Local Linearization Scheme Research Articles

Numerical Investigation of a Combustible Polymer in a Rectangular Stockpile: A Spectral Approach

Despite the wide application of combustion in reactive materials, one of the challenges faced globally is the auto-ignition of such materials, resulting in fire and explosion hazards. To avoid this unfortunate occurrence, a mathematical model describing the thermal decomposition of combustible polymer material in a rectangular stockpile is formulated. A nonlinear momentum equation is provided with the assumption that the combustible polymer follows a Carreau constitutive relation. The chemical reaction of the polymer material is assumed to be exothermic; therefore, Arrhenius’s kinetic theory is considered in the energy balance equation. The bivariate spectral local linearization scheme (BSLLS) is utilized to provide a numerical solution for the dimensionless equations governing the problem. The obtained results are validated by the collocation weighted residual method (CWRM), and a good agreement is achieved. Dimensionless velocity, temperature, and thermal stability results are presented and explained comprehensively with suitable applications. Some of the obtained results show that thermal criticality increases with increasing power law index (n) and radiation (Ra) values and decreases with increasing variable viscosity (β1) and material parameter (We) values.

Develop blood oxygen level dependent signal by metabolic/hemodynamic model using numerical methods

Background and objectiveThe metabolic/hemodynamic (MH) model describes the blood flow mechanisms as well as the coupling between the hemodynamic responses and the metabolic activities in a blood vessel in the human brain. In the existing MH model, the blood flow out from a blood vessel is formulated as dependent only on the capillary volume. In fact, the blood flow out from a blood vessel depends not only on the capillary volume but also on the blood flow into the capillary bed. For this reason, the blood flow out formula of the existing model has been modified. In addition to implementing existing model modification to obtain better accuracy, we have used new methods to solve the model instead of conventional methods. MethodThe MH model describes physical phenomena of a blood vessel by eight processes equations (PEs). These PEs are often solved by using a local linearization (LL) scheme and the Taylor series method. In addition to the previously used Taylor series method, we have also used the Euler method and the Runge–Kutta (RK) method to solve the model instead of a LL scheme for estimating dynamical variables (DVs). By using these DVs, a Blood Oxygen Level Dependent (BOLD) signal is generated through a well-defined observation equation (OE). There are two OEs, called Obata and Friston. The Friston OE produces a BOLD signal from the cerebral blood volume and deoxy-hemoglobin content with their nonlinear properties; conversely, the Obata OE produces a BOLD signal without considering nonlinear properties. For this reason, we have used the Friston OE instead of the Obata OE to estimate the BOLD signal perfectly. ResultsAt 20% resting oxygen extraction fraction (ROEF), the BOLD signals of the modified and the existing model are identical, but when the ROEF increased up to 50% at its standard value, the modified model accuracy is increased by 16.12%–23.07% more than that of the existing model. The Euler and RK methods generate a BOLD signal 6.95% more accurately than that of Taylor series method from the modified model. ConclusionIn the model inversion process, this research will be helpful to estimate the model parameters and hidden states accurately.

Numerical analysis of activation energy on MHD nanofluid flow with exponential temperature-dependent viscosity past a porous plate

The current investigation scrutinized the incompressible steady flow with temperature-dependent viscosity of magnetohydrodynamics nanofluid through a vertically stretched porous sheet. The Reynolds exponential model is employed to formulate the mathematical modeling. The momentum equation is further devised utilizing the Darcy–Brinkman–Forchheimer model. Electrically conducting nanofluids encompass uniformly suspended nanoparticles in the viscous base fluid. The Buongiorno model is adopted that aspects the behavior of thermophoretic forces and Brownian motion. The momentum, mass conservative, energy, and nanoparticle concentration equations are defined with magnetic body force. The looming nonlinear coupled differential equations are resolved numerically by employing the spectral local linearization method (SLLM). The SLLM algorithm is straightforward to develop and apply, as it is based on a smooth univariant linearization of nonlinear functions. The numerical performance of SLLM is more impressive as it grows a set of equations; those are successively solved by operating the results from the one equation into the subsequent equation. To accelerate and improve the convergence for the SLLM scheme, the successive over relaxation scheme has been utilized. The accuracy of the SLLM will be confirmed through the known methods, and convergence analysis is also presented. Graphical conduct for all the emerging parameters across temperature, velocity, and concentration distributions, as well as the Nusselt number, skin friction, and Sherwood number, is presented and discussed in detail. A comparative study of the novel proposed technique along with the preceding explored literature is also granted. It is costly to affirm that the spectral local linearization scheme is uncovered to be much stable and adaptable to solve the nonlinear problems.

On local linearization method for stochastic differential equations driven by fractional Brownian motion

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method.

A weak Local Linearization scheme for stochastic differential equations with multiplicative noise

In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The scheme is explicit, preserves the first two moments of the solution of SDEs with linear drift and diffusion coefficients in state and time, and inherits the mean-square stability or instability that such solution may have. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme.

Convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise

There exists a diversity of weak Local Linearization (LL) schemes for the integration of stochastic differential equations with additive noise, which differ in the algorithms employed for the numerical implementation of the weak Local Linear discretizations. Despite convergence results for these discretizations have been already developed, the convergence of the weak LL schemes has not been considered up to date. In this work, a general result concerning the convergence rate of the weak LL schemes is presented, as well as specificities for a number of particular schemes. As an application, the convergence of weak LL schemes for equations driven by Poisson processes is presented in addition.

Local Linearization—Runge–Kutta methods: A class of A-stable explicit integrators for dynamical systems

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge–Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.

Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.

Efficient quantile estimation for functional-coefficient partially linear regression models

The quantile estimation methods are proposed for functional-coefficient partially linear regression (FCPLR) model by combining nonparametric and functional-coefficient regression (FCR) model. The local linear scheme and the integrated method are used to obtain local quantile estimators of all unknown functions in the FCPLR model. These resulting estimators are asymptotically normal, but each of them has big variance. To reduce variances of these quantile estimators, the one-step backfitting technique is used to obtain the efficient quantile estimators of all unknown functions, and their asymptotic normalities are derived. Two simulated examples are carried out to illustrate the proposed estimation methodology.

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.

Rate of convergence of Local Linearization schemes for random differential equations

Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.

Numerical simulation of nonlinear dynamical systems driven by commutative noise

The local linearization (LL) approach has become an effective technique for the numerical integration of ordinary, random and stochastic differential equations. One of the reasons for this success is that the LL method achieves a convenient trade-off between numerical stability and computational cost. Besides, the LL method reproduces well the dynamics of nonlinear equations for which other classical methods fail. However, in the stochastic case, most of the reported works has been focused in Stochastic Differential Equations (SDE) driven by additive noise. This limits the applicability of the LL method since there is a number of interesting dynamics observed in equations with multiplicative noise. On the other hand, recent results show that commutative noise SDEs can be transformed into a random differential equation (RDE) by means of a random diffeomorfism (conjugacy). This paper takes advantages of such conjugacy property and the LL approach for defining a LL scheme for SDEs driven by commutative noise. The performance of the proposed method is illustrated by means of numerical simulations.

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.

Rate of convergence of local linearization schemes for initial-value problems

There is a large variety of Local Linearization (LL) schemes for the numerical integration of initial-value problems, which differ with respect to the algorithm that is used in the numerical implementation of the Local Linear Discretization. However, in contrast with the LL Discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general result about that matter is given. In addition, a brief survey of the main implementations of the LL method is also presented.

A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations

In this paper, a simple algebraic expression to evaluate the local linearization scheme for stochastic differential equations is introduced. In this way, the computation of the deterministic part of LL schemes is reduced to the evaluation of a matrix exponential, which reduces considerably the computational cost of these schemes.

Dynamic properties of the local linearization method for initial-value problems

Some dynamic properties of the local linearization (LL) scheme for the numerical integration of initial-value problems in ordinary differential equations (ODEs) are investigated. Specifically, the general conditions under which this scheme preserves the stationary points and periodic orbits of the ODEs and the local stability at these steady states are studied. These dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes. In addition, a brief review of the computational implementations of the LL scheme is also presented.

The local linearization scheme for nonlinear diffusion models with discontinuous coefficients

Abstract The local linearization scheme, defined in Shoji and Ozaki [1997, J. Time Ser. Anal. 18, 485–506] for diffusions with smooth coefficients, is studied under some regularity conditions when the coefficients are not necessarily continuous. It is shown that the local scheme converges weakly to the diffusion itself. These results are applied to continuous-time threshold autoregressive moving-average processes and multi-dimensional continuous-time threshold AR models.

Simulation of Stochastic Differential Equations Through the Local Linearization Method. A Comparative Study

A new local linearization (LL) scheme for the numerical integration of nonautonomous multidimensional stochastic differential equations (SDEs) with additive noise is introduced. The numerical scheme is based on the local linearization of the SDE's drift coefficient by means of a truncated Ito–Taylor expansion. A comparative study with the other LL schemes is presented which shows some advantanges of the new scheme over other ones.

Computing the noise covariance matrix of the local linearization scheme for the numerical solution of stochastic differential equations

An algorithm is given that computes the covariance matrix of the noise term of the local linearization scheme for the numerical integration of stochastic differential equations. The order of convergence of the resulting approximation is studied. An example is presented that illustrates the performance of the algorithm.