Method For Numerical Solution Research Articles

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval − 1 , 1 is solved. The discontinuous solution on the domain − 1 , 1 is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.

Study on the dynamic characteristics of the suspender with additional dampers and a frequency-based multiple parameter identification method for the system

The measurement of tensile forces inside suspenders plays an important role in safety inspection and monitoring during construction and operation of large structural engineering projects, including arch bridges and suspension bridges. The frequency-based method is one of the most widely used methods for tension force measurement of such structure components, but it is affected by many factors, such as the additional damping from a damper and various boundary conditions. In this study, the equation of damped motion for a suspender-damper system is derived, a numerical solution method based on the finite difference scheme is given, and a frequency-based multiple parameter identification method is proposed based on the derivative of the system eigenvalue. The study shows that the position and damping coefficient of the damper exert an important effect on the frequency and the plural mode of the suspender-damper system. The influence mechanism is discussed in the study. The numerical examples indicate that the proposed parameter identification algorithm can identify multiple system parameters of the suspender with a high precision and the maximum error is no more than 1%. The influence of the errors in the frequencies is also discussed in this study.

An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

Numerical Analysis of Target Scattering Characteristics Based on Singularity Expansion Method

Based on the physical meaning of wave equation and Singularity Expansion Method, a numerical solution method is designed to extract the pole distributions of infinite cylinder, sphere and ellipsoid. By analysing the wave equation and boundary conditions of the target scattering characteristics, the extractions of poles of target scattering field are transformed into the solution of Hankel function and its derivative roots. Then, the T-matrix method is improved to make the numerical solution process of roots more accurate and faster. The numerical results are compared with strict analytical solutions and experimental results to fully verify the correctness of the method mentioned in this paper this study provides a basis for understanding the scattering mechanism of surface waves, and also provides a method and idea for solving problems such as acoustic scattering and electromagnetic scattering in engineering practice.

Characterization of high-power mechanical quality factor of piezoelectric ceramic discs under self-heating condition

We have developed a method to characterize high-power mechanical quality factor Qm of a thickness-poled piezoelectric disc under self-heating condition. In order to obtain Qm from temperature and vibration measurements, a heat transfer model for a thin piezoelectric disc vibrating in its fundamental radial mode was built, and a numerical solution method based on the finite difference approach was developed to investigate the temperature rise of the piezoelectric disc as a function of position and time. Numerical simulations were performed to verify the validity of the model and to quantify the influence of different material parameters on the temperature rise. A theoretical formula for characterizing Qm of a piezoelectric disc was deduced according to the definition of the mechanical quality factor, in which the relationship among the self-heating parameters (hg), the sample vibration level (v0) and Qm was established. Furthermore, using a self-designed experimental equipment, we have characterized the high-power mechanical quality factor Qm of a PZT8 disc sample under different vibration levels using a tone-burst excitation, which can increase the operating electric field strength limits. The disc results show that Qm degrades drastically with increasing vibration amplitude. Especially, the Qm values measured under self-heating conditions have lower magnitudes than those measured using transient/burst method, indicating self-heating effects do significantly contribute to Qm characterization.

Symmetry analysis, laws of conservation, and numerical and approximate analysis of Burger’s fractional order differential equation

The computation of fractional order differential and integration equations is highly utilized in numerous fields, such as mathematics in biology, ecology, physics, chemistry, and so on. This article aims at presenting methods for analytical and numerical solutions, Lie’s symmetry analysis, computing conservation laws for Burger’s fractional order differential equation (FODE), and reducing time-fractional cylindrical-Burgers equation order. Hence, we have employed a generalized (G′/G) expansion method for analytical solutions and a non-standard finite difference scheme for numerically solving Burger’s FODE. Additionally, the fractional derivative generalization of Noether's theorem has been utilized to compute the equation’s conservation laws. Numerical results have been reported here to approve theoretical results acquired in non-standard finite difference schemes. The proposed generalized (G′/G) expansion method is a simple, accurate, and practical approach to problem solving. Additionally, this method can be employed to execute non-linear wave equations. Programs including MATLAB and Maple have been utilized to simplify the process of solving complex equations.

Numerical solution of critical force of n-step telescopic boom with superlift device

The installation of a superlift device on the telescopic boom of a crane is the main method to improve the stress condition of the telescopic boom and to effectively improve the lifting performance of the crane. However, there is a lack of an effective and accurate numerical method to solve the critical force of a crane telescopic boom with a superlift device. Therefore, the telescopic boom with a superlift device is reasonably simplified as a stepped column and double cable model. Based on the longitudinal and transverse bending theory, the differential equations of the deflection of the n-stepped column with double cables model are established, the buckling characteristic equation and its recurrence formula of the n-stepped column with double cables model are derived, and the combined superlift equations are established. The Levenberg–Marquardt optimization algorithm is used to solve the critical force and length coefficient. A large amount of data were compared and analyzed between the numerical solution results and the ANSYS finite element simulation results. The error result analysis proved the correctness of the deduced buckling characteristic equation and its derived formula and the accuracy of the numerical solution. The numerical solution method in this paper can be used to calculate the critical force and length coefficient under any number of boom sections, boom section length, sectional moment of inertia, cable length, and included angle between the two cables of the telescopic boom with a superlift device. The numerical solution method can provide a technical support for the structural design of a telescopic boom with a superlift device in practical projects.

Mesh-free least-squares reverse-time migration

AbstractThe reflectivity of the subsurface can be precisely determined using least-squares reverse-time migration (LSRTM). As LSRTM necessitates solving the wave equation, the numerical solution method of the wavefield directly determines the quality of the migration image. The conventional LSRTM method usually uses the finite difference method based on a regular grid to calculate the wavefield. Due to the stepwise approximation of an irregular surface with a regular grid, scattering noise may occur in the propagation of the wavefield, which affects the quality of the image. In addition, the conventional LSRTM cannot effectively handle the models with rugged topography. The finite difference method generated by radial basis functions (FD-RBF) is a mesh-free method and can construct interpolation functions to solve the wave equations numerically according to arbitrarily distributed spatial coordinate points. Therefore, we use the FD-RBF method to develop a mesh-free LSRTM approach to eliminate the influence of the inherent limitation of a regular grid on the imaging. Numerical examples show that the mesh-free LSRTM method can better represent the curved or steep interface within the model and is also suitable for models with rugged topography. The LSRTM method can provide higher-quality images and effectively reduce the memory required for calculations.

Series Solution of the Time-Dependent Schrödinger–Newton Equations in the Presence of Dark Energy via the Adomian Decomposition Method

The Schrödinger–Newton model is a nonlinear system obtained by coupling the linear Schrödinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper, we investigate the effects of dark energy on the time-dependent Schrödinger–Newton equations by including a new source term with energy density proportional to the cosmological constant Λ, in addition to the particle-mass source term. The resulting Schrödinger–Newton–Λ (S-N-Λ) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-Λ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-Λ system to be obtained quickly, even with minimal computational resources. The extensive use of the Adomian Decomposition Method in the field of quantum mechanics and quantum field theory may open new mathematical, and physical, perspectives on obtaining semi-analytical solutions for some complex problems of quantum theory.

An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order 1/4 in L1 and L2 for the parameter regime κθ>σ2. We then extend the new method to cover all parameter values by introducing a soft zero region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order 1 when κθ>σ2 rather than 1/4. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.

CCGnet: A deep learning approach to predict Nash equilibrium of chance-constrained games

This paper proposes a novel method for efficiently finding the Nash equilibrium in a chance-constrained game (CCG). Conventional numerical solution methods require significant computational time when solving multiple instances of CCG. We introduce CCGnet, a deep learning approach that is capable of efficiently solving multiple instances of CCG in a one-shot manner. CCGnet employs a specialized network structure and training algorithm based on neurodynamic optimization. We demonstrate the strong performance of CCGnet in practice and show that our proposed method outperforms conventional methods.

Продольная динамическая магнитная восприимчивость одноосных антиферромагнитных наночастиц: влияние подмагничивающего поля

This paper develops a kinetic theory of the low-frequency (far from the microwave range) longitu-dinal linear magnetic response of uniaxial antiferromagnetic nanoparticles. The proposed formal-ism is based on the Fokker-Planck-type equation for the distribution function of orientations of the antiferromagnetic vector. For this equation, a method of numerical solution is developed for the situation when a harmonic linearly polarized magnetic field and a constant bias field are co-aligned and applied along the anisotropy axis of the particle. Frequency dependencies of the real and imag-inary parts of the longitudinal dynamic magnetic susceptibility of the particle are calculated for dif-ferent values of the bias field magnitude. It is shown that in the presence of a constant field the magnetic response of the particle to a probing field decreases and the absorption maximum shifts to higher frequencies. Provided that the bias field is non-zero, the position of the specified maximum is not fixed but changes with the value of the uncompensated magnetization. It has been established that in the low frequency range, dispersion of the longitudinal dynamic magnetic susceptibility of the uniaxial antiferromagnetic nanoparticle is determined by a single Debye-type term.

Providing mechanisation of loading and unloading operations in hopper rail stores

Grain and fertiliser transportation are the fastest growing segments of the transportation services market, including railway transportation. Presidential Decree No. 204 of 7 May 2018 “On the national goals and strategic objectives for the development of the Russian Federation for the period until 2024” tasked the Russian government with developing a programme of measures to increase the productivity of economic sectors, including the agroindustrial complex. The increase in the productivity of the latter should entail the development of both domestic consumption and the export component, which places an additional burden on rail transport. The aim of the research is to improve loading and unloading operations in bunker-type railway yards by using auxiliary constructions to speed up and facilitate the loading and unloading of rolling stock by bulk cargo. The research is based on the principles of the system approach using methods of mathematical modelling of technical systems, numerical methods of problem solution and rheology engineering methods. The results of the research are resource-saving ways of bulk cargo loading and unloading from containers of various purposes, allowing to increase the degree of their uniform distribution in the container, and to ensure smooth release of material.

SIMULATING THE BEHAVIOR OF THE POPULATION DYNAMICS USING THE NON-LOCAL FRACTIONAL CHAFFEE–INFANTE EQUATION

In recent years, there has been growing interest in fractional differential equations, which extend the concept of ordinary differential equations by including fractional-order derivatives. The fractional Chaffee–Infante ([Formula: see text]) equation, a nonlinear partial differential equation that describes physical systems with fractional-order dynamics, has received particular attention. Previous studies have explored analytical solutions for this equation using the method of solitary wave solutions, which seeks traveling wave solutions that are localized in space and time. To construct these solutions, the extended Khater II ([Formula: see text]) method was used in conjunction with the properties of the truncated Mittag-Leffler ([Formula: see text]) function. The resulting soliton wave solutions demonstrate how solitary waves propagate through the system and can be used to investigate the system’s response to different stimuli. The accuracy of the solutions is verified using the variational iteration [Formula: see text] technique. This study demonstrates the effectiveness of analytical and numerical methods for finding accurate solitary wave solutions to the [Formula: see text] equation, and how these methods can be used to gain insights into the behavior of physical systems with fractional-order dynamics.

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

Modeling One Dimensional Two-Cell Model with Tumor Interaction Using Krylov Subspace Methods

A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.

Performance of Inexpensive Local Error Estimation Techniques for Integral Equation Numerical Solutions

The performance of several inexpensive local error estimation techniques is evaluated in connection with the Rao-Wilton-Glisson method of moments numerical solutions of the electric field integral equation. Results for 18 perfectly conducting test targets are used to evaluate the performance of the estimators. Two of the estimators produce error maps that consistently exhibit high correlations with reference solutions. These estimators are also suitable for “goal-oriented” estimation of secondary quantities, such as identifying cells that contribute the most error to the radar cross section of the target.

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function