Numerical method to solve chemical differential‐algebraic equations

a b s t r a c tIn this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane-Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane-Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Padé series form, we obtain an approximate polynomial of arbitrary order for solving Lane-Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

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“…. , M; such as [16]. Therefore, we have constructed an [L/M] Padé approximation, which agrees with ∞ i=0 a i x i through order x L+M .…”

Section: The Proposed Methodsmentioning

confidence: 95%

“…Notice that in (16) there are L + 1 numerator coefficients and M + 1 denominator coefficients. They have a more or less irrelevant common factor, and for definiteness we take q 0 = 1.…”

Section: The Proposed Methodsmentioning

confidence: 99%

On the numerical solution of differential equations of Lane–Emden type

Vanani

Aminataei

2010

Computers & Mathematics with Applications

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“…The effectiveness of the UKF and URTSS algorithms proposed here for nonlinear DAE systems is demonstrated by an electrochemical reaction considering a galvanostatic charge process of a thin-film nickel hydroxide electrode [35]. The problem was investigated in [29], [30] to examine the performance of EKF and UKF algorithms.…”

Section: Illustrative Examplementioning

confidence: 99%

“…Motivated by the need for state estimation in several engineering applications [5], [6], [30], [35], this work attempts to provide a filter and smoother for nonlinear descriptor systems that yield the mean and covariance of all state estimates. The unscented transform (UT) [36]- [39] is particularly attractive in this context, namely due to its superior performance in providing Gaussian approximations to random variables that undergo nonlinear transformations.…”

Section: Introductionmentioning

confidence: 99%

Unscented Rauch-Tung-Striebel smoothing for nonlinear descriptor systems

Mercieca

Aram

Kadirkamanathan

2015

2015 European Control Conference (ECC)

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This paper extends the application of the unscented Rauch-Tung-Striebel smoother to nonlinear descriptor systems, where the mean and covariance estimates of the algebraic states can also be computed. The consistency in the smoother solution is ensured through the unscented transformation of the differential state estimates. The method also allows for measurements that are functions of both differential and algebraic states. The performance of the proposed smoother is demonstrated by an electrochemical case study. I. INTRODUCTIONDescriptor models, or differential-algebraic equations (DAEs), provide a natural mathematical representation for systems that cannot be modelled as entirely dynamic or entirely static [1]. This enables modelling several timeevolutionary phenomena such as ordinary state-space equations, combinations of static and dynamic equations and noncausal systems [1], and has therefore been a topic of research over recent decades.DAE systems are clearly distinguished from systems of ordinary differential equations (ODEs) [2] and are generally characterized by their differentiation index, typically defined as the minimum number of differentiations required in order to obtain an explicit ODE formulation [3]. Such systems frequently arise in applications spanning fields as diverse as chemical engineering [4], fluid dynamics [5], electronic network modelling [6] and robotics [7], to name a few. Taking chemical kinetics as an example, reaction rates are described by differential equations, while algebraic equations generally represent equilibrium properties, pseudo-stationary assumptions and charge or mass balances [4].The problem of state estimation for linear descriptor systems has been well developed by several authors [8]- [12]. By contrast, most of the research efforts in designing observers and filters for the nonlinear case are more recent [13], [14]. By dividing the system into a dynamic system and a static one, a full-order observer is reported in [15] for a class of nonlinear descriptor systems subject to unknown inputs and faults. Other researchers have endeavoured to design observers for nonlinear DAE systems described in terms of a linear part and a nonlinearity which is assumed to be Lipschitz [16]-[18]. This restriction is relaxed in the work of Yang et al. [19], [20], however the nonlinearity is assumed to obey a quadratic inequality so that the observer error system is represented as a Lur'e descriptor system.

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“…In [11], the implicit Runge-Kutta method has been applied to solve the DAEs numerically. Ç elik et al, in [12], have presented approximate solutions for chemical DAEs using the Padé series method. Also, Ç elik and Yeloglu [13] have used the Chebyshev series approximation for solving other kinds of DAEs.…”

Section: Introductionmentioning

confidence: 99%

A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-Series Method

Moaddy

Al‐Smadi

Hashim

2015

Discrete Dynamics in Nature and Society

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We implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula. The solution methodology is based on generating the residual power series expansion solution in the form of a rapidly convergent series with easily computable components. The residual power series method (RPSM) can be used as an alternative scheme to obtain analytical approximate solution of different types of differential algebraic equations system applied in mathematics. Simulations and test problems were analyzed to demonstrate the procedure and confirm the performance of the proposed method, as well as to show its potentiality, generality, viability, and simplicity. The results reveal that the proposed method is very effective, straightforward, and convenient for solving different forms of such systems.

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Numerical method to solve chemical differential‐algebraic equations

Cited by 32 publications

References 14 publications

On the numerical solution of differential equations of Lane–Emden type

On the numerical solution of differential equations of Lane–Emden type

Unscented Rauch-Tung-Striebel smoothing for nonlinear descriptor systems

A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-Series Method

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