Nonlinear Stiff Systems Research Articles

Exploring the performance of the optimized inductive linearization solver for simulation of the Van der Pol oscillator (a stiff non-linear system)

Exploring the performance of the optimized inductive linearization solver for simulation of the Van der Pol oscillator (a stiff non-linear system)

Physics-informed neural networks for the reaction-diffusion Brusselator model

In this work, we are interesting in solving the 1D and 2D nonlinear stiff reaction-diffusion Brusselator system using a machine learning technique called Physics-Informed Neural Networks (PINNs). PINN has been successful in a variety of science and engineering disciplines due to its ability of encoding physical laws, given by the PDE, into the neural network loss function in a way where the network must not only conform to the measurements, initial and boundary conditions, but also satisfy the governing equations. The utilization of PINN for Brusselator system is still in its infancy, with many questions to resolve. Performance of the framework is tested by solving some one and two dimensional problems with comparable numerical or analytical results. Validation of the results is investigated in terms of absolute error. The results showed that our PINN has well performed by producing a good accuracy on the given problems.

Neuro-Heuristic Computational Intelligence Approach for Optimization of Electro-Magneto-Hydrodynamic Influence on a Nano Viscous Fluid Flow

In this investigative study, the electro-magneto hydrodynamic (EMHD) influence on a nano viscous fluid model is scrutinized by designing an artificial neural network (ANN) paradigm using a neuro-heuristic approach (NHA) through the combination of GAs (genetic algorithms) and one of the most efficient locally searching solver SQP (sequential quadratic programming), i.e., NHA-GA-SQP. The fluid flow for the proposed problem is initially interpreted in the form of PDEs and then utilization of suitable similarity transformation on these PDEs yields in terms of a stiff nonlinear system of ODEs. The numerical results of the suggested fluidic model based on the variation of its physically existing parameters are calculated through the NHA-GA-SQP solver to detect the variation in velocity, thermal gradient, and concentration during the fluid flow. A detailed analysis of obtained outcomes through the NHA-GA-SQP algorithm and their comparison with the reference results estimated via the Adams method are presented. The calculation of the proposed solver’s accuracy, stability, and consistency through various statistical operators is also involved in the current inspection.

A computational stochastic procedure for solving the epidemic breathing transmission system

This work provides numerical simulations of the nonlinear breathing transmission epidemic system using the proposed stochastic scale conjugate gradient neural networks (SCGGNNs) procedure. The mathematical model categorizes the breathing transmission epidemic model into four dynamics based on a nonlinear stiff ordinary differential system: susceptible, exposed, infected, and recovered. Three different cases of the model are taken and numerically presented by applying the stochastic SCGGNNs. An activation function ‘log-sigmoid’ uses twenty neurons in the hidden layers. The precision of SCGGNNs is obtained by comparing the proposed and database solutions. While the negligible absolute error is performed around 10–06 to 10–07, it enhances the accuracy of the scheme. The obtained results of the breathing transmission epidemic system have been provided using the training, verification, and testing procedures to reduce the mean square error. Moreover, the exactness and capability of the stochastic SCGGNNs are approved through error histograms, regression values, correlation tests, and state transitions.

On the applications of collocation method for numerically analyzing the nonlinear Degasperis–Procesi and Benjamin–Bona–Mahony equations

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by [Formula: see text]-weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of [Formula: see text]. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique.

Penalty function method for geometrically nonlinear buckling analysis of imperfect truss with multi-freedom constraints based on mixed FEM

This paper is concerned with the approach to implementing the Penalty function method for imposing multi-freedom constraints in geometrically nonlinear analysis of imperfect trusses based on mixed finite element formulation. Using a finite element model based on displacement formulation, it is required to incorporate both the dependent boundary relations and initial length imperfection to the nonlinear master stiffness system of equations for solving the geometrically nonlinear problem of imperfect truss with multi-freedom constraints. For decreasing the mathematical complexion of the incorporating process, the author proposes a novel mixed finite truss element considering initial imperfection, used in building the model for solving the geometrically nonlinear problem of truss with multi-freedom constraints. The modified nonlinear stiffness equation is constructed by employing the penalty function method to convert a constrained problem into an unconstrained problem by extremizing the augmented energy function established based on the proposed mixed finite element formulation. For solving the nonlinear equilibrium equation of imperfect trusses with multi-freedom constraints, the incremental equilibrium equation is constructed, and the incremental-iterative algorithm for calculation is established utilizing the arc-length method, used for writing the calculation program for investigating geometrically nonlinear behavior of imperfect truss with multi-freedom constraints. The results of the numerical test show the influence of initial imperfection and choosing weight values on the equilibrium path of the truss.

An Experimental Investigation of the Displacement Transmissibility for a Two-Stage HSLD Stiffness System

Vibration isolation across the frequency spectrum is a challenge in many applications, particularly at low frequencies where linear oscillators amplify excitation forces. To overcome this, nonlinear high static low dynamic (HSLD) stiffness oscillators have been proposed with the aim of reducing the resonant frequency while maintaining the high load capacities of much stiffer linear systems. A two-degree of freedom (2DOF) HSLD stiffness system is proposed to investigate the effectiveness of such systems. Experiments reveal that a 2DOF non-linear HSLD stiffness system outperforms a similar single-degree of freedom (SDOF) HSLD stiffness system, as well as similar SDOF and 2DOF linear systems. Three performance criteria are used to assess these systems, including (1) minimizing the resonant frequency and maximizing the isolation zone, (2) minimizing the magnitude of amplification at resonance, and (3) maximizing the ability to isolate large input frequencies. Exact numerical and approximate analytical simulations are validated using these experimental data. A sensitivity analysis of system parameters reveals that it is necessary to incorporate adjustability into the geometry of a design to counteract unavoidable manufacturing tolerances. Changes of less than 2% to the stiffness or geometry of a system can drastically change its dynamic response.

Two Methods for the Implicit Integration of Stiff Reaction Systems

Abstract We present two methods for the implicit integration of nonlinear stiff systems. Direct application of the Newton method to backward Euler discretization of such systems may diverge. We observe that the solution is recovered by smoothing out certain eigenvalues in the Jacobian matrix. To this end, we introduce a solution-dependent matrix-weighted combination of backward and forward Euler methods. The weight is tuned on each Newton iteration to reproduce the solution with an exponential integrator, whereby a weight function for smoothing eigenvalues is obtained. We apply the proposed techniques, namely quasi-Newton backward Euler and matrix-weighted Euler, to several stiff systems, including Lotka–Volterra, Van der Pol’s, and a blood coagulation cascade.

Dynamic state estimation in nonlinear stiff systems using implicit state space models

Dynamic state estimation in nonlinear stiff systems using implicit state space models

Nonlinear hydrodynamic analysis and optimization of oscillating wave surge converters under irregular waves

To evaluate the mean annual capture width ratio (CWR) of the oscillating wave surge converters (OWSCs) under irregular waves, the time-domain dynamic equation was derived by considering the nonlinear hydrostatic stiffness, drag, and nonlinear power take-off (PTO) system. A Python code was developed to solve the response and capture performance with 4th-order Runge–Kutta integration. The wave spectrum was corrected according to the water depth in shallow water. Unlike the inflexible drag coefficients in steady flow, the drag coefficients in irregular waves (calibrated with OpenFOAM) are strongly affected by OWSC thickness. Multi-objective genetic algorithm (MOGA) optimization of OWSC geometric sizes, internal water filling, and PTO parameters was conducted for two objective functions: maximizing the mean annual CWR and minimizing the structural mass per unit width. The optimized result presents a slender OWSC that neutrally balances these two objectives. The effects and local sensitivities of the width, thickness, axis depth, water filling, PTO stiffness, damping, and friction were comprehensively discussed. The results show that the axis depth has the greatest positive influence on the CWR, and the increase of thickness creates significant economical disadvantage due to a heavier structure. However, inertia adjustment by filling water does not benefit the mean annual CWR.

Solutions of Stiff Systems of Ordinary Differential Equations Using Residual Power Series Method

The stiff differential equations occur in almost every field of science. These systems encounter in mathematical biology, chemical reactions and diffusion process, electrical circuits, meteorology, mechanics, and vibrations. Analyzing and predicting such systems with conventional numerical techniques require more time and memory; still accurate solution is completely uneconomical and uncertain. Most of the numerical techniques have stability issues while dealing with stiff systems. To overcome these limitations, residual power series method (RPSM) is proposed for stiff systems of differential equations (DEs). RPSM is applied to various linear and nonlinear stiff systems, and closed‐form solutions are achieved. This indicates the effectiveness of proposed scheme for stiff family of DEs. Since this method leads to better results with less computational cost, it can be extended for more complex systems which arise in different areas of engineering and sciences.

Vibration isolation with passive linkage mechanisms

This study is to present a novel way to achieve superior passive vibration isolation by employing a specially designed and compact linkage mechanism. The proposed anti-vibration system has beneficial nonlinear inertia, inspired by swinging motion of human arms, and is constructed with an adjustable nonlinear stiffness system inspired by animal or human leg skeleton. It is shown with comprehensive theoretical analysis and consequently validated by a series of well-designed experiments that the nonlinear stiffness, nonlinear damping and nonlinear inertia of the proposed system are very helpful for significantly reducing resonant frequency and enhancing damping effect in a beneficial nonlinear way. This results in excellent vibration isolation performance with lower resonant frequency and resonant peak of faster decay rate. This study provides an innovative solution to a cost-efficient vibration control demanded in various engineering systems.

Semi-Implicit and Explicit Runge Kutta Methods for Stiff Ordinary Differential Equations

In this work, we study the A[α] – stability of the additive methods of Runge- Kutta kind of orders ranging from 2 up to 4 that will be applied for determining some stiff nonlinear system of the ODEs. Moreover, we find the stability function for the additive Runge-Kutta method and some methods of this type of order 2,3, and 4. Where the method (A,B1 ) is A-stable and semi-implicit and method (A,B2 ) is explicit. Furthermore, the stiff term is managed by the semi-implicit Runge-Kutta method while no stiff term is treated by the explicit Runge Kutta method. Those methods are suitable for solving chemical reactions problems that include stiff and non-stiff terms.

High order extended boundary value methods for the solution of stiff systems of ODEs

A class of high order extended boundary value methods (HEBVMs) suitable for the numerical approximation of stiff systems of ordinary differential equations (ODEs) is constructed. This class of BVMs is based on the second derivative class of linear multistep formulas (LMF) and it provides a set of very highly stable methods that can produce considerably accurate solutions to stiff systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The class of BVMs derived herein is of high order, small error constants and large region of absolute stability. Specifically, it is Ok1,k2-stable, Ak1,k2-stable with (k1,k2) -boundary conditions and order p=k+4 for values of the step length k≥1. The numerical results obtained from standard linear and non-linear stiff systems indicate that this scheme is highly competitive with existing methods.

Design of Spline-Evolutionary Computing Paradigm for Nonlinear Thin Film Flow Model.

A novel design of stochastic numerical computing method is introduced for computational fluid dynamics problem governed with nonlinear thin film flow (TFF) system by exploiting the competency of polynomial splines for discretization and optimization with evolutionary computing aided with brilliance of local search. The TFF model of second grade fluid is represented with nonlinear second-order differential system. The aim of the present work is to exploit the cubic spline approach (CSA) to transform the differential equations for TFF model into an equivalent set of nonlinear equations. The approximation in mean squared error sense is introduced for the formulation of cost function for solving the nonlinear system of equations representing TFF model. The optimization of the decision variables of the cost function is carried out with global search efficacy of evolution by genetic algorithms (GAs) integrated with sequential quadratic programming (SQP) for speedy adjustments. The designed spline–evolutionary computing paradigm, CSA–GA–SQP, is evaluated for different scenarios of TFF model by variation of second grade and magnetic parameters, as well as variation in the length of splines. Results endorsed the worth of CSA–GA–SQP solver as an efficient alternative, reliable, stable, and accurate framework for the variants of nonlinear TFF systems on the basis of multiple autonomous executions. The design computing spline paradigm CSA–GA–SQP is a promising alternative numerical solver to be implemented for the solution of stiff nonlinear systems representing the complex scenarios of computational fluid dynamics problems.

A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

In this article, the Elzaki homotopy perturbation method is applied to solve fractional stiff systems. The Elzaki homotopy perturbation method (EHPM) is a combination of a modified Laplace integral transform called the Elzaki transform and the homotopy perturbation method. The proposed method is applied for some examples of linear and nonlinear fractional stiff systems. The results obtained by the current method were compared with the results obtained by the kernel Hilbert space KHSM method. The obtained result reveals that the Elzaki homotopy perturbation method is an effective and accurate technique to solve the systems of differential equations of fractional order.

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

On A Hybrid Concept for Approximating Self‐Excited Periodic Oscillations of Large‐Scaled Dynamical Systems

AbstractConcerning the approximation of self‐excited periodic oscillations in large‐scaled mechanical systems involving strong nonlinearities, this contribution suggests a concept for an efficient treatment. The presented Hybrid FD‐HB method takes the advantages of both schemes Harmonic Balance and Finite Difference to enhance the ratio of computational cost and accuracy for mechanical systems with many degrees of freedom. Within this contribution the residual equations, required when applying a NEWTON‐RAPHSON‐scheme, are derived and the method is applied to a stiff nonlinear mechanical system.

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general Lévy noise.

Solving Stiff Systems by using Symbolic - Numerical Method

In this paper, an efficient symbolic-numerical procedure based on the power series method is presented for solving a system of differential equations. The basic idea is to substitute power series into the differential equations and to find a polynomial system of coefficients, where a powerful symbolic computation technique (i.e., Grobner basis) is used to solve the system. In fact, the proposed method is an excellent bridge between symbolic and numeric computation and specially, enables us to find the solution of linear and non-linear stiff systems. Numerical experiments were performed to justify our new approach.