Differential Operators Research Articles

A fractional-order generalized Richards growth model and its implementation to COVID-19 data

In this article, we include the memory effect in the generalized Richards model (GRM), which is proposed in the form of a fractional-order GRM. The fractional-order GRM is developed by replacing the first-order derivative in the GRM with a fractional-order derivative and then taking care of model homogeneity of dimension. We consider two fractional-order differential operators: Caputo and Atangana–Baleanu in the Caputo sense (ABC). The proposed fractional-order models are then implemented to fit the COVID-19 data in East Java, Indonesia from March 25 until October 31, 2020. The fitting is performed by minimizing the sum of the squared residual between the numerical solutions of each fractional-order model and the daily data of a cumulative number of COVID-19 cases. The numerical solutions of both fractional-order models are determined by the predictor-corrector method. The performance of the two fractional models is measured by two performance metrics: coefficient of determination (R 2) and root mean square error (RMSE). By considering that the order of fractional derivative as an extra degree of freedom, we perform data fitting for several orders of fractional derivative and evaluate the two performance metrics. It is observed that the fractional-order model with the ABC operator generally has the best performance in calibrating and forecasting both the cumulative number of COVID-19 cases and daily new cases of COVID-19.

Input‐to‐state stability of impulsive stochastic systems with state‐dependent impulses and regime‐switching

AbstractThis article introduces a unified criterion for input‐to‐state stability (ISS), integral input‐to‐state stability (iISS) and ‐input‐to‐state stability (‐ISS) of impulsive stochastic system with switching. The criterion demonstrates that the premise of a switching‐impulse system to achieve three types of ISS is that a mutually constraining relationship between switching, impulse and continuous dynamics needs to be satisfied. Furthermore, using it we know that switching can stabilize a system containing stabilizing factors by affecting both continuous dynamics and impulses, that switching itself is one of the factors in system instability, and that impulses have a dual effect on the stability of the system. The coefficients of the upper bound of Lyapunov functional differential operators are time‐varying functions and the impulses contain stable and unstable impulses, including the case of constants, which advances and improves the existing results. Finally, an example and its simulation results are given to verify the validity of theoretical analysis.

Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators

AbstractWe consider essential self-adjointness on the space $$C_0^{\infty }((0,\infty ))$$ C 0 ∞ ( ( 0 , ∞ ) ) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type $$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$ τ 2 n ( c ) = ( - 1 ) n d 2 n d x 2 n + c x 2 n , x > 0 , n ∈ N , c ∈ R , in $$L^2((0,\infty );dx)$$ L 2 ( ( 0 , ∞ ) ; d x ) . While the special case $$n=1$$ n = 1 is classical and it is well known that $$\tau _2(c)\big |_{C_0^{\infty }((0,\infty ))}$$ τ 2 ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self-adjoint if and only if $$c \ge 3/4$$ c ≥ 3 / 4 , the case $$n \in {{\mathbb {N}}}$$ n ∈ N , $$n \ge 2$$ n ≥ 2 , is far from obvious. In particular, it is not at all clear from the outset that $$\begin{aligned} \begin{aligned}&\textit{there exists }c_n \in {{\mathbb {R}}}, n \in {{\mathbb {N}}}\textit{, such that} \\&\quad \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \, \textit{ is essentially self-adjoint}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (*)\\ {}&\quad \textit{ if and only if } c \ge c_n. \end{aligned} \end{aligned}$$ there exists c n ∈ R , n ∈ N , such that τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self - adjoint ( ∗ ) if and only if c ≥ c n . As one of the principal results of this paper we indeed establish the existence of $$c_n$$ c n , satisfying $$c_n \ge (4n-1)!!\big /2^{2n}$$ c n ≥ ( 4 n - 1 ) ! ! / 2 2 n , such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question, $$\begin{aligned} \textit{for which values of }c\textit{ is }\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}{} \textit{ bounded from below?}, \end{aligned}$$ for which values of c is τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) bounded from below ? , which permits the sharp (and explicit) answer $$c \ge [(2n -1)!!]^{2}\big /2^{2n}$$ c ≥ [ ( 2 n - 1 ) ! ! ] 2 / 2 2 n , $$n \in {{\mathbb {N}}}$$ n ∈ N , the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, $$\begin{aligned} c_{1}&= 3/4, \quad c_{2 }= 45, \quad c_{3 } = 2240 \big (214+7 \sqrt{1009}\,\big )\big /27, \end{aligned}$$ c 1 = 3 / 4 , c 2 = 45 , c 3 = 2240 ( 214 + 7 1009 ) / 27 , and remark that $$c_n$$ c n is the root of a polynomial of degree $$n-1$$ n - 1 . We demonstrate that for $$n=6,7$$ n = 6 , 7 , $$c_n$$ c n are algebraic numbers not expressible as radicals over $${{\mathbb {Q}}}$$ Q (and conjecture this is in fact true for general $$n \ge 6$$ n ≥ 6 ).

Extensions of Some Statistical Concepts to the Complex Domain

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone.

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤp

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind.

Local invariants of noncommutative tori

The notion of a generic curved noncommutative torus is considered, which extends the notion of a conformally deformed noncommutative torus introduced by Connes and Tretkoff. For this manifold, an asymptotic expansion is established for the heat semigroup generated by the Laplace–Beltrami operator (in fact, for an arbitrary selfadjoint positive elliptic differential operator of order 2 2 ) and an algorithm is provided to compute the local invariants that arize as the coefficients in the expansion. This allows one to extend a series of previous results by several authors beyond the conformal case and/or for multidimensional tori.

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable.

This study suggests a strategy for calculating the fuzzy analytical solutions to a two-dimensional fuzzy fractional-order heat problem including a diffusion variable connected externally. We propose Sawi residual power series scheme (SRPSS) which is the amalgamation of Sawi transform and residual power series scheme under the Caputo fractional differential operator. We demonstrate three different examples to derive the fuzzy fractional series solution which is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity. The most significant aspect of this scheme is that it derives the results without time effort compared with the traditional residual power series approach. Our findings confirm that the SRPSS is a robust and valuable method for approximating the solution of fuzzy fractional problems. Furthermore, we provide 2D and 3D symbolic representations to present the physical behavior of fuzzy fractional problems under the lower and upper bounded solutions.

Two-dimensional optical multiple-order differentiations based on spatial spectrum modulation.

Optical differential operations have recently attracted considerable attention owing to the capabilities of ultrafast speed and low power consumption. The transfer function, which embodies the frequency-domain characteristics of differential systems, plays an important role in differentiator design. Here, we report a super-Gaussian aperture differential filter, and we reveal unique characteristics of odd- and even-order transfer functions and corresponding differential effects via spatial spectrum modulation. We show that the feature of the transfer function is well maintained, and more precise differentiation can be achieved using the designed filter. Two-dimensional first- to fifth-order full and partial differentiations are implemented both theoretically and experimentally. Our work provides an approach for engineering customized multiple-order differentiators and promotes the advancements of related areas such as optical analog computing and image processing.

Subclass of Analytic Functions Associated with Differential Operator

Subclass of Analytic Functions Associated with Differential Operator

Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and $q$-deformed rational numbers defined by Morier-Genoud and Ovsienko and lead to $q$-deformed Möbius transformations acting on the hyperbolic plane.

A Certain <i>q</i>-Sălăgean Differential Operator and Its Applications to Subclasses of Analytic and Bi-Univalent Functions Involving (<i>p</i>, <i>q</i>)-Chebyshev Polynomia

In the present investigation, we make use of the q- analogue of the Sălăgean differential operator and introduce a new subclass of analytic and bi-univalent functions involving the (P, Q)-Chebyshev polynomials. Furthermore, we derive coefficient inequalities and obtain the Fekete-Szegö problem for this new function class f ∈ of functions.

A comprehensive mathematical analysis of fractal–fractional order nonlinear re-infection model

In recent times, various real word problems including infectious diseases, engineering problems, and chemical processes, etc are modelled by using differential equations with fractals and fractional orders. The mentioned area provides a powerful tool to investigate the aforesaid problems from different perspectives like analysis, numerical investigation, and qualitative theory. Similarly, one of the global issue is devoted to re-infection of viral diseases like COVID-19. The said behaviour has produced very saviour impact on human life worldwide. Also, this is very critical for health system of a country as well as for its economics situation. To study the aforesaid phenomenon of re-infection, a hybrid type fractal–fractional three compartments model is formulated. The basic results related to boundedness, positivity, equilibrium points, basic reproductive number, and sensitivity analysis of the basic threshold are included in the theoretical aspect. Additionally, the Volterra–Lyapunov approach is used to establish both local and global stability analysis under fractals and fractional derivative. Furthermore, utilizing techniques from numerical functional analysis and fixed point theory, the existence and Hyers–Ulam stability of the solutions are also examined. A numerical algorithm based on Lagrange’s interpolation polynomials is extended to simulate the results of various compartments. Graphical interpretations corresponding to different fractals–fractional order values are presented. For verification of our numerical results, a comparison is provided between the real and simulated data. Here it should be kept in mind that the differential operator of fractals–fractional order is considered with Mittag-Leffler kernel. The mentioned operator is more general among the available fractals fractional differential operators in literature.

STEF: a Swin Transformer-Based Enhanced Feature Pyramid Fusion Model for Dongba character detection

The Dongba manuscripts are a unique primitive pictographic writing system that originated among the Naxi people of Lijiang, China, boasting over a thousand years of history. The uniqueness of the Dongba manuscripts stems from their pronounced pictorial and ideographic characteristics. However, the digital preservation and inheritance of Dongba manu manuscripts face multiple challenges, including extracting its rich semantic information, recognizing individual characters, retrieving Dongba manuscripts, and automatically interpreting the meanings of Dongba manuscripts. Developing efficient Dongba character detection technology has become a key research focus, wherein establishing a standardized Dongba detection dataset is crucial for training and evaluating techniques. In this study, we have created a comprehensive Dongba manuscripts detection dataset covering various commonly used Dongba characters and vocabularies. Additionally, we propose a model named STEF. Firstly, the Swin Transformer extracts the complex structures and diverse shapes of Dongba manuscripts’ features. Then, by introducing a Feature Pyramid Enhancement Module, features of different sizes are cascaded to preserve multi-scale information. Subsequently, all features are fused in a FUSION module, resulting in features of various Dongba manuscript styles. Each pixel’s binarisation threshold is dynamically adjusted through a differentiable binarisation operation, accurately distinguishing between foreground Dongba manuscripts and background. Lastly, deformable convolution is introduced, allowing the model to dynamically adjust the convolution kernel’s size and shape based on the Dongba manuscripts’ size, thereby better capturing the detailed information of Dongba characters of different sizes. Experimental results show that STEF achieves a recall rate of 88.88%, a precision rate of 88.65%, and an F-measure of 88.76%, outperforming other text detection algorithms. Visualization experiments demonstrate that STEF performs well in detecting Dongba manuscripts of various sizes, shapes, and styles, especially in blurred handwriting and complex backgrounds.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

Spectral reordering for faster elasticity simulations

We present a novel method for faster physics simulations of elastic solids. Our key idea is to reorder the unknown variables according to the Fiedler vector (i.e., the second-smallest eigenvector) of the combinatorial Laplacian. It is well known in the geometry processing community that the Fiedler vector brings together vertices that are geometrically nearby, causing fewer cache misses when computing differential operators. However, to the best of our knowledge, this idea has not been exploited to accelerate simulations of elastic solids which require an expensive linear (or non-linear) system solve at every time step. The cost of computing the Fiedler vector is negligible, thanks to an algebraic Multigrid-preconditioned Conjugate Gradients (AMGPCG) solver. We observe that our AMGPCG solver requires approximately 1 s for computing the Fiedler vector for a mesh with approximately 50K vertices or 100K tetrahedra. Our method provides a speed-up between 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} – 30%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$30\\%$$\\end{document} at every time step, which can lead to considerable savings, noting that even modest simulations of elastic solids require at least 240 time steps. Our method is easy to implement and can be used as a plugin for speeding up existing physics simulators for elastic solids, as we demonstrate through our experiments using the Vega library and the ADMM solver, which use different algorithms for elasticity.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

On Some Properties of a Class of Analytic Functions Defined by Opoola Differential Operator

The Fekete-Szego functional upper bounds and the Second Hankel Determinant upper bounds for a class of analytical functions defined by the Opoola Differential Operator are found in this study. The estimations made are accurate with the theorems proved.

Solving second order homogeneous differential equations in terms of Heun's general function

In this paper, we present an algorithm that checks if a second-order differential operator L∈C(x)[∂] can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in C(x)[∂] that transfer the general Heun's differential operator to the operator L whose solutions are of the form(1)exp(∫rdx)⋅HeunG(a,q;α,β,γ,δ;f(x)), where {α,β,δ,γ}∈Q∖Z, the functions r,f(x)∈C(x), and C(f(x)) is a subfield of C(x) of index 2 or 3 or f(x)=axn+bcxn+d for some n in N∖{1}.

Global Gevrey solvability of a complex of differential operators associated with an involutive system

We consider a complex of differential operators associated with an involutive system of vector fields on the torus. We present a complete characterization of the global Gevrey solvability in terms of Diophantine conditions involving exponential Liouville forms.

A peridynamic differential operator modeling approach for ceramic matrix composites microstructure with uniform or non-uniform discretization

Based on the theory of peridynamic differential operator (PDDO), a novel three-dimensional (3D) peridynamic (PD) model for isotropic and orthotropic material is proposed to investigate the elastic mechanical behaviors and stress distribution of Ceramic Matrix Composites (CMC) microstructure. The non-local expressions of strain and stress tensors are derived by employing PDDO and the classical constitutive equations. The bond forces in interface region crossing two different constituent materials are obtained by converting the partial differential terms into the non-local forms. The weak form of PD equations of motion is derived by using the principle of virtual work and PDDO. The validity of this approach is verified by predicting the elastic response and stress distributions of isotropic and orthotropic materials under uniaxial tension and pure shear deformation. Finally, simulations are conducted on a CMC microstructure with fiber and matrix under periodic boundary conditions. It is demonstrated that the current approach can effectively build 3D CMC microstructure with uniform or non-uniform discretization, and accurately predict the stress fields and effective elastic properties.