Discrete Formula Research Articles

An Elegant Formula Derivation for The Circumference of An Ellipse

It is a widely accepted fact that the circumference of ellipse has no discrete formula. This happens, in part, because the orthodox methods of calculating circumference of an ellipse run into a system of infinite series and, in part, since there have been little concerted efforts to explore other ways to calculate an explicit formula for ellipse. However, this article claims that some minor revisions in a concerted manner, if proceeded in a different direction, can result in a full-fledged, distinct and explicit formula for the circumference of an ellipse, that is elegant yet simple. The article progresses in an orthodox manner, at first, and then displays, how a solution becomes possible in relatively simple calculus. The formula is a generalized form for all ellipses and as a special case, derives the regular formula for a circle. The formula has been tested on various mathematical calculation platforms available on internet and the results are very close to what the website claims. However, this article claims that the actual calculations result from the formula derived in the article and not from those available on the internet.

CAFE-AMR: a computational MHD solar physics simulation tool that uses AMR

ABSTRACT The study of our Sun holds significant importance in space weather research, encompassing a diverse range of phenomena characterized by distinct temporal and spatial scales. To address these complexities, we developed CAFE-AMR, an implementation of an adaptive mesh refinement (AMR) strategy coupled with a magnetohydrodynamics (MHD) equation solver, aiming to tackle solar-physics-related problems. CAFE-AMR employs standard fluid dynamics methods, including finite-volume discretization, HLL and Roe class flux formulas, linear order reconstructors, second-order Runge–Kutta, and corner transport upwind time stepping. In this paper, we present the core structure of CAFE-AMR, discuss and evaluate mesh refinement criteria strategies, and conduct various tests, including simulations of idealized solar wind models, relevant for space weather applications.

Continuous and Discrete ZND Models with Aid of Eleven Instants for Complex QR Decomposition of Time-Varying Matrices

The problem of QR decomposition is considered one of the fundamental problems commonly encountered in both scientific research and engineering applications. In this paper, the QR decomposition for complex-valued time-varying matrices is analyzed and investigated. Specifically, by applying the zeroing neural dynamics (ZND) method, dimensional reduction method, equivalent transformations, Kronecker product, and vectorization techniques, a new continuous-time QR decomposition (CTQRD) model is derived and presented. Then, a novel eleven-instant Zhang et al discretization (ZeaD) formula, with fifth-order precision, is proposed and studied. Additionally, five discrete-time QR decomposition (DTQRD) models are further obtained by using the eleven-instant and other ZeaD formulas. Theoretical analysis and numerical experimental results confirmed the correctness and effectiveness of the proposed continuous and discrete ZND models.

A Predictor–Corrector Compact Difference Scheme for a Nonlinear Fractional Differential Equation

In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and the Caputo derivative term is discretized by the L1 discrete formula. Through the first and second derivatives of the matrix under the compact difference, we improve the precision of this scheme. Then, the existence and uniqueness are proved, and the numerical experiments are presented.

A novel well‐balanced numerical model for water‐solute mixture underflows coupled with ambient water

AbstractThis study considers a water‐solute mixture underflow coupled with ambient water. Such physical phenomenons are modelled by a two‐layer stratified Saint‐Venant system with a scalar transport equation in this paper. The governing system is first relaxed to a hyperbolic system discretized by a path‐conservative algorithm. In such framework, a novel discrete formula of nonconservative and source terms, is proposed to be able to guarantee the well‐balanced property for the hydrodynamic model and preserve the uniform concentration in underflows, especially at wet‐dry fronts. The performances of the developed numerical schemes are verified by several tests.

Data-Driven Control for Continuum Robots Based on Discrete Zeroing Neural Networks

The effectiveness of continuous-time zeroing neural network (CZNN) in continuum robot control has been preliminarily verified. However, CZNN is not friendly to digital devices and hardware implementation. Although discrete ZNN (DZNN) has been investigated in other tasks, all the existing DZNNs are designed with fixed time steps, which cannot meet the needs of different tasks. This motivates us to develop a generic DZNN model with variable number of time steps and apply it to continuum robot control. In this paper, we present a data-driven scheme based on CZNNs to learn the unknown kinematics of continuum robots and solve the kinematic control problem. Further, a unified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -step forward discretization formula is derived to discretize the CZNN-based scheme into DZNN-based control algorithms. Finally, we take the 1-step and the 3-step algorithms as examples to show the discretization process, and verify their efficacy by simulations and experiments.

An efficient hybrid boundary-type weak-form meshfree algorithm for 2D thermal analysis in inhomogeneous functionally graded materials

Exponential, quadratic, and trigonometric 2D thermal analysis in inhomogeneous functionally graded materials (FGMs) are solved through an efficient hybrid boundary-type weak-form meshfree algorithm, which is the virtual boundary meshfree Galerkin method with fundamental solutions (FS-VBMGM). Three types of temperature fundamental solutions for FGMs are introduced. The corresponding thermal flow fundamental solutions are derived in detail. Temperature and thermal flux are obtained by utilizing the virtual boundary element method (VBEM) based on the continuous virtual loads. The continuous virtual loads are constructed through the radial basis function interpolation (RBFI) of meshfree algorithm. Using the Galerkin method, calculating formula of FS-VBMGM for thermal analysis in inhomogeneous FGMs is deducted. The benefits of VBEM, Galerkin method, and meshfree algorithm are thus presented in FS-VBMGM. Exponential, quadratic, and trigonometric inhomogeneous numerical examples of FGMs are calculated and contrasted to the exact solutions as well as other numerical methods. The reliability and exactness are proved. The specific discrete formula of FS-VBMGM for thermal analysis in inhomogeneous FGMs is simple to program and extended to calculate other particularly composite FGMs examples.

A new ZNN model for finding discrete time-variant matrix square root: From model design to parameter analysis

In recent years, artificial neural network technology has developed very rapidly. More and more experts and scholars use neural network technology to solve the related problems of applied mathematics. As a representative of the Hopfield neural network, zeroing neural network (ZNN) which can deal with the time-variant problem that may not be solved by traditional methods effectively. In this paper, to solve the problem of time-variant matrix square root, the continuous-time ZNN (CT-ZNN) model is presented based on the ZNN design formula. Next, to meet the requirements of computer simulation, the general square-pattern discretization (SPD) formula is applied to obtain the general discrete-time ZNN (G-DT-ZNN) model. Moreover, by changing the value of the selected parameter in the G-DT-ZNN model, it is presented that the effect of different values of the selected parameter on the precision of the G-DT-ZNN model. Finally, in the numerical experiments, three discrete time-variant matrix square root problems are presented as examples for verifying such theoretical results of the proposed G-DT-ZNN model.

From Dual Unitarity to Generic Quantum Operator Spreading.

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of nongeneric behavior. By investigating the effect of weakly broken dual unitarity on the spreading of local operators, we study whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and recover a butterfly velocity smaller than the light-cone velocity, v_{B}<v_{LC}, and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. The butterfly velocity and diffusion constant are determined by a small set of microscopic quantities, and the operator entanglement of the gates has a crucial role.

A fast BDF2 Galerkin finite element method for the one-dimensional time-dependent Schrödinger equation with artificial boundary conditions

In this paper, we propose an efficient numerical scheme with linear complexity for the one-dimensional time-dependent Schrödinger equation on unbounded domains. The artificial boundary method is used to address the unboundedness of the domain. By applying the two-step backward difference formula for time discretization and performing the Z-transform, we derive an exact semi-discrete artificial boundary condition of the Dirichlet-to-Neumann type. To expedite the discrete temporal convolution involved in the exact semi-discrete artificial boundary conditions, we design a fast algorithm based on the best relative Chebyshev approximation of the square-root function. The Galerkin finite element method is used for spatial discretization. By introducing a constant damping term to the original Schrödinger equation, we present a complete error estimate for the fully discrete problem. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed numerical scheme.

Explicit Linear Left-and-Right 5-Step Formulas With Zeroing Neural Network for Time-Varying Applications.

In this article, being different from conventional time-discretization (simply called discretization) formulas, explicit linear left-and-right 5-step (ELLR5S) formulas with sixth-order precision are proposed. The general sixth-order ELLR5S formula with four variable parameters is developed first, and constraints of these four parameters are displayed to guarantee the zero stability, consistence, and convergence of the formula. Then, by choosing specific parameter values within constraints, eight specific sixth-order ELLR5S formulas are developed. The general sixth-order ELLR5S formula is further utilized to generate discrete zeroing neural network (DZNN) models for solving time-varying linear and nonlinear systems. For comparison, three conventional discretization formulas are also utilized. Theoretical analyses are presented to show the performance of ELLR5S formulas and DZNN models. Furthermore, abundant experiments, including three practical applications, that is, angle-of-arrival (AoA) localization and two redundant manipulators (PUMA560 manipulator and Kinova manipulator) control, are conducted. The synthesized results substantiate the efficacy and superiority of sixth-order ELLR5S formulas as well as the corresponding DZNN models.

Generalizing the German Tank Problem

The German Tank Problem dates back to World War II when the Allies used a statistical approach to estimate the number of enemy tanks produced or on the field from observed serial numbers after battles. Assuming that the tanks are labeled consecutively starting from 1, if we observe k tanks from a total of N tanks with the maximum observed tank being m, then the best estimate for N is m(1 + 1/k) - 1. We refer to an estimate as "best" when the estimate is closest to the actual number of tanks. We explore many generalizations; first, we looked at the discrete and continuous one-dimensional case. We attempted to improve the original formula by using different estimators such as the second largest and Lth largest tank, and applied motivation from portfolio theory by seeing if a weighted average of different estimators would produce less variance; however, the original formula, using the largest tank proved to be the best; the continuous case was similar. Then, we looked at the discrete and continuous square and circle variants where we pick pairs instead of points, which were more complex as we dealt with problems in geometry and number theory, such as dealing with curvature issues in the circle, and the problem that not every number is representable as a sum of two squares. In some cases, when we could not derive precise formulas, we derived approximate formulas. For the discrete and continuous square, we tested various statistics, but found that the largest observed component of our pairs is the best statistic to look at; the scaling factor for both cases is (2k+1)/2k. For the circle we used motivation from the equation of a circle; for the continuous case, we looked at the square root of X2+Y2 and for the discrete case, we looked at X2+Y2 and took a square root at the end to estimate for r. Interestingly, the scaling factors, a number, generally a little greater than 1, that we multiplied to scale up to get our estimation, were different for the cases. Lastly, we generalized the problem into L-dimensional squares and circles. The discrete and continuous square proved to be similar to the two-dimensional square problem. However, for the Lth dimensional circle, we had to use formulas for the volume of the L-ball, and had to approximate the number of lattice points inside it. The discrete circle formula was particularly interesting, as there was no L dependence in the formula.

Improved Replica Bounds for the Independence Ratio of Random Regular Graphs

Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter lambda tending to infinity. Finding the independence ratio of random d-regular graphs for some fixed degree d has received much attention both in random graph theory and in statistical physics. For d ge 20 the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large d in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method shows that this 1-RSB formula is an upper bound for each d ge 3. For d le 19 this bound is not tight and full-RSB is expected. In this work we use numerical optimization to find good substituting parameters for discrete r-RSB formulas (r=2,3,4,5) to obtain improved rigorous upper bounds for the independence ratio for each degree 3 le d le 19. As r grows, these formulas get increasingly complicated and it becomes challenging to compute their numerical values efficiently. Also, the functions to minimize have a large number of local minima, making global optimization a difficult task.

Research on the Calculation and Analysis of Ski-Track Areas Based on Laser Point Clouds

To address the long-term statistical problem of ski-track area in the construction and operation of ski resorts, we propose a new ski-track point cloud boundary extraction method that improves the accuracy of boundary extraction and minimizes the offset of the area error. In this method, all point clouds are first projected onto the fitting plane using the random sample consensus (RANSAC) method. An improved point cloud boundary extraction algorithm is used to triangulate and extract the high-precision ski-track boundary. A discrete Green formula is then used to calculate and count the ski track’s exact area. It is demonstrated through five sets of test experiments that the error offset of the method proposed in this paper is smaller than that of other classical methods, which confirms its benefits and feasibility.

Error and stability estimates of a time-fractional option pricing model under fully spatial–temporal graded meshes

To price vanilla European and American options via the fractional Black–Scholes model, first a (2−α)-order discretization scheme for the Caputo fractional derivative based upon graded meshes along time is presented. This is fruitful for problems having nonsmooth data at the initial time. Second, a nonuniform discretization of space is also considered and the coefficient weights based upon meshless radial basis function generated finite difference (RBF-FD) methodology are contributed under the inverse quadratic function. The fully space–time nonuniform solution method is then constructed via the presented discretization formulas. Numerical tests are given to show effectiveness and accuracy of our numerical scheme.

Multi-output physics-informed neural network for one- and two-dimensional nonlinear time distributed-order models

&lt;abstract&gt;&lt;p&gt;In this article, a physics-informed neural network based on the time difference method is developed to solve one-dimensional (1D) and two-dimensional (2D) nonlinear time distributed-order models. The FBN-$ \theta $, which is constructed by combining the fractional second order backward difference formula (BDF2) with the fractional Newton-Gregory formula, where a second-order composite numerical integral formula is used to approximate the distributed-order derivative, and the time direction at time $ t_{n+\frac{1}{2}} $ is approximated by making use of the Crank-Nicolson scheme. Selecting the hyperbolic tangent function as the activation function, we construct a multi-output neural network to obtain the numerical solution, which is constrained by the time discrete formula and boundary conditions. Automatic differentiation technology is developed to calculate the spatial partial derivatives. Numerical results are provided to confirm the effectiveness and feasibility of the proposed method and illustrate that compared with the single output neural network, using the multi-output neural network can effectively improve the accuracy of the predicted solution and save a lot of computing time.&lt;/p&gt;&lt;/abstract&gt;

An efficient boundary meshfree computational approach for 3-D multi-domain transient thermal analysis with variable thermal sources in non-homogeneous media

Solutions of 3-D multi-domain transient thermal analysis with variable thermal sources in non-homogeneous media are separated into homogeneous and special solutions by an efficient boundary meshfree computational approach, namely virtual boundary meshfree Galerkin method. Homogeneous solutions are expressed by the virtual boundary element method. The virtual source functions of homogeneous solutions and the unknowable coefficients of special solutions can be formed by the radial basis function interpolation. Considering the control equation, the boundary and continuous conditions, and using the Galerkin method, the discrete formula for 3-D multi-domain transient thermal analysis with variable thermal sources in non-homogeneous media can be obtained. This discrete equation has symmetry. Meanwhile, in order to illustrate the steps of implementation more clearly, the final detailed implementation process is given. The numerical results of two calculation examples are obtained and compared to other methods and exact solutions. The proposed method?s stability and exactness are validated for 3-D multi-domain transient thermal analysis with variable thermal sources in non-homogeneous media.

NUMERICAL METHODS FOR THE CAPUTO-TYPE FRACTIONAL DERIVATIVE WITH AN EXPONENTIAL KERNEL

In the present article, several typical numerical discrete formulas for the Caputo-type fractional derivative with an exponential kernel (call "exponential Caputo derivative" for brevity) with order $ \alpha\in(0,1) $ and $ \alpha\in(1,2) $ are constructed, which are L1, L1-2, L2-1$ _\sigma $ formulas for $ \alpha\in(0,1) $, and H2N2 and L2$ _{1} $ formulas for $ \alpha\in(1,2) $, respectively. And the estimates of the truncation errors are determined. Meanwhile, the properties of the coefficients in these formulas are studied. Finally, some numerical examples are displayed which support the theoretical analysis.

BDF2 schemes for optimal parameter control problems governed by bilinear parabolic equations

SummaryIn this paper, BDF2 schemes combined with the finite element method are proposed to solve control constrained optimal parameter control problems governed by bilinear parabolic equations, where the control variable acts as a reaction coefficient. The discretize‐then‐optimize (DO) approach is adopted. First, the state equation is discretized and two different discrete formulas are utilized for the objective functional. Then, by using the Lagrange multiplier method, two fully discrete first‐order necessary optimality systems are deduced and named as DO1 and DO2 systems, respectively. It can be found that the optimize‐then‐discretize (OD) and discretize‐then‐optimize (DO) approaches coincide in the case of the DO2 system. Furthermore, a priori error estimates are derived for the state, co‐state and control variables of the DO2 system, which show that the DO2 system has second‐order accuracy in time. Finally, three numerical examples are presented to illustrate the theoretical findings.

A second-order modular grad-div stabilized scheme for the Darcy–Brinkman model

In this article, a fully discrete modular grad-div stabilized finite element scheme for the Darcy–Brinkman equations are considered. This fully discrete scheme includes two steps: the first step is a combination of a mixed finite element approximation for space discretization, the second-order backward differentiation formula for temporal discretization, and extrapolated treatments in linearization for the nonlinear terms. Then, in the second step, a modular grad-div stabilized technique is applied, which can improve solution accuracy without increasing computational time for large stabilized parameters. Moreover, we show that the scheme is unconditionally stable and convergent with second-order accuracy with respect to time step. Finally, several numerical examples are provided to support the derived theoretical results, and demonstrate the efficiency of the presented scheme.