Search Results (99)

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

A multiplexing method for orbital angular momentum (OAM) beams was proposed. The aperture size as a new information carrier was provided, and it could be modulated by the external variable aperture. The field of the beams propagating through turbulence was derived and discretized with Gauss–Legendre quadrature formulas. Based on this, the demultiplexing method was improved, and the beam OAM states, amplitude, Gaussian spot radius and aperture radius were decoded. Moreover, the influence of turbulence on the multiplexing parameters was also analyzed, and the decoding precision of the aperture radius was higher than that of other parameters. The aperture radius was recommended as an extra carrier for multiplexing communication. This study provides a simple method to modulate the information carried by OAM beams, and it has promising applications in large capacity laser communication. Full article

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors. Full article

In this paper, the remainder term of anti-Gaussian quadrature rules for analytic integrands with respect to Jacobi weight functions ${\omega }_{a,b}\left(x\right)={(1-x)}^a{(1+x)}^b$, where $a,b>-1$, is analyzed, and sharp estimates of the error are provided. These kinds of quadrature formulas were introduced by D.P. Laurie and have been recently studied by M.M. Spalević for the case of Jacobi-type weight functions $\omega$. Full article

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish several new biparametrized fractal integral inequalities for functions whose local fractional derivatives are of a generalized convex type. In addition to employing tools from local fractional calculus, our approach utilizes the Hölder inequality, the power mean inequality, and a refined version of the latter. Further results are also derived using the concept of generalized concavity. To support our theoretical findings, we provide a graphical example that illustrates the validity of the obtained results, along with some practical applications that demonstrate their effectiveness. Full article

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter $H\in (1/2,1)$. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order $\alpha \in (0,1)$ and the Riemann–Liouville time-fractional integral of order $\gamma \in (0,1)$, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order $O({\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}),\epsilon >0$. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

We explore the equivalence between the low-energy dynamics of strained graphene and a quantum mechanical framework for the 2D Dirac equation in flat space with a deformed momentum operator. By considering some common forms of the anisotropic Fermi velocity tensor emerging from the elasticity theory, we associate such tensor forms with a deformation of the momentum operator. We first explore the bound states of charge carriers in a background uniform magnetic field in this framework and quantify the impact of strain in the energy spectrum. Then, we use a quadrature algebra formula as a mathematical tool to analyze the impact of the deformation attached to the momentum operator and identify physical consequences of such deformation in terms of energy modifications due to the applied strain. Full article

A family of strongly nonlinear nonstationary equations of mathematical physics with three independent variables is investigated, which contain an arbitrary degree of the first derivative with respect to time and a quadratic combination of second derivatives with respect to spatial variables of the Monge–Ampère type. Individual PDEs of this family are encountered, for example, in electron magnetohydrodynamics and differential geometry. The symmetries of the considered parabolic Monge–Ampère equations are investigated by group analysis methods. Formulas are obtained that make it possible to construct multiparameter families of solutions based on simpler solutions. Two-dimensional and one-dimensional symmetry and non-symmetry reductions are considered, which lead to the original equation to simpler partial differential equations with two independent variables or ordinary differential equations or systems of such equations. Self-similar and other invariant solutions are described. A number of new exact solutions are constructed by methods of generalized and functional separation of variables, many of which are expressed in elementary functions or in quadratures. To obtain exact solutions, the principle of the structural analogy of solutions was also used, as well as various combinations of all the above-mentioned methods. In addition, some solutions are constructed by auxiliary intermediate-point or contact transformations. The obtained exact solutions can be used as test problems intended to check the adequacy and assess the accuracy of numerical and approximate analytical methods for solving problems described by highly nonlinear equations of mathematical physics. Full article

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings. Full article

Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results and examples. In addition to several types of standard orthogonality, the concept of orthogonality on arbitrary radial rays is introduced, some or all of which may be infinite. A general method for numerical constructing, the so-called discretized Stieltjes–Gautschi procedure, is described and several interesting examples are presented. The main properties, zero distribution and some applications are also given. Special attention is paid to completely symmetric cases. Recurrence relations for such kinds of orthogonal polynomials and their zero distribution, as well as a connection with the standard polynomials orthogonal on the real line, are derived, including the corresponding linear differential equation of the second order. Finally, some applications in physics and electrostatics are mentioned. Full article

A damped iterative explicit guidance (DIEG) algorithm is proposed to address the problem of the insufficient convergence of classical explicit guidance methods in the event of thrust drop faults in multistage rockets. Based on the iterative guidance mode (IGM) and powered explicit guidance (PEG), this method is enhanced in three aspects: (1) an accurate transversality condition is derived and applied in the dimension-reduction framework instead of using a simplified assumption; (2) the Gauss–Legendre quadrature formula (GLQF) is adopted to increase the accuracy of the method by addressing the issue of excessive errors in calculating thrust integration using linearization methods based on a small quantity assumption under fault conditions; and (3) a damping factor for solving the time-to-go is introduced to avoid the chattering phenomenon and enhance convergence. A numerical simulation was conducted in single- and multistage mission scenarios by gradually reducing the engine thrust to compare the performance of DIEG and PEG. The results show that DIEG has a much larger convergence range than PEG and has fuel optimality similar to that of the optimization method in most fault scenarios. Finally, the robustness of DIEG under various deviations is verified through Monte Carlo simulation. Full article

The limitations of cloning and discriminating quantum states are related to the non-orthogonality of the states. Hence, understanding the collective features of quantum states is essential for the future development of quantum communications technology. This paper investigates the non-orthogonality of different coherent-state signal constellations used in quantum communications, namely phase-shift keying (PSK), quadrature-amplitude modulation (QAM), and a newly defined signal named the sunflower-like (SUN) coherent-state signal. The non-orthogonality index (NOI) and the average probability of correct detection (detection probability) are numerically computed. Results show that PSK NOI increases faster than QAM and SUN as the number of signals increases for a given number of signal photons. QAM and SUN exhibit similar NOI and detection probability, behaving similarly to randomly generated signals for a larger number of signals. Approximation formulas are provided for the detection probability as a function of NOI for each signal type. While similar to QAM, SUN signal offers potential advantages for applications requiring uniform signal-space distribution. The findings provide valuable insights for designing useful quantum signal constellations. Full article

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain ${[a,b]}^2=[a,b]\times [a,b]$. Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on ${[a,b]}^2$, they involve two or four univariate integrals. A useful property of these cubature formulae is that they are definite of order $(2,2)$, that is, they provide one-sided approximation to the double integral for real-valued integrands from the class ${\mathcal{C}}^{2,2}[a,b]=\{f(x,y):{\displaystyle \frac{{\partial }^{4}f}{\partial x^2\partial y^2}}\mathrm{continuous}\mathrm{and}\mathrm{does}\mathrm{not}\mathrm{change}\mathrm{sign}\mathrm{in}{(a,b)}^2\}$. For integrands from ${\mathcal{C}}^{2,2}[a,b]$ we prove monotonicity of the remainders and derive a posteriori error estimates. Full article

Counterparty risk, which combines market and credit risks, gained prominence after the 2008 financial crisis due to its complexity and systemic implications. Traditional management methods, such as netting and collateralization, have become computationally demanding under frameworks like the Fundamental Review of the Trading Book (FRTB). This paper explores the combined application of Gaussian process regression (GPR) and Bayesian quadrature (BQ) to enhance the efficiency and accuracy of counterparty risk metrics, particularly credit valuation adjustment (CVA). This approach balances excellent precision with significant computational performance gains. Focusing on fixed-income derivatives portfolios, such as interest rate swaps and swaptions, within the One-Factor Linear Gaussian Markov (LGM-1F) model framework, we highlight three key contributions. First, we approximate swaption prices using Bachelier’s formula, showing that forward-starting swap rates can be modeled as Gaussian dynamics, enabling efficient CVA computations. Second, we demonstrate the practical relevance of an analytical approximation for the CVA of an interest rate swap portfolio. Finally, the combined use of Gaussian processes and Bayesian quadrature underscores a powerful synergy between precision and computational efficiency, making it a valuable tool for credit risk management. Full article