Solvability Of Nonlinear Equations Research Articles

Analytical and numerical methods for calculation the deep of penetration the welding seam formed by the electron beam generated by glow discarge electron guns

The article is devoted to the problem of defining the focal diameter of electron beam, formed by the glow discharge electron guns, as well as the necessary pressure in the gun chamber for realising the welding process. Taking into account, that glow discharge electron guns are widely used in industry for welding of different metals, and that for providing the high quality of welding joints estimation of energetic parameters in beam focus is very important, proposed methods are very important for effective elaboration and designing of the novel glow discharge electron guns constructions for specific technological operations. With known focal beam deameter and thermodinamic parameters of welding details material the deep of penetration of welding seam, as well as the necessary pressure in discharge chamber have been estimatied. Two proposed methods are generally based on the analytical solving of explite equation and on numeracal solving of sophisticated non-linear equation. Obtained simulation results with and without taking into account the spsace charge of own beam electrons are also given

Strain demand of elastic pipes subjected to permanent ground displacements using the finite difference method

Long-distance pipelines are one of the primary means of oil and gas transportation. During the construction process, long-distance pipelines are inevitably buried across geohazard zones, which potentially generate permanent ground displacements. These ground displacements can potentially induce excessive strains in the pipe posing a great threat to the pipe’s safety and integrity. In this study, a new numerical methodology for the response analysis of pipes subjected to ground displacements is proposed based on the finite difference method. Simulating the pipeline as a large deformation Euler Bernoulli beam, the finite difference method is used to solve the two interacting nonlinear differential equations of equilibrium in the longitudinal and lateral directions considering the nonlinear pipe-soil interaction induced by the ground displacement. Implemented using the nonlinear equation solver of FindRoot by Mathematica for solving nonlinear equations, the longitudinal strain along the pipeline can be subsequently derived, and the tensile and compressive strain demands can be therefore determined for engineering reference. Finally, the applicability of this method is validated based on two hypothetical study cases involving symmetric and non-symmetric soil resistance on the lateral direction of the pipe. Comparing the results with the finite element analysis solver Abaqus, we demonstrate that this present methodology has excellent predictive capabilities. Our study is carried out for elastic response calculation, but the proposed method shows a great promise for further development involving material nonlinearity, which is appropriate for the preliminary safety evaluation for the design of new pipelines or for risk prescreening of existing pipelines.

A Gibbs Sampling-based approach for parameter estimation of the EGK distribution

We present a novel approach for estimating the parameters of the extended generalized-K (EGK) distribution commonly used as a fading model in wireless and optical communications links. The proposed method is based on the Gibbs sampling technique and does not require solving nonlinear equations nor performing numerical integrations. Numerical and simulation results are presented showing that the estimated and original distributions are virtually indistinguishable and formal metrics like Kullback-Leibler (KL) divergence, the mean integrated squared bias (MISB), the mean integrated variance (MIV) and the mean integrated squared error (MISE) all show excellent agreement between the two as well.

Zeroing neural network for bound-constrained time-varying nonlinear equation solving and its application to mobile robot manipulators

A typical class of recurrent neural networks called zeroing neural network (ZNN) has been considered as a powerful alternative for time-varying problems solving. In this paper, a new ZNN model is proposed and studied to solve the bound-constrained time-varying nonlinear equation (BCTVNE). Specifically, by introducing a time-varying nonnegative vector, the BCTVNE is reformulated as a combined system of nonlinear equations. On the basis of two indefinite error functions and the exponential decay formula, the new ZNN model is thus developed, which can zero in on the combined system. Theoretical analysis and simulation results are provided to verify the effectiveness of the proposed ZNN model. The applicability is further indicated under the simulations on an omnidirectional mobile robot manipulator via the proposed ZNN model.

A novel robust fixed-time convergent zeroing neural network for solving time-varying noise-polluted nonlinear equations

ABSTRACT Solving nonlinear equations is a crucial step in the territories of science and engineering, as many practical problems could be mathematically described by nonlinear equations. In this paper, a novel robust fast convergence zeroing neural network (RFCZNN) by utilizing a reconstructed activation function (AF) is presented and investigated for the dynamic nonlinear equations (DNE) solving problems in predictable period. Comparing with recently reported finite-time nonlinear recurrent neural network, the presented RFCZNN solves the DNE in settled theoretical time and possesses better robustness in noise-polluted environments. Unlike the finite-time convergent neural network models, the time consumption of the presented RFCZNN in convergence process can be calculated directly by mathematics without considering modelling initial states. The comparative experimental results for solving high-order (second and third order) DNE and tracking robotic motional trail are presented separately to further represent that the proposed fixed-time convergent RFCZNN model is more robust and efficient.

Computational methods for pore-scale simulation of rarefied gas flow

Direct simulation at the pore-scale is crucial to unravel rarefaction effect on gas transport in tight porous media. To satisfy the dual demands on modeling accuracy and computation effort, an appropriate method must be chosen. This work, therefore, evaluates four numerical methods for pore-scale rarefied gas flows in a two-dimensional (2D) model porous media over a wide range of rarefaction. These methods are the incompressible Navier-Stokes equation with the first-order velocity-slip boundary condition, and three gas-kinetic solvers i.e. a finite-difference (FD) iterative solver for the linearized Bhatnagar, Gross and Crook (BGK) model kinetic equation, a finite-volume (FV) solver for the non-linear Shakhov model, and an open-source direct simulation Monte Carlos (DSMC) solver. The benchmark cases cover the characteristic Knudsen number ranging from 0.0231 to 4.62 while the characteristic Reynolds numbers are kept to be less than 1.0. All the solvers are developed in OpenFOAM, except the FD solver, allowing us to investigate the effect of local grid refinements using the automatic Cartesian grid generator in OpenFOAM. The flow fields and the apparent permeabilities predicted by all the solvers have been compared in detail. Besides, the computational time of these solvers is measured and analyzed to demonstrate the relative cost of the three kinetic solvers. It is found that the FD solver is the most efficient one and gives accurate results over the whole range of Knudsen number. Finally, this study also evaluates the feasibility of the recently developed discrete unified gas-kinetic scheme (DUGKS), the algorithm in the aforementioned non-linear Shakhov model equation solver, for pore-scale rarefied gas flow. It is found the results predicted by this solver agree well with the other two kinetic solvers.

Multiwave interaction solutions for a (3 + 1)-dimensional generalized BKP equation

ABSTRACT The direct algebraic method, together with the inheritance solving strategy, is utilized to construct multiwave interaction solutions among solitons, rational waves and periodic waves for a -dimensional generalized BKP equation. In addition, an N-soliton decomposition algorithm derived from the simplified Hirota method, the conjugated parameters assignment and long-wave limit techniques is introduced. By the N-soliton decomposition algorithm, higher-order interaction solutions among solitons, breathers and lump waves for this equation are generated. The highlight of N-soliton decomposition algorithm is that it bypasses the solving of (super) large-scale nonlinear algebraic equations, so it can be used to obtain much higher-order multiwave interaction solutions.

A Fast Convergent Homotopy Perturbation Method for Solving Selective Harmonics Elimination PWM Problem in Multi Level Inverter

Pulse width modulation (PWM) control for power converters have been vastly investigated and used in many industrial application. For medium voltage and high power applications, low switching frequency PWM techniques are preferred over high switching frequency-based PWM techniques. The preprogrammed low switching frequency-based PWM technique known as selective harmonics elimination (SHE) gives the better quality waveform at a lower switching frequency. The main difficulty in applying SHE PWM is in solving of non-linear transcendental equations for obtaining switching angles. Several methods have been proposed for computation of switching angles that include numerical techniques, optimization techniques, and algebraic techniques. However, the computation of switching angles is still a challenging task in the application of SHE PWM. In this paper, a novel fast convergent homotopy perturbation method (HPM) is proposed to compute switching angles for a multilevel inverter at a faster rate. The solutions obtained by the proposed technique is as accurate as obtained by the algebraic methods with no dependency on the initial guess. The proposed technique can compute the higher number of switching angles with multiple solutions in some modulation index range. A prototype is developed and the computed switching angles have been verified using field-programmable gate arrays (FPGA) controller to validate the results for practical applications.

About Exact Solution of Some Non Linear Partial Integro-differential Equations

Data on solving of nonlinear integro-differential equations using Laplace-SBA method are scarce. The objective of this paper is to determine exact solution of nonlinear 2 dimensionnal Voltera-Fredholm differential equation by this method. First, SBA method and Laplace SBA method are described. Second, three nonlinear Voolterra-Fredholm integro-differential equations are solved using each method. Application of each method give an exact solution. However, application of Laplace-SBA method permits for solve integro-differential equation compared with SBA method. This proves that this last method can be fruitfully applied in the resolution of integro-differential equations.

Highly efficient solvers for nonlinear equations in Banach space

Highly efficient solvers for nonlinear equations in Banach space

Optimal Eighth-Order Solver for Nonlinear Equations with Applications in Chemical Engineering

A new iterative technique for nonlinear equations is proposed in this work. The new scheme is of three steps, of which the first two steps are based on the sixth-order modified Halley’s method presented by the authors, and the last is a Newton step, with suitable approximations for the first derivatives appeared in the new scheme. The eighth-order of convergence of the new method is proved via Mathematica code. Every iteration of the presented scheme needs the evaluation of three functions and one first derivative. Therefore, the scheme is optimal in the sense of Kung-Traub conjecture. Several test nonlinear problems are considered to compare the performance of the proposed method according to other optimal methods of the same order. As an application, we apply the new scheme to some nonlinear problems from the field of chemical engineering, such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.

Modeling Issues in Problems of the Elasticity and Viscoelasticity Theory

The development of computer technology and information technology does not exclude preliminary analytical modeling in complex problems of mechanics. The approach proposed in this work allows one to obtain reasonable simplified equations that admit analytical or numerical-analytical solutions. Authors research are devoted to method elaboration for space problems of viscoelasticity. The approach for solving problems of nonlinear elasticity theory, when final deformations or physical nonlinear properties of materials are taken into consideration, is proposed. New perturbation method for solving of nonlinear equations in particular derivatives is suggested. Such approach is allowed to reduce the solution of complicated problems of linear elasticity and viscoelasticity to subsequently solved boundary problems of potential theory. New linear problems are investigated, in particular, problems on load transference from stringer to single layer or multilayer solids, on stress-strain state of fibrous composite with crack in matrix (plate and axisymmetric problems), numerous contact problems (pressing of hard stamp in an orthotropic plate with different anisotropy). Problems of nonlinear elasticity theory on stress-strain state of a plate with a circular hole under different types of loading are solved.

Inequality-constrained free-surface evolution in a full Stokes ice flow model (<i>evolve_glacier v1.1</i>)

Abstract. Like any gravitationally driven flow that is not constrained at the upper surface, glaciers and ice sheets feature a free surface, which becomes a free-boundary problem within simulations. A kinematic boundary condition is often used to describe the evolution of this free surface. However, in the case of glaciers and ice sheets, the naturally occurring constraint that the ice surface elevation (S) cannot fall below the bed topography (B) (S-B≥0), in combination with a non-zero mass balance rate complicates the matter substantially. We present an open-source numerical simulation framework to simulate the free-surface evolution of glaciers that directly incorporates this natural constraint. It is based on the finite-element software package FEniCS solving the Stokes equations for ice flow and a suitable transport equation, i.e. “kinematic boundary condition”, for the free-surface evolution. The evolution of the free surface is treated as a variational inequality, constrained by the bedrock underlying the glacier or the topography of the surrounding ground. This problem is solved using a “reduced space” method, where a Newton line search is performed on a subset of the problem (Benson and Munson, 2006). Therefore, the “constrained” non-linear problem-solving capabilities of PETSc's (Portable, Extensible Toolkit for Scientific Computation, Balay et al., 2019) SNES (Scalable Non-linear Equations Solver) interface are used. As the constraint is considered in the solving process, this approach does not require any ad hoc post-processing steps to enforce non-negativity of ice thickness and corresponding mass conservation. The simulation framework provides the possibility to divide the computational domain into different subdomains so that individual forms of the relevant equations can be solved for different subdomains all at once. In the presented setup, this is used to distinguish between glacierised and ice-free regions. The option to chose different time discretisations, spatial stabilisation schemes and adaptive mesh refinement make it a versatile tool for glaciological applications. We present a set of benchmark tests that highlight that the simulation framework is able to reproduce the free-surface evolution of complex geometries under different conditions for which it is mass-conserving and numerically stable. Real-world glacier examples demonstrate high-resolution change in glacier geometry due to fully resolved 3D velocities and spatially variable mass balance rate, whereby realistic glacier recession and advance states can be simulated. Additionally, we provide a thorough analysis of different spatial stabilisation techniques as well as time discretisation methods. We discuss their applicability and suitability for different glaciological applications.

A nonlinear solver based on an adaptive neural network, introduction and application to porous media flow

Sensitivity to derivatives and a need for proper initial guesses are the main disadvantages of classic nonlinear solvers like Newton's method. To overcome the obstacles, a numerical solver for second-order nonlinear Partial Differential Equations (PDEs) based on an Adaptive Neural Network (AdNN) is introduced. While using Newton's method needs to form the Jacobian matrix and its inversion, which both of them are time-consuming and dependent on the nature of the problem, the proposed solver tries to find the roots of the algebraic equations by changing the weights of AdNN by the help adaptive laws. The proposed approach has been applied to solve the governing PDEs of the gas flow in shale resources and the immiscible two-phase flow of water and oil in hydrocarbon reservoirs as two highly nonlinear phenomena. The generated profiles of pressures and saturations show a satisfying match with the outputs of Newton's method. However, using the presented algorithm not only removes the former necessities but also helps the community to solve the relevant PDEs governing the critical elements of the energy market in the future with a higher level of confidence.

Theoretical Study of Non-Iterative Holomorphic Embedding Methods for Solving Nonlinear Power Flow Equations: Algebraic Property

Solving nonlinear algebraic equations is a core task in many fields of engineering and the sciences. Iterative methods such as Newton's method have been popular in solving power flow equations. Recently, the non-iterative holomorphic embedding (HE) methods that employ polynomial embedded systems have been proposed to solve power flow equations. It is advantageous for the solution functions obtained by the HE method to possess an algebraic property. The present paper points out that algebraic solution functions can have several desired properties for the HE method to work. This paper also exemplifies that the algebraic property cannot be ensured by the elimination process, which corrects an assertion in the literature. It is hence essential to study required conditions for ensuring the existence of algebraic solution functions. The present paper is devoted to the theoretical foundation of HE methods using a polynomial embedded system and develops a novel general theorem and sufficient conditions to ensure the required algebraic property. Examples and numerical results are presented to better explain the derived theoretical results.

Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

A clustering-based differential evolution with different crowding factors for nonlinear equations system

Solving nonlinear equations systems (NESs) is one of the most important tasks in numerical computation. It is common that most NESs contain more than one root. Generally, these roots are equally important. Therefore, locating as many of these roots as possible is extremely useful; however, it is a difficult task. Recently, the use of evolutionary algorithms (EAs) for NESs has been given more consideration. There are several issues that need to be considered when applying EAs to solve NESs: (1) How to guide the population towards multiple roots, (2) how to tackle individuals which trip into local optimum, and (3) how to minimize replacement errors. In this paper, we deal with the first issue by introducing a one-step k-means clustering method combined with niching. In this way, individuals will search for different roots in their respective directions. For the second issue, we propose two methods to form species, the goal of which is to promote individuals to get rid of local optima. Finally, different crowding factors are adopted to reduce replacement errors. By assembling these improvements, a one-step k-means clustering based differential evolution, namely KSDE, is proposed. To evaluate the effectiveness of the proposal, we use 30 NES problems from the literature as the test suite. Experimental results demonstrate KSDE has great potential to locate multiple roots in a single run. Furthermore, according to the evaluation criteria, KSDE shows better performance compared with other state-of-the-art methods.

Base level changes based on Basin Filling Modelling: a case study from the Paleocene Lishui Sag, East China Sea Basin

Estimation of base level changes in geological records is an important topic for petroleum geologists. Taking the Paleocene Upper Lingfeng Member of Lishui Sag as an example, this paper conducted a base level reconstruction based on Basin Filling Modelling (BFM). The reconstruction was processed on the ground of a previously interpreted seismic stratigraphic framework with several assumptions and simplification. The BFM is implemented with a nonlinear diffusion equation solver written in R coding that excels in shallow marine stratigraphic simulation. The modeled results fit the original stratigraphy very well. The BFM is a powerful tool for reconstructing the base level, and is an effective way to check the reasonableness of previous interpretations. Although simulation solutions may not be unique, the BFM still provides us a chance to gain some insights into the mechanism and dynamic details of the stratigraphy of sedimentary basins.

A THREE-STEP NINTH ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS

A THREE-STEP NINTH ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS

Thermal-Hydraulic Calculation and Analysis on Evaporator System of a 1000 MW Ultra-Supercritical Pulverized Combustion Boiler with Double Reheat

A double reheat ultra-supercritical boiler is an important development direction for high-parameter and large-capacity coal-fired power plants due to its high thermal efficiency and environmental value. China has developed a 1000 MW double reheat ultra-supercritical boiler with steam parameters of 35 MPa at 605°C/613°C/613°C. Reasonable water wall design is one of the keys to safe and reliable operation of the boiler. In order to examine the thermal-hydraulic characteristics of the double reheat ultra-supercritical boiler, the water wall system was equivalent to a flow network comprising series-parallel circuits, linking circuits and pressure nodes, and a calculation model was built on account of the conservation equations of energy, momentum and mass. Through the iterative solving of nonlinear equations, the prediction parameters of the water wall at boiler maximum continue rate (BMCR), 75% turbine heat-acceptance rate (THA) and 30% THA loads, including total pressure drops, flow distribution, outlet steam temperatures, fluid and metal temperatures were gotten. The results of calculation exhibit excellent thermal-hydraulic characteristics and substantiate the feasibility of the water wall design of the double reheat ultra-supercritical boiler.