Solvability Of Nonlinear Equations Research Articles

Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations

BackgroundNumerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping.ResultsThe results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s.ConclusionsThe solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.

Self-Mixing Interferometer for Acoustic Measurements through Vibrometric Calibration.

The Self-Mixing Interformeter (SMI) is a self-aligned optical interferometer which has been used for acoustic wave sensing in air through the acousto-optic effect. This paper presents how to use a SMI for the measurement of Sound Pressure Level (SPL) in acoustic waveguides. To achieve this, the SMI is first calibrated in situ as a vibrometer. The optical feedback parameters C and α in the strong feedback regime (C≥4.6) are estimated from the SMI vibrometric signals and by the solving of non-linear equations governing the SMI behaviour. The calibration method is validated on synthetic SMI signals simulated from SMI governing equations for C ranging from 5 to 20 and α ranging from 4 to 10. Knowing C and α, the SMI is then used as an acoustic pressure sensor. The SPLs obtained using the SMI are compared with a reference microphone, and a maximal deviation of 2.2 dB is obtained for plane waves of amplitudes ranging from 20 to 860 Pa and frequencies from 614 to 17,900 Hz. The SPL measurements are carried out for C values ranging from 7.1 to 21.5.

Response surface method and finite element-based investigation of uniform/nonuniform hydromagnetic convection of nanofluids inside a quarter-circular enclosure

Under the influence of magnetohydrodynamics (uniform, periodic, and inclined), the current research evaluates the two-dimensional unsteady dynamics of natural convection flow and heat transportation in the quarter-circular domain considering the Brownian motion of nanoparticles. The nonlinear partial differential equations (PDEs) solver finite element method (FEM) of the Galerkin type is used for the numerical simulation. In addition, the central composite design-based response surface methodology (RSM) is applied to analyze the optimization of the outcomes. To assess the models’ significance in learning the optimum thermal transport efficiency of nanofluids, an analysis of variance is performed. The impacts of several physical characteristics such as Rayleigh number, nanoparticles volume fraction, mean Nusselt number, Brownian effects, size, and shape of the nanoparticles, and magnetic effects (uniform, periodic, and inclined) are examined. The heat transfer rate is significantly influenced by the magnetic field’s period and inclined angle. As an illustration, the optimal heat transmission is noted at λ = 0.75 and γ = 0°. When Brownian effects are included, the heat transfer rate increases by 23.44% and it rises by 39.34% as nanoparticle size drops from 100 nm to 10 nm. Moreover, compared to spherical nano-sized particles, blade-shaped nano-sized particles can improve the transportation of heat by 9.56%. Furthermore, the optimization of the independent factors (Ra, Ha, ϕ, and d p) on the response function Nu (average) is demonstrated by the statistical RSM technique, and a novel correlation is proposed based on FEM numerical data. The optimal thermal transport (Nu av = 24.045) is found when Ra = 10, Ha = 0, ϕ = 0.05, and d p = 10 nm. The model’s average Nusselt number’s R 2-value of 0.9927 indicates that the results are effective.

Physics-informed ConvNet: Learning physical field from a shallow neural network

We introduce a novel methodology for solving nonlinear partial differential equation (PDE) on regular or irregular domains using physics-informed ConvNet, which we call the PICN. The network structure consists of three parts: 1) a convolutional neural network for physical field generation, 2) a pre-trained convolutional layer corresponding to the finite-difference filters to estimate differential fields of the generated physical field, and 3) an interpolation network for loss analysis in irregular geometry domains. From a CNN perspective, the physical field is generated by a deconvolution layer and a convolution layer. Unlike the standard Physics-informed Neural Network (PINN) approach, the convolutions corresponding to the finite-difference filters estimate the spatial gradients forming the physical operator and then construct the PDE residual in a PINN-like loss function. The total loss function involving boundary conditions and the physical constraints in irregular geometry domains can be calculated from an efficient linear interpolation network. The theoretical analysis of PICN convergence is performed on a simplified case for solving a one-dimensional physical field, and several examples of nonlinear PDE of solutions with multifrequency characteristics are executed. The theory and examples confirm the effective learning capability of PICN for the physical field solution with high-frequency components, compared to the standard PINN. A series of numerical cases are performed to validate the current PICN, including the solving (and estimation) of nonlinear physical operator equations and recovering physical information from noisy observations. First, the ability of PICN to solve nonlinear PDE has been verified by executing three nonlinear problems including ODE with sine nonlinearity, PDE involving nonlinear sine-square operators, and Schrödinger equation. The proposed PICN has been assessed by solving some nonlinear PDE on irregular domains such as star-shaped domain, bird-like domain, and starfish domain. Moreover, PICN is applied to identify the thermal diffusivity parameters in an anisotropic heat transfer problem from noisy data, and a denoising display of the temperature field from strong noisy data with standard deviations ranging from 0.1 to 0.4. The numerical results demonstrate the high accuracy approximation and fast convergence performance of PICN. The potential advantage in approximating complex physical field with multi-frequency components indicates that PICN may become an alternative efficient neural network solver in physics-informed machine learning. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.10967, Jan 26, 2022). The data and code accompanying this paper are publicly available at https://github.com/zengzhi2015/PICN.

Advancing Convergence Rates in Nonlinear Equation Solvers

It is crucial to achieve high-order convergence for nonlinear equations in numerical analysis and scientific computing. Newton-Raphson and other conventional techniques frequently converge slowly. High-order algorithms offer quicker convergence and less expensive processing.This abstract outlines important ideas and innovations, such as improved Newtonian methods for better convergence. To hasten convergence, these techniques make use of enhanced Jacobian matrix approximations and higher-order derivatives. Additionally, it covers useful topics like dealing with stability problems and computing higher-order derivatives, which could lead to efficiency and accuracy gains in scientific and engineering applications. Keywords: Non- linear Equations, High Order Convergence, Numerical Methods, Jacobian Matrix, High Order Derivatives

A predictive model for the techno-economic assessment of CO[formula omitted] chemisorption processes applicable to a large number of amine solvents

With the growing need to reduce carbon dioxide (CO2) emissions, there have been substantial efforts to identify new solvents that can improve the overall performance of chemical-absorption CO2 capture processes. Given the large number of potential solvents, computer-aided molecular and process design (CAMPD) approaches can play a critical role in accelerating the search for optimal solvents by enabling the systematic exploration of solvent candidates and process conditions. One of the challenges in developing such a framework is the requirement for a process model that can be used to capture the interactions between process performance and solvent structure without significantly increasing its numerical complexity. In the current work, a model for the absorption–desorption of CO2 is developed using the predictive SAFT-γ-Mie group-contribution approach, allowing the performance of numerous solvents to be assessed without the need for extensive experimental data. In order to avoid convergence difficulties when solving this highly nonlinear model, a tailored initialization strategy is established, using an adapted inside-out algorithm to prime a nonlinear equation solver for each column, providing a good initial guess for the whole flowsheet. Tests on three solvents confirm the robustness of the approach. Building on this enhanced numerical stability, the model is validated by comparison against pilot-plant data, showing good accuracy. A detailed parametric study of the effect of the key process variables is undertaken; the important role of the CO2 capture rate, of the lean solvent temperature and loading, and of the desorber pressure is highlighted. The results of the parametric study are used to formulate an optimization problem which is successfully solved for four solvents. A large reduction in total annualized cost and energy requirements is achieved by tuning the operating conditions to each solvent considered. The predictive capability, robustness, and reliability of the proposed model and associated initialization strategy open the way for the evaluation of a large number of novel solvents.

Combination of Optimal Three-Step Composite Time Integration Method with Multi-Point Iterative Methods for Geometric Nonlinear Structural Dynamics

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.

A CONSTRUCTIVE INVESTIGATION BY THE METHODS OF TWO-SIDED APPROXIMATIONS OF BOUNDARY PROBLEMS FOR SECOND-ORDER SEMILINEAR ELLIPTIC EQUATIONS

The paper considers the first boundary value problem for a second-order semilinear elliptic equation. Problems of this class often arise in the modeling of processes occurring in chemistry, physics, biology, etc. A special place among the methods of analysis of the problems under consideration is occupied by the so-called constructive research methods, which allow not only to prove the existence of a problem solution, but also offer an algorithm for finding it with a given accuracy. For a constructive study of a nonlinear boundary value problem, it is proposed to use two versions of the method of two-sided approximations. Both methods are based on the transition from a differential problem to an equivalent nonlinear integral equation (using the Green’s function or using the Green-Rvachev’s quasi-function), which is analyzed by methods of the theory of nonlinear operators in semi-ordered Banach spaces. Conclusions about the existence of positive solutions of the constructed integral equations and the two-sided convergence of successive approximations to these solutions are made on the basis of the results of V.I. Opojcev on the solvability of nonlinear equations with a heterotone operator. The practical implementation of the method of two-sided approximations based on the use of the Green's function has certain limitations associated with the need to have an explicit expression for this function, which narrows the range of areas in which the method can actually be applied. The method of two-sided approximations, based on the use of the Green-Rvachev’s quasi-function, which can be constructed using the apparatus of the R-functions theory for regions of sufficiently arbitrary geometry, is free from this shortcoming. The proposed methods are illustrated by computational experiments for elliptic equations with Laplace and Helmholtz operators and heterotone power nonlinearity in a number of two- and three-dimensional domains. The results of both methods of two-sided approximations were compared with each other.

Multipurpose modified iterative solver for nonlinear equations

Non-linear Eq.s occur as a sub-problem in a wide variety of engineering and scientific domains. To deal with the complexity of Non-linear Eq.s, it is often required to use numerical procedures, which are the most suitable method to employ in certain circumstances. Many classic iterative approaches have been regularly employed for various situations; nevertheless, the convergence rate of those methods is low. In many cases, an iterative approach with a faster convergence rate is needed. This is something that classical methods like the Newton-Raphson Method (NRM) cannot provide. As part of this investigation, a modification to the NRM has been suggested to speed up convergence rates and reduce computational time. Ultimately, this research aims to improve the NRM, resulting in a Modified Iterative Method (MIM). The proposed method was thoroughly examined. According to the research, the convergence rate is higher than that of NRM. The proposed method delivers more accurate results while reducing computational time and requiring fewer iterations than earlier methods. The numerical findings confirm that the promised performance is correct. The results include the number of iterations, residuals, and computing time. This innovative technique, which is appropriate to any Non-linear equation, produces more accurate approximations with less iteration than conventional methods, and it is appropriate to any Non-linear equation.

Two-Stage Reinforcement Learning-Based Differential Evolution for Solving Nonlinear Equations

Solving nonlinear equations (NEs) requires the algorithm to locate multiple roots of NEs in one run. In this article, a generic framework based on two-stage reinforcement learning (RL) and differential evolution (DE) is proposed to effectively deal with NEs problems. The major advantage of our approach are: 1) different niching methods and mutation strategies are integrated into the DE algorithm to assist evolution; 2) the diversity is maintained by utilizing the search characteristics of different niching methods at population level; 3) additionally, each individual is regarded as an agent, and three classical mutation strategies are used as the agent’s alternative actions; and 4) different state settings and reward function can be easily integrated into this framework. To verify the performance of our approach, 30 problems and 18 new NEs are selected as the test suite. The experimental results demonstrate that RL can facilitate the algorithm and improve the problem-solving ability. Moreover, the proposed method also obtains competitive performance compared with other peer algorithms.

Efficient Solvers for Nonlinear Partial Differential Equations in Computational Fluid Dynamics

Improving the computational efficiency of solving nonlinear partial differential equations (PDEs) is a key area of focus for this research, which has the ability to significantly impact computational fluid dynamics (CFD) simulations. The complexities of fluid flow phenomena can be difficult for traditional methods to handle, leading to simulations that are resource-intensive and computationally slow. The urgent requirement for improved computational tools capable of quickly and precisely simulating complex fluid dynamics scenarios is the driving force behind the significance of this research. New approaches are needed to solve nonlinear partial differential equations (PDEs) in computational fluid dynamics (CFD), which are characterized by turbulent flows, fluid-structure interactions, and high computing resource intensity. Adaptive mesh optimization and high-performance computing architectures are combined in the AMO-HPI technique, which provides a strategic solution. Adaptive Mesh Optimization based on High-Performance Integration (AMO-HPI) is suggested in this research as a way to improve the accuracy of solutions by focusing computational resources on regions of interest through adaptive mesh refinement and coarsening. Making effective use of shared-memory parallelism and message-passing interfaces (MPI) to divide up computational tasks. Improvements in engineering system performance, reliability, and safety can be achieved by the effective simulation of fluid dynamics processes, which opens up new design areas that were previously unreachable. When compared to other methods, AMO-HPI performs better in simulations due to its optimal resource utilization and decreased processing time. In addition to improving CFD methodology, this research could have far-reaching effects on engineering design processes in several industries, which would be great for innovation and sustainable development.

New submodels describing attenuation of the ultrasonic beams in 3-d cubic nonlinear medium in the absence of dissipation

For a three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, we studied the attenuation of ultrasonic beams after the formation of shock fronts. Earlier, in the work of one of the authors for this model, submodels were obtained and studied, described by non-stationary solutions, invariant with respect to some three-dimensional subgroups of the main ten-parameter group of the differential equation that defines this model. In our paper we obtained and studied four non-stationary submodels of this model that are invariant with respect to four-parameter subgroups of the main group of this differential equation. These submodels are new submodels and have not been previously noted in the literature. They are given by invariant solutions of rank 1. Among these 4 submodels, 2 submodels describe axisymmetric ultrasonic beams, the remaining 2 describe one-dimensional ultrasonic beams. The search for invariant solutions that define these 4 submodels is reduced to solving of nonlinear integral equations, the implicit solutions of which are obtained in the form of nonlinear algebraic equations containing transcendental functions. These submodels are used to study the propagation of ultrasonic beams, for which either the acoustic pressure and its rate of change or the acoustic pressure and its gradient are given at the initial time at a fixed point. Conditions are obtained that ensure the existence and uniqueness of solutions to boundary value problems describing these processes. This makes it possible to correctly carry out numerical calculations in the study of these processes. As a result of the numerical solution of these boundary value problems for some values of the parameters characterizing these processes, pressure distribution graphs were obtained. In all cases, ultrasonic beams are weaken monotonically with time and completely fade away in a finite time. At each point, the time of complete attenuation of ultrasonic beams is found.The obtained and studied new submodels are another step towards the creation of a database of physically significant submodels of the three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, which describes the attenuation of ultrasonic beams after the formation of shock fronts.

New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations

This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to 43≈1.587 and 84≈1.681, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods.

An accurate and efficient numerical solution for the generalized Burgers–Huxley equation via Taylor wavelets method: Qualitative analyses and Applications

This paper aims to propose a highly accurate and simple algorithm based on the Taylor wavelet methods for obtaining the approximate solution of the generalized Burgers–Huxley equation. Additionally, various qualitative analyses including positivity-preservation, monotonicity-preservation, boundedness of the obtained solutions as well as convergence analysis of the proposed method have been provided. Furthermore, the applicability and validity of the proposed method are demonstrated on a benchmark equation. By comparing the approximate solutions with the exact solution, it is observed that the proposed method has recorded better results than the other methods in the literature. Although the generalized Burgers–Huxley equation has been studied throughout the study, it is worth emphasizing that the proposed method is a good solver for such nonlinear equations.

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

A Dimension-reduction method for the finite-horizon spacecraft pursuit-evasion game

<p style='text-indent:20px;'>The finite-horizon two-person zero-sum differential game is a significant technology to solve the finite-horizon spacecraft pursuit-evasion game (SPE game). Considering that the saddle point solution of the differential game usually results in solving a high-dimensional (24 dimensional in this paper) two-point boundary value problem (TPBVP) that is challengeable, a dimension-reduction method is proposed in this paper to simplify the solution of the 24-dimensional TPBVP related to the finite-horizon SPE game and to improve the efficiency of the saddle point solution. In this method, firstly, the 24-dimensional TPBVP can be simplified to a 12-dimensional TPBVP by using the linearization of the spacecraft dynamics; then the adjoint variables associated with the relative state variables between the pursuer and evader can be expressed in the form of state transition; after that, based on the necessary conditions for the saddle point solution and the adjoint variables in the form of state transition, the 12-dimensional TPBVP can be transformed into the solving of 6-dimensional nonlinear equations; finally, a hybrid numerical algorithm is developed to solve the nonlinear equations so as to obtain the saddle point solution. The simulation results show that the proposed method can effectively obtain the saddle point solution and is robust to the interception time, the orbital altitude and the initial relative states between the pursuer and evader.</p>

A fast and efficient Newton-type iterative scheme to find the sign of a matrix

<abstract><p>This work proposes a new scheme under the umbrella of iteration methods to compute the sign of an invertible matrix. To this target, a review of the exiting solvers of the same type is given and then a new scheme is derived based on a multi-step Newton-type nonlinear equation solver. It is shown that the new method and its reciprocal converge globally with wider convergence radii in contrast to their competitors of the same order from the general Padé schemes. After investigation on the theoretical parts, numerical experiments based on complex matrices of various sizes are furnished to reveal the superiority of the proposed solver in terms of elapsed CPU time.</p></abstract>

Two Practical Methods for the Forward Kinematics of 3-3 Type Spatial and 3-RRR Planar Parallel Manipulators

The forward kinematics in parallel manipulators is a mathematically challenging issue, unlike serial manipulators. Kinematic constraint equations are non-linear transcendental equations that can be reduced to algebraic equations with appropriate transformations. For this reason, sophisticated and time-consuming methods such as the Bezout method, the Groebner bases method, and the like, are used. In this paper, we demonstrate that these equations can be solved by non-complicated mathematical methods for some special types of manipulators such as the 3-3 and 6-3 types of Stewart platforms, and the 3-RRR planar parallel manipulator. Our first method is an analytical approach that exploits the special structure of kinematic constraint equations and yields polynomials of 32nd and 16th order, as mentioned in the previous works. In the second method, an error function is defined. This error function is employed to find the most appropriate initial values for the non-linear equation solver which is used for solving kinematic constraint equations. Determining the initial values in this manner saves computation time and guarantees fast convergence to real solutions.

Resonance in bounded nonlinear pendulum-type systems

Solving nonlinear differential equations with external forces is important for understanding resonant phenomena in the physics of oscillations. The article analyzes this problem basing on example of an ordinary second-order differential equation of the pendulum type, where the nonlinearity is described by a sinusoidal term. The phase plane of such an oscillator is constructed and its periodic trajectories are studied. It is illustrated that bounded nonlinearity matters only at intermediate amplitudes. The excitation of a nonlinear oscillator is carried out using a limited two–component force; the first its component corresponds to an oscillation at the resonant frequency of a linear oscillator, and the second is a limited function with a variable frequency. It is shown that with the appropriate choice of an external force, it is possible to obtain unlimited amplification of oscillations in a pendulum-type oscillator with amplitude linearly proportional to time. Spectral composition of the external force is investigated using short-time Fourier transform. It is demonstrated that in order to maintain the resonant mode, the frequency of the external force must continuously increase. Energy estimates of the external force and oscillator fluctuations depending on time are performed. The considered example is important for understanding resonant conditions in nonlinear problems.

Updated and Weaker Convergence Criteria of Newton Iterates for Equations

Newton iteration is often used as a solver for nonlinear equations in abstract spaces. Some of the main concerns are general: Criteria for convergence, error estimations on consecutive iterates, and the location of a solution. A plethora of authors has addressed these concerns by presenting results based on the celebrated Kantorovich theory. This article contributes in this direction by extending earlier results but without additional conditions. These extensions become possible using a more precise majorization than the one given in earlier articles. Numerical experimentation complements the theoretical results involving a partial differential and an integral equation.