Differentiation Formula Research Articles

A machine learning approach for entropy due to natural convection flow of a nanofluid under the uniform inclined magnetic field

In this study, entropy generation of natural convection flow of Cu-water nanofluid under the influence of a uniform inclined magnetic field (MF) is firstly modeled by a machine learning approach using the data obtained from a numerical process. The two dimensional, time dependent dimensionless equations are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second order backward differentiation formula (BDF2) in time derivatives. In a set of combined dimensionless problem parameters, numerical simulations are performed. In the obtained data, the inputs are a set of combined dimnesionless problem parameters and the outputs are entropy indicators. Then, global and local entropy indicators are modeled by Neural Network (NN). In view of smaller mean squared error metric results, small number of layers or layer sizes in NN seems enough in smaller data while trilayer NN (TNN) is a better modeler in case of larger data.

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

Nonlinear vibration characteristics of axially moving anisotropic membrane with gas-thermal-elastic coupling

In the actual printing production process, the printing electronic motion membrane is susceptible to transverse vibration caused by interference from air drag, temperature changes, and other external conditions, resulting in membrane wrinkles, slippage, and other phenomena in the transmission process. We studied the bifurcation and chaos movement properties of anisotropic membranes under air drag and temperature. According to D’Alembert’s theory and von Kármán’s principle, the nonlinear dynamic differential formulas of axially moving anisotropic membranes with gas-thermal-elastic coupling are established. The Galerkin method is applied to discretize the formulas to obtain the state equation of the system. Finally, numerical simulations are performed by applying the fourth-order Runge–Kutta method to analyze the bifurcation and chaos of the system in terms of orthotropic coefficient, dimensionless air drags, and dimensionless temperature. The bifurcation diagrams, Lyapunov exponent diagrams, displacement time-history diagrams, phase-trajectory plane diagrams, and Poincaré diagrams of the membrane system are obtained. The results show that the anisotropic coefficient, dimensionless air drag, and dimensionless temperature significantly impact the investigated nonlinear dynamic of the anisotropic membrane, which provides a theoretical basis for production efficiency and high-quality printing equipment.

Bernstein collocation technique for a class of Sturm-Liouville problems

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

Application of the Fourier Modulation Analysis Method to the Problem of Derivatives Recovery

Abstract. In this work, formulas are obtained for finding higher derivatives of a function (dependence), expressed through its Fourier coefficients (harmonic amplitudes). These formulas were found by differentiating both a function with a harmonically modulated argument and its Fourier series. The expressions given make it possible to find higher derivatives of functions numerically with any accuracy by means of different methods. For example, numerical integration of the Euler-Fourier formulas may be used for harmonic amplitudes. Derivatives also can be found experimentally when studying a nonlinear physical process by digitally recording the harmonic amplitudes of the dependence under study under simultaneous static and harmonic influences. The problem of reconstructing derivatives from Fourier coefficients is posed and an analysis of its correctness is performed. Formulas for estimating recovery errors are defined and recommendations are given to reduce these errors. Examples are given with different smoothness of analytical functions and of functions used to explain experiments: 1) an analytical function used to explain the magnetic properties of superconductors, whose Fourier coefficients are determined numerically with the error of the computational algorithm in the Mathcad software environment; 2) current-voltage characteristic (CVC) of 2 back-to-back semiconductor diodes, whose harmonic amplitudes were determined experimentally with a given measurement error of the device. The resulting derivative of the current-voltage characteristic is compared with the derivative obtained using numerical differentiation formulas; 3) an analytical function whose derivative has a discontinuity of the first kind. Errors in measuring Fourier coefficients were added artificially using a random number generator.

Modified generalized second derivative extended backward differentiation formulas for highly stiff and stiffly oscillatory systems of ODEs

Modified generalized second derivative extended backward differentiation formulas for highly stiff and stiffly oscillatory systems of ODEs

Superconvergence analysis of a two-grid BDF2-FEM for nonlinear dispersive wave equation

The aim of this paper is to study the superconvergent behavior of a two-grid finite element method (FEM) with 2-step backward differential formula (BDF2) for nonlinear dispersive wave equation. By introducing an auxiliary variable q=ut for the original variable u, the problem is transformed into a parabolic system. With the help of the combination technique of the interpolation and Ritz projection, the superclose and superconvergent estimates for the above two variables in H1-norm are obtained under the lower regularity of the exact solution compared with the standard Galerkin FEM. Finally, some numerical results are provided to show the correctness of the theoretical predictions and good performance of the proposed method.

Modern management of leukemoid reactions

ABSTRACT. Leukemoid reaction (LR) is a transient hematological syndrome of reactive nature characterized by persistent leukocytosis of more than 50,000 cells/µL. LR can be caused by a variety of factors, including infections, intoxications, malignant neoplasms, severe hemorrhages, or acute hemolysis.
 The most common form of LR is neutrophilic myeloid reaction, which occurs in 80-90 % of cases. The diagnosis of LR is based on clinical examination and laboratory test results, including complete blood count with differential leukocyte formula, determination of leukocyte alkaline phosphatase level, aspiration and biopsy of bone marrow.
 Differential diagnosis of LR from other forms of leukemia is a challenging task. For this, it is necessary to conduct a thorough examination, including cytogenetic and molecular analysis of peripheral blood and bone marrow granulocytes. Despite certain advances in the differential diagnosis of LR, the mortality rate of patients in the world remains high, which requires the attention of doctors of various profiles.
 Treatment of LR depends on the underlying cause that caused it. The article proposes a diagnostic and treatment algorithm for LR in the case of its toxico-infectious origin.

Fractional Block Method for the Solution of Fractional Order Differential Equations

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be \textit{A}-stable for different values of α within the interval (0,1) . Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order α lies within the range of 0 and 1 . Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods.

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

SHoP: A Deep Learning Framework for Solving High-Order Partial Differential Equations

Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is widely used to approximate the solutions of PDEs. However, existing works are incapable of solving high-order PDEs due to insufficient calculation accuracy of higher-order derivatives, and the final network is a black box without explicit explanation. To address these issues, we propose a deep learning framework to solve high-order PDEs, named SHoP. Specifically, we derive the high-order derivative rule for neural network, to get the derivatives quickly and accurately; moreover, we expand the network into a Taylor series, providing an explicit solution for the PDEs. We conduct experimental validations four high-order PDEs with different dimensions, showing that we can solve high-order PDEs efficiently and accurately. The source code can be found at https://github.com/HarryPotterXTX/SHoP.git.

Oscillator Simulation with Deep Neural Networks

The motivation behind this study is to overcome the complex mathematical formulation and time-consuming nature of traditional numerical methods used in solving differential equations. It seeks an alternative approach for more efficient and simplified solutions. A Deep Neural Network (DNN) is utilized to understand the intricate correlations between the oscillator’s variables and to precisely capture their dynamics by being trained on a dataset of known oscillator behaviors. In this work, we discuss the main challenge of predicting the behavior of oscillators without depending on complex strategies or time-consuming simulations. The present work proposes a favorable modified form of neural structure to improve the strategy for simulating linear and nonlinear harmonic oscillators from mechanical systems by formulating an ANN as a DNN via an appropriate oscillating activation function. The proposed methodology provides the solutions of linear and nonlinear differential equations (DEs) in differentiable form and is a more accurate approximation as compared to the traditional numerical method. The Van der Pol equation with parametric damping and the Mathieu equation are adopted as illustrations. Experimental analysis shows that our proposed scheme outperforms other numerical methods in terms of accuracy and computational cost. We provide a comparative analysis of the outcomes obtained through our proposed approach and those derived from the LSODA algorithm, utilizing numerical techniques, Adams–Bashforth, and the Backward Differentiation Formula (BDF). The results of this research provide insightful information for engineering applications, facilitating improvements in energy efficiency, and scientific innovation.

On the Two-Variable Analogue Matrix of Bessel Polynomials and Their Properties

In this paper, we explore a study focused on a two-variable extension of matrix Bessel polynomials. We initiate the discussion by introducing the matrix Bessel polynomials involving two variables and derive specific differential formulas and recurrence relations associated with them. Additionally, we present a segment detailing integral formulas for the extended matrix Bessel polynomials. Lastly, we introduce the Laplace–Carson transform for the two-variable matrix Bessel polynomial analogue.

New Summation and Integral Representations for 2-Variable (p,q)-Hermite Polynomials

In this paper, we introduce and study new features for 2-variable (p,q)-Hermite polynomials, such as the (p,q)-diffusion equation, (p,q)-differential formula and integral representations. In addition, we establish some summation models and their (p,q)-derivatives. Certain parting remarks and nontrivial examples are also provided.

Implementation of Caputo type fractional derivative chain rule on back propagation algorithm

Fractional gradient computation is a challenging task for neural networks. In this study, the brief history of fractional derivation is abstracted, and the core framework of the Faà di Bruno formula is implemented to the fractional gradient computation problem. As an analytical approach to solve the chain rule problem of fractional derivatives, the use of the Faà di Bruno formula for the Caputo-type fractional derivative is proposed. When the fractional gradient is calculated using the proposed approach, the problem of determining the bounds of the formula for calculating the Caputo derivative is addressed. As a consequence, the fundamental problem with the fractional back-propagation algorithm is solved analytically, paving the way for the use of any differentiable and integrable activation function in case they are stable. The development of the algorithm and its practical implementation on the photovoltaic fault detection data-set is investigated. The reliability of the neural network metrics is also investigated using analysis of variance (ANOVA), the results are then presented to decide which are the best metrics and the best fractional order. Ordinary back-propagation, fractional back-propagation with and without L2 regularization and momentum are presented in the results to show the advantages of using L2 regularization and momentum in fractional order gradient. Consequently, the test metrics are reliable but cannot be separated from each other. By changing the batch size from 2 to full batches, the proposed system performs better with bigger batches, but adaptive moment estimation (ADAM) performs better with small batches. The cross validation shows the performance of back propagate ion neural networks have better performance than the ordinary neural networks. It is interesting that the best order for a specific data-set cannot be determined because it depends on the batch size, number of epochs and the cross-validation folds .

Dynamic modeling and characteristics analysis of sliding vane in a high-speed rotary compressor considering groove distribution and multi-body coupling

As one significant component in high-speed rotary compressors, sliding vane undergoes complex motions within the vane slot. The requirements for high-performance compressors result in the applications of low-viscosity lubrication techniques and high working frequency, thus worsening the operation conditions of the vane. To investigate the service performance of the sliding vane, a new tribo-dynamic model considering groove distribution and multi-body coupling is established in this study. Based on the solution with the modified extended backward differentiation formulae (MEBDF), the tribo-dynamic behaviors of the vane-related friction pairs are revealed. After comparison, the optimal scheme of the vane width is determined as 3.188 mm, with which the maximum wear load of the vane slot on the suction side is reduced obviously and the wear volume changes little. With the conical structure scheme of Lcs=1.5mm and hcs=1μm, the maximum wear load of the vane slot can be greatly reduced, thus increasing the wear uniformity. To improve the operation performance of the vane-slot conjunction, the lower suction pressure is suggested. This work provides comprehensive information for manufacturers to understand the working state of the sliding vane and conduct the optimal design.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

Partial Soft Derivative

The concept of soft derivative, introduced by Molodtsov in 1999, is one of the fundamental concepts in soft analysis. The handled paper defines partial soft derivative and studies some of its basic properties, such as the relation between partial soft derivative and boundedness, some basic partial soft derivative rules, e.g., sum rule, constant multiple rule, and difference rule, the relation between soft derivative and partial soft derivative, the relation between classical partial derivative and partial soft derivative, and the geometric interpretation of partial soft derivative. Moreover, it exemplifies the theoretical part of the study and provides figures for the geometric interpretation. Finally, this study discusses the need for further research.

Model order reduction based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.