Stiff Differential Systems Research Articles

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.

A variable step-size implementation of the hybrid Nyström method for integrating Hamiltonian and stiff differential systems

A variable step-size implementation of the hybrid Nyström method for integrating Hamiltonian and stiff differential systems

HIGH ORDER SECOND DERIVATIVE DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR ODES

Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.

A new one-step method with three intermediate points in a variable step-size mode for stiff differential systems

This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, in particular, in problems derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder in an article on problems of chemical kinetics (Hirschfelder, Proc Natl Acad Sci USA 38:235–243, 1952). The techniques of interpolation and collocation are used in the construction of the scheme. We consider a suitable polynomial to approximate the theoretical solution of the problem under consideration. The basic properties of the new scheme are analyzed. An embedded strategy is adopted to formulate the proposed scheme in a variable stepsize mode to get better performance. Finally, some models of initial-value problems, including ordinary and time-dependent partial differential equations, are solved numerically to assess the performance and efficiency of the proposed technique, with applications to real-world problems.

Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy

Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.

Time-efficient reformulation of the Lobatto III family of order eight

Implicit block methods for solving initial value problems in ordinary differential equations are well-known among the contemporary scientific community, since they are cost-effective, self-starting, consistent, stable, and usually converge fast when applied to solve particularly stiff models. These characteristics of block methods are the primary reasons for the one-step optimized block method devised in the present research study with three off-grid points. Theoretical analysis, including the order of convergence, consistency, zero-stability, A-stability, order stars, and the local truncation error, are considered. The obtained method may be categorized as the well-known Lobatto IIIA Runge–Kutta method. The superiority of the devised method over various existing approaches having similar features is proved via numerical simulations of stiff and nonlinear differential systems. Furthermore, a suitable reformulation of the devised method results in considerable savings in computation time, as revealed through the efficiency plots. This turns out in a strategy to reformulate Runge–Kutta type methods in order to get a better performance.

Second Derivative Block Hybrid Methods for the Numerical Integration of Differential Systems

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators.

Using scientific machine learning to develop universal differential equation for multicomponent adsorption separation systems

AbstractUniversal differential equations are a concept in scientific machine learning that leverages the potential of the universal approximator theorem and the physical knowledge of a given system. Creating this level of hybridization within a stiff partial differential equation system is a challenge. On the other hand, adsorption phenomenological models have sink/source terms that describe the adsorption equilibrium through a well‐known simplified model (e.g., Langmuir; Sips; and Brunauer, Emmet, Teller [BET]). These suitable mechanistic assumptions are identified through experiments, providing the parameters of the sink/source model. However, these mechanistic assumptions are a simplification of the system phenomenology. Therefore, the resulting model is limited by its premises. In this scenario, the universal ordinary differential equations (UODE) is presented as an approach that conciliates the potential of artificial neural networks to learn given phenomena without conceptual simplifications. On the other hand, keeping into consideration the system physics. This work proposes a UODE system to solve the multicomponent separation by adsorption in a fixed bed column. Experimental data is used to identify the hybrid model. The required amount of data used in the model identification demonstrates that hybrid models can use a few data points to precisely describe the system. Furthermore, the obtained model can describe competitive adsorption with higher precision than the Langmuir model.

A Family of A -Stable Optimized Hybrid Block Methods for Integrating Stiff Differential Systems

In this article, a family of one-step hybrid block methods having two intrastep points is developed for solving first-order initial value stiff differential systems that occur frequently in science and engineering. In each method of the family, an intrastep point controls the order of the main method and a second one has a control over the stability features of the method. The approach used to develop the class of A -stable methods is based on interpolation and collocation procedures. The methods exhibit hybrid nature and produce numerical solutions at several points simultaneously. These methods can also be formulated as Runge-Kutta (RK) methods. Comparisons between the RK and block formulations of the proposed methods reveal a better performance of the block formulation in terms of computational efficiency. Furthermore, the efficiency of the methods is improved when they are formulated as adaptive step-size solvers using an error-control approach. Some methods of the proposed class have been tested to solve some well-known stiff differential systems. The numerical experiments show that the proposed family of methods performs well in comparison with some of the existing methods in the scientific literature.

The Direct Two-Point Block One-Step Method Which is efficient and Suitable for Solving Second-Order Differential Equation

Abstract: A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size.Two point four step direct implicit block method is developed for solving directly the second order system of ordinary differential equations (ODEs) using variable step size. The method will estimate the solutions of Initial Value Problems (IVPs) at two points simultaneously by using four backward steps. The method developed is suitable for the numerical integration of non stiff and mildly stiff differential systems. Numerical results are given to compare the efficiency of the developed method to the existence block method.

Artificial neural network scheme to solve the nonlinear influenza disease model

The aim of this study is to present the numerical simulations of the influenza disease nonlinear system (IDNS) using the stochastic artificial neural networks (ANNs) procedures supported with Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The IDNS is constructed with four classes, susceptible S(t), infected I(t), recovered R(t) and cross-immune people C(t), based stiff nonlinear ordinary differential system. The numerical computations have been performed through the stochastic ANNs-LMB for solving six different variations of the IDNS. The obtained numerical solutions through the stochastic ANNs-LMB for solving the IDNS have been presented using the training, verification and testing measures to reduce mean square error (MSE) from data-based reference solutions. To observed the correctness, efficiency, competence and proficiency of the designed computing paradigm ANNs-LMB, an exhaustive analysis is presented using the correlation studies, error histograms (EHs), mean squared error (MSE), regression and state transitions (STs) information. The worth and significance of ANNs-LMB is substantiated through comparisons of the outcomes admitted the good agreement from data derived results with 5–7 decimal places of accuracy for each scenario of IDNS.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and SomeError Analysis

In this paper, an efficient multi-step scheme is presented based on reproducing kernel Hilbert space (RKHS) theory for solving ordinary stiff differential systems. The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform form for a rapidly convergent series in the posed Sobolev space. Using the Gram-Schmidt orthogonality process, complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction. Consequently, by applying the standard RKHS method to each subinterval, approximate solutions that converge uniformly to the exact solutions are obtained. For this purpose, several numerical examples are tested to show proposed algorithm’s superiority, simplicity, and efficiency. The gained results indicate that the multi-step RKHS method is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

Balanced-Euler approximation schemes for stiff systems of stochastic differential equations

This paper aims to design new families of balanced-Euler approximation schemes for the solutions of stiff stochastic differential systems. To prove the mean-square convergence, we use some fundamental inequalities such as the global Lipschitz condition and linear growth bound. The meansquare stability properties of our new schemes are analyzed. Also, numerical examples illustrate the accuracy and efficiency of the proposed schemes.

Determination of single-phase transformer saturation characteristic by using Nelder–Mead optimization method

Correctly modelling the nonlinear saturation characteristics of the transformer iron core is of great importance in steady-state calculations (harmonics and power quality) and calculations of the electromagnetic transients (inrush current, ferroresonance, temporary overvoltages). Thus, this paper presents an original estimation algorithm for determining the transformer nonlinear saturation characteristic from the measured inrush and steady-state current waveforms in entire windows of occurrences. The proposed algorithm is based on the minimization of a specifically defined cost function by applying the Nelder–Mead method, which was used to create a balance between the transformer steady-state current and the inrush current at zero residual flux. The algorithm uses the BDF numerical method in the background to solve the stiff differential equation system. By applying the appropriate indexes of quality estimation, it was demonstrated that, out of a total of twelve proposed analytical functions, the best optimization results were achieved with three proposed functions (the Frolich extended, three-term polynomial and special hyperbolic function). Detailed performances and accuracy suggested that the estimation algorithm has been verified by a very close agreement between the simulation results and measured transformer current waveforms in the steady-state and different inrush current scenarios.

Qualitative behavior of stiff ODEs through a stochastic approach

In the last few decades, stiff differential equations have attracted a great deal of interest from academic society, because much of the real life is covered by stiff behavior. In addition to importance of producing model equations, capturing an exact behavior of the problem by dealing with a solution method is also handling issue. Although there are many explicit and implicit numerical methods for solving them, those methods cannot be properly applied due to their computational time, computational error or effort spent for construction of a structure. Therefore, simulation techniques can be taken into account in capturing the stiff behavior. In this respect, this study aims at analyzing stiff processes through stochastic approaches. Thus, a Monte Carlo based algorithm has been presented for solving some stiff ordinary differential equations and system of stiff linear ordinary differential equations. The produced results have been qualitatively and quantitatively discussed.

A new class of efficient general linear methods for ordinary differential equations

We describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.

Testing the Multi-Step Single-Stage Method on Stiff Problems

A multi-step single-stage method is considered, which allows one to integrate stiff differential equations and systems of equations with high accuracy and low computational costs. The examples show that the proposed method is in solving stiff problems as good as the best available methods. The calculation results allow us to determine the absolute stability domains for the multi-step single-stage method, where it is possible to vary integration step within a wide range while maintaining the computational stability of the method.

Study on split-step Rosenbrock type method for stiff stochastic differential systems

ABSTRACT In this paper, we present a split-step Rosenbrock type method for stiff stochastic differential systems. The method is proved to be mean-square (MS) convergent with strong order . For one- and two-dimensional Itô test equations with multiplicative noise, we analysis asymptotically MS stability and plot asymptotic MS stability regions. Numerical examples and simulations are given to illustrate the effectiveness of theoretical results.

Explicit time integration of eddy current problems using a selective matrix update strategy

PurposeDiscretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.Design/methodology/approachThe resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.FindingsNumerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

Probabilistic analysis of wind-induced vibration mitigation of structures by fluid viscous dampers

The high-rise buildings usually suffer from excessively large wind-induced vibrations, and thus vibration control systems might be necessary. Fluid viscous dampers (FVDs) with nonlinear power law against velocity are widely employed. With the transition of design method from traditional frequency domain approaches to more refined direct time domain approaches, the difficulty of time integration of these systems occurs sometimes. In the present paper, firstly the underlying reason of the difficulty is revealed by identifying that the equations of motion of high-rise buildings installed with FVDs are sometimes stiff differential equations. Thus, an approach effective for stiff differential systems, i.e., the backward difference formula (BDF), is then introduced, and verified to be effective for the equation of motion of wind-induced vibration controlled systems. Comparative studies are performed among some methods, including the Newmark method, KR-alpha method, energy-based linearization method and the statistical linearization method. Based on the above results, a 20-story steel frame structure is taken as a practical example. Particularly, the randomness of structural parameters and of wind loading input is emphasized. The extreme values of the responses are examined, showing the effectiveness of the proposed approach, and also necessitating the refined probabilistic analysis in the design of wind-induced vibration mitigation systems.