Finite Series Research Articles

The Extended Weierstrass Transformation Method for the Biswas–Arshed Equation with Beta Time Derivative

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.

Eliminating blowing-ups and evanescent waves when using the finite series technique in evaluating beam shape coefficients for some T-matrix approaches, with the example of Gaussian beams

When evaluating beam shape coefficients which encode the description of laser beams, for use in some T-matrix approaches such as generalized Lorenz-Mie theory or Extended Boundary Condition Method for structured beams, by using finite series, blowing-ups are observed. When numerical inaccuracies are ruled out, it has been firmly demonstrated that such blowing-ups correspond to genuine physical phenomena, namely they describe evanescent waves. We propose a method to eliminate these blowing-ups and the corresponding evanescent waves (at least most of them).

Binomial summation formulae concerning harmonic numbers

This study establishes various summation formulas involving harmonic and generalized harmonic numbers, as well as binomial coefficients. Additionally, several finite series identities and their limits related to $H_{2k}$, $H^{(2)}_{2k}$ and $O_k$ are derived as specific instances.

Reservoir computing with logistic map.

Recent studies on reservoir computing essentially involve a high-dimensional dynamical system as the reservoir, which transforms and stores the input as a higher-dimensional state for temporal and nontemporal data processing. We demonstrate here a method to predict temporal and nontemporal tasks by constructing virtual nodes as constituting a reservoir in reservoir computing using a nonlinear map, namely, the logistic map, and a simple finite trigonometric series. We predict three nonlinear systems, namely, Lorenz, Rössler, and Hindmarsh-Rose, for temporal tasks and a seventh-order polynomial for nontemporal tasks with great accuracy. Also, the prediction is made in the presence of noise and found to closely agree with the target. Remarkably, the logistic map performs well and predicts close to the actual or target values. The low values of the root mean square error confirm the accuracy of this method in terms of efficiency. Our approach removes the necessity of continuous dynamical systems for constructing the reservoir in reservoir computing. Moreover, the accurate prediction for the three different nonlinear systems suggests that this method can be considered a general one and can be applied to predict many systems. Finally, we show that the method also accurately anticipates the time series of the all the three variable of Rössler system for the future (self-prediction).

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

A study on yield calculation method using suspension power chart in beam pumping systems

Abstract Accurate production prediction plays a crucial role in well fault diagnosis and parameter optimization, significantly contributing to the advancement of oilfield digital construction. This paper focuses on the dynamic analysis of beam pumping machines to establish a calculation model for the power diagram of rod pumping systems. The solution involves employing the finite difference method and Fourier series method, facilitating the transition from the suspension point work diagram to the pump work diagram. The effective stroke of the plunger is determined by analyzing the opening and closing points of the Ver using the maximum curvature analysis method. Subsequently, the leakage and volume coefficients are derived by integrating on-site production data, culminating in the realization of an oil well production calculation method based on the suspension work diagram. The compilation of the pump power chart and oil well production calculation program was executed using QT 6.2.3. Comparative analysis with measured production data revealed an average accuracy of 90.05% using the finite difference method, surpassing the performance of the Fourier series algorithm. The established model not only offers a reliable means of predicting production but also provides valuable technical support for the digital and intelligent applications within the oilfield.

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

Trajectory planning of a closed five-bow-shaped bar linkage based on finite Fourier series

With aim to reduce the energy consumption, a trajectory planning method is presented for a closed five-bow-shaped bar linkage, which can be propelled itself by morphing configuration. The objective herein is to optimize the driving joints trajectories within the global feasible region when the linkage rolls along the ground with a desired acceleration. The driving joint trajectories were represented by finite Fourier series, whose coefficients were solved by genetic algorithm to ensure a minimal energy consumption of the linkage. The impact of the number of terms of finite Fourier series on the energy consumption was also discussed through numerical examples. As a result, the energy consumption based on this strategy had been reduced by 19%, comparing with the constant potential energy strategy. A number of terms between six and eight using to denote the joint trajectories are appropriate, because that a small number of terms is incapable of expressing the joint trajectories accurately, whereas, a large number makes the joints to be subjected to vibration shock. At last, simulation on a virtual model and experiments on a prototype were carried out to verify the effectiveness of the proposed method.

Approximate analytical solutions for three-dimensional ascent trajectory of a solid-fuel launch vehicle with time-varying mass flow rate

In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management, Approximate Analytical Solutions (AAS) for the three-dimensional (3D) ascent flight states are derived, which are the only solutions capable of considering time-varying Mass Flow Rate (MFR) at present. The uneven MFR makes the thrust vary nonlinearly and thus increases the difficulty of the problem greatly. The AAS are derived based on a 3D Generalized Ascent Dynamics Model (GADM) with a normalized mass as the independent variable. To simplify some highly nonlinear terms in the GADM, several approximate functions are introduced carefully, while the errors of the approximations relative to the original terms are regarded as minor perturbations. Notably, a finite series with positive and negative exponents, called Exponent-Symmetry Series (ESS), is proposed for function approximation to decrease the highest exponent in the AAS so as to reduce computer round-off errors. To calculate the ESS coefficients, a method of seeking the Optimal Interpolation Points (OIP) is proposed using the least-squares-approximation theory. Due to the artful design of the approximations, the GADM can be decomposed into two analytically solvable subsystems by a perturbation method, and thus the AAS are obtained successfully. Finally, to help implement the AAS, two indirect methods for measuring the remaining mass and predicting the burnout time in flight are put forward using information from accelerometers. Simulation results verify the superiority of the AAS under the condition of time-varying MFR.

Chebyshev polynomial-based Fourier transformation and its use in low pass filter of gravity data

In this paper, we introduce the novel Chebyshev Polynomials Least-Squares Fourier Transformation (C-LSQ-FT) and its robust variant with the Iteratively Reweighted Least-Squares technique (C-IRLS-FT). These innovative techniques for Fourier transformation are predicated on the concept of inversion, and the C-LSQ-FT method establishes an overdetermined inverse problem within the realm of Fourier transformation. However, given the LSQ approach’s vulnerability to data outliers, we note the potential for considerable errors and potentially unrepresentative model estimations. To circumvent these shortcomings, we incorporate Steiner’s Most Frequent Value method into our framework, thereby providing a more reliable alternative. The fusion of the Iteratively Reweighted Least-Squares (IRLS) algorithm with Cauchy-Steiner weights enhances the robustness of our Fourier transformation process, culminating in the C-IRLS-FT method. We use Chebyshev polynomials as the basis functions in both methods, leading to the approximation of continuous Fourier spectra through a finite series of Chebyshev polynomials and their corresponding coefficients. The coefficients were obtained by solving an overdetermined non-linear inverse problem. We validated the performance of both the traditional Discrete Fourier Transform (DFT) and the newly developed C-IRLS-FT through numerical tests on synthetic datasets. The results distinctly exhibited the reduced sensitivity of the C-IRLS-FT method to outliers and dispersed noise, in comparison with the traditional DFT. We leveraged the newly proposed (C-IRLS-FT) technique in the application of low-pass filtering in the context of gravity data. The results corroborate the technique’s robustness and adaptability, making it a promising method for future applications in geophysical data processing.

Finite and high-temperature series expansion via many-body perturbation theory: application to Heisenberg spin-1/2 XXZ chain

We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.

Evaluating the beam shape coefficients of Bessel–Gauss beams with radial quadrature: a comparison with angular spectrum decomposition and finite series methods

The Bessel–Gauss beam (BGB) stands as a physically realizable beam extensively employed in applications such as micromanipulation and optical trapping. In these contexts, the assessment of beam shape coefficients (BSCs) becomes imperative. Previous research reveals that the BSCs of the BGBs obtained with different methods deviate from each other under certain circumstances. In this paper, the formulation of BSCs employs the radial quadrature method, and a comparative analysis is conducted with counterparts formulated using the angular spectrum decomposition and the finite series technique. Contributions stemming from evanescent waves and the situation of the BSC blowing-ups are discussed, offering a deep insight of pertinent BSC evaluation methods. The paper provides an alternative approach for calculating the BSCs of the BGBs.

Haar wavelet collocation technique for numerical solution of porous media equations

This paper focuses on using efficient Haar wavelet collocation method to nonlinear parabolic differential equations in order to solve steady state pore-scale creeping fluid flow in porous medium, substituting finite Haar series for space derivatives, incontrast forward difference approach is used to estimate the time derivative for porous media equation. These equations has vast number of application in engineering and various other fields such as, in petroleum engineering, there are often situations where fluids are near critical conditions. As oil pressure reduces below the bubble point when gas forms liquid condensations, or when the surface tension become very low i.e, transition from miscibility to immiscibility. To model gas production by pressure decline requires an accurate description of physical mechanism involved in the appearance of the gas bubbles. The observed stability with theoretical convergence assertion, the conservation law of mass and energy are stated. This supports the convergence of the preferred paradigm. A convergent approximation to the porous media equation is obtained by suggested approach. By calculating the maximum error norm and the experimental rate of convergence for various problems, the method efficiency and dependability are demonstrated. The method is accurate, easily applicable, and efficient.

Robust Recursive Least-Squares Fixed-Point Smoother and Filter using Covariance Information in Linear Continuous-Time Stochastic Systems with Uncertainties

This study develops robust recursive least-squares (RLS) fixed-point smoothing and filtering algorithms for signals in linear continuous-time stochastic systems with uncertainties. The algorithms use covariance information, such as the cross-covariance function of the signal with the observed value and the autocovariance function of the degraded signal. A finite Fourier cosine series expansion approximates these functions. Additive white Gaussian noise is present in the observation of the degraded signal. A numerical simulation compares the estimation accuracy of the proposed robust RLS filter with the robust RLS Wiener filter, showing similar mean square values (MSVs) of the filtering errors. The MSVs of the proposed robust RLS fixed-point smoother are also compared to those of the proposed robust RLS filter.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Exact solutions for travelling waves using Tanh method for two dimensional stochastic Allen–Cahn equation with multiplicative noise

AbstractThe research outlined here focuses on the hyperbolic tangent (tanh) method for deriving analytical solutions for travelling wave to the two-dimensional stochastic Allen–Cahn equation with multiplicative noise. The novelty of our work is to derive these exact solutions since the previous studies focused on the solutions of numerical nature. The tanh approach, which employs a finite tanh power series, is particularly adept at modeling travelling wave profiles. A key area of interest in this study is the effect of multiplicative noise on these travelling waves dynamics, especially how high levels of noise can lead to propagation failure of waves. The results demonstrate that for weak noise, the propagation of the travelling wave is basically unaffected, while the wave fails to propagate in the strong noise regime.

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{\mu-1}}{x^\mu}[u(t)]^\beta dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu>1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{\gamma k_1}}{\gamma k_1+\mu}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu>1$. We extend the parameter range to include $\mu>1$ and $0<\mu\leq 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta\geq 2$, and positive rational parameter value $\mu>0$ and for any finite value of $n\geq2$.