejer’s inequality, and product Hermite–Hadamard’s like inequality. We establish an interesting relation between means by means of Hermite–Hadamard’s inequality. We utilize the numerical and graphical approaches to showcase the significance and effectiveness of primary findings. Also, the proposed results are powerful tools to evaluate the bounds for stochastic Riemann–Liouville fractional operators in different scenarios for a larger space of processes.

Hermite–Hadamard-type inequalities arising from tempered fractional integrals including twice-differentiable functions

UDC 517.5 We propose a new method for the investigation of integral identities according to tempered fractional operators. In addition, we prove the midpoint-type and trapezoid-type inequalities by using twice-differentiable convex functions associated with tempered fractional integral operators. We use the well-known Hölder inequality and the power-mean inequality in order to obtain inequalities of these types. The resulting Hermite–Hadamard-type inequalities are generalizations of some investigations in this field, involving Riemann–Liouville fractional integrals.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Existence theory for ψ -Caputo fractional differential equations

UDC 517.9 The target of this paper is to handle a nonlocal boundary-value problem for a specific kind of nonlinear fractional differential equations encapsuling a collective fractional derivative known as the ψ -Caputo fractional operator. The applied fractional operator generated by the kernel is of the following kind: k ( t , s ) = ψ ( t ) - ψ ( s ) . The existence of the solutions of the above-mentioned equations is tackled by using Mönch's fixed-point theorem combined with the technique of measures of noncompactness. In addition, we discuss the problem of stability within the scope of the Ulam–Hyers stability criteria for the main fractional system. Finally, an example is given to illustrate the viability of the reported results.

Mathematical analysis of Ebola considering transmission at treatment centres and survivor relapse using fractal-fractional Caputo derivatives in Uganda

In this article, we seek to formulate a robust mathematical model to study the Ebola disease through fractal-fractional operators. The study thus incorporates the transmission rate in the treatment centers and the relapse rate, since the Ebola virus persists or mostly hides in the immunologically protected sites of survivors. The Ebola virus disease (EVD) is one of the infectious diseases that has recorded a high death rate in countries where it is endemic, and Uganda is not an exception. The world at large has suffered from this deadly disease since 1976 when it was declared epidemic by the World Health Organization. The study employed fractal-fractional operators to identify the epidemiological patterns of EVD, especially in treatment centers and relapse. Memory loss and relapse are mostly observed in EVD survivors and this justifies the use of fractional operators to capture the true dynamics of the disease. Through dynamical analysis, the model is proven to be positive and bounded in the region. The model is further explicitly shown to have a solution that is unique and stable. The reproduction number was duly computed by using the next-generation matrix approach. By taking EVD epidemic cases in Uganda, the study fitted all parameters to real data. It has been shown through sensitivity index analysis that the transmission rate outside treatment centers and relapse have a significant effect on the endemic state of the disease, as they lead to an increase in the basic reproduction ratio.