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# A novel technique to solve nonlinear higher-index Hessenberg differential–algebraic equations by Adomian decomposition method

- Brahim Benhammouda
^{1}Email author

**Received:**30 December 2015**Accepted:**21 April 2016**Published:**11 May 2016

## Abstract

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential–algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

## Keywords

- Differential–algebraic equations
- Adomian decomposition method
- Analytical methods

## Mathematics Subject Classification

- 34A09
- 65L80
- 37Mxx

## Background

Systems of differential–algebraic equations (DAEs) are often used to model many important problems in real applications. These equations arise, for instance, in electrical networks, optimal control, mechanical systems, incompressible fluids dynamics and chemical process simulations. DAEs are characterized by the so called index, which has several definitions (Günther and Wagner 2001; Kunkel and Mehrmann 1996; Martinson and Barton 2000). The most used index is the differentiation index, which is the minimum number of times that all or a part of the DAE must be differentiated with respect to time, in order to obtain an ordinary differential equation (Martinson and Barton 2000). In real applications, higher-index DAEs (differentiation index greater than one) arise naturally in many important application problems like constrained multibody systems (Benhammouda and Vazquez-Leal 2015; Simeon 1996; Simeon et al. 1994), vehicle system dynamics (Simeon et al. 1991), space shuttle simulation (Brenan 1983) and incompressible fluids dynamics. Unfortunately, higher-index DAEs are known to be difficult to solve. Therefore, they are usually transformed to ordinary differential systems (index-zero) or index-one DAEs before solving them. This transformation, called index-reduction, can be computationally very expensive and may also change the properties of the solution. Therefore, new techniques are required to solve higher-index DAEs efficiently.

The Adomian decomposition method (ADM) and its modifications (Adomian and Rach 1985; Adomian 1988; Almazmumy et al. 2012; Fatoorehchi et al. 2015; Hendi et al. 2012; Pue-on and Viryapong 2012; Ramana and Raghu Prasad 2014; Wazwaz 2001) are known to be efficient methods in solving a large variety of linear and nonlinear problems in science and engineering. Among these problems, we mention algebraic equations (Adomian and Rach 1985), ordinary differential equations (Almazmumy et al. 2012; Fatoorehchi et al. 2015; Hendi et al. 2012; Pue-on and Viryapong 2012; Ramana and Raghu Prasad 2014; Wazwaz 2001), partial differential equations (Adomian 1984) and integral equations (Bakodah 2012).

However, for the application of the ADM to DAEs, one finds only four pieces of work in the literature (Duan and Sun 2014; Çelika et al. 2006; Hosseini 2006a, b). In Duan and Sun (2014), an index-one DAE is transformed to a second order ordinary differential equation before applying the ADM to it. In Çelika et al. (2006), the ADM is applied to simple semi-explicit index-one DAEs, where the DAE is first transformed to a system of ordinary differential equations before applying the ADM. In Hosseini (2006a), the ADM is applied to linear higher-index Hessenberg DAEs after transforming them to index-one DAEs. In Hosseini (2006b), index-one and index-two DAEs with linear constraints are solved where these DAEs are pre-processed by a transformation that relies much on the special forms they have. Therefore, in all these previous works, the ADM is not applied directly to the DAEs but rather to the transformed equations. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions.

In this work, we present a new procedure for solving a class of nonlinear higher-index Hessenberg DAEs based on the ADM. The ADM is first applied directly to the DAE where the nonlinear terms are expanded using the Adomian polynomials (Duan 2010a, b, 2011, Rach 2008, 1984; Wazwaz 2000). Based on the index condition, a nonsingular algebraic recursion system is derived for the expansion components of the solution. Also, it is important to note that unlike previous works (Duan and Sun 2014; Çelika et al. 2006; Hosseini 2006a, b), our procedure does not make transformations to the DAEs before applying the ADM to them. To demonstrate the effectiveness of the proposed technique, we solve a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. Further, our technique is based on a simple algorithm that can be programmed in Maple or Mathematica to simulate real application problems.

This paper is organized as follows: in “Review of the Adomian decomposition method” section, we review the ADM for solving ordinary differential equations. Next, in “The proposed method” section, we present our method for the solution of nonlinear higher-index Hessenberg DAEs systems. Then in “Application to a nonlinear index-3 DAEs system” section, we apply the developed technique to solve a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. Finally, a discussion and a conclusion are given in “Discussion” and “Conclusion” sections, respectively.

## Review of the Adomian decomposition method

*L*is an easily invertible operator (usually taken as the highest-order derivative),

*R*is an operator grouping the remaining lower-order derivatives, \(N\left( u\right)\) is the nonlinear term and

*f*is a given analytical function.

*Lu*then applying the inverse operator \(L^{-1}\) to both sides, we obtain

*t*and \(L^{-1}Lu=u-c_{0}.\) If \(Lu=\ddot{u}=u^{(2)}=d^{2}u/dt^{2}\) and the initial conditions \(u\left( t_{0}\right) =c_{0}\) and \(\dot{u}\left( t_{0}\right) =c_{1}\) are given, then \(L^{-1}\) is the double fold integral from \(t_{0}\) to

*t*and \(L^{-1}Lu=u-c_{0}-c_{1}\left( t-t_{0}\right)\). In this case from (2), we have

*u*of (1) to have the infinite series form

*N*(

*u*) is expanded in an infinite series in terms of the Adomian polynomials \(N_{n}\) (Duan 2010a, b, 2011; Rach 2008, 1984; Wazwaz 2000) as

*K*to solution can be obtained from

*m*-variable case is recently proposed in Duan (2011)

## The proposed method

In this section, we present our method for solving a class of nonlinear higher-index Hessenberg differential–algebraic equations (DAEs). This technique is based on the Adomian decomposition method (ADM). To solve this class of DAEs, we first apply the ADM directly to it and expand the nonlinear terms using the Adomian polynomials. Then, an algebraic recursion system for the solution expansion components is derived. Taking account of the index of the DAE, this system is shown to be uniquely solvable for the solution expansion components. Also, it is important to note that unlike previous works (Çelika et al. 2006; Duan and Sun 2014, 2006a, b), our technique does not make transformations to DAEs before applying the ADM to them.

*v*. We also assume that DAE initial-value problem (13)–(14) has a unique analytical solution.

*u*and

*v*are called the differential and the algebraic variables respectively. System (13) is index \((m+1)\) if the square matrix

Systems (13) arise frequently in many important applications like Navier–Stokes equations in incompressible fluids dynamics or Euler–Lagrange equations in constrained multibody systems. In what follows, we assume that system (13) is index \((m+1)\) that is \((m+1)\)-index condition (15) holds.

*m*-fold integral from \(t_{0}\) to

*t*) to both sides of the first equation of (13) to get

*u*and

*v*of the solution of (13) to have the infinite series form

*N*(

*u*) in infinite series using the Adomian polynomials \(M_{n}\) and \(N_{n}\) as

*m*terms \(u_{n}\) as

*v*and this system is nonlinear with respect to \(v_{0}\) if \(M\left( u,v\right)\) is nonlinear with respect to

*v*. Since index condition (15) holds, the Jacobian matrix

*m*times with respect to

*t*, we determine the unknown \(v_{n-m}\)

*K*is the order of approximation of

*u*(

*t*).

It is worth noting that Eq. (21) is linear with respect to \(v_{n-m}\) for all values of \(n\ge m\), except for the case \(n=m\) where (21) is nonlinear with respect to \(v_{0}\) if the given function *M*(*u*, *v*) is nonlinear with respect to *v*. This linearity property of Eq. (21) has a great positive impact on the simplicity of our method and its efficiency. Note here also that many important problems arising from applications like constrained mechanical systems and the semi-discrete form of Navier–Stokes equations lead to DAEs systems of the form of (13 ), where *M*(*u*, *v*) is linear with respect to *v*. For these problems, system ( 21) is linear for all values of \(n\ge m\).

## Application to a nonlinear index-3 DAEs system

In this section, we illustrate and demonstrate the effectiveness of our technique to solve nonlinear higher-index Hessenberg DAEs systems, which are known to be difficult to solve even numerically. Following the procedure developed in the previous section, we first apply the ADM directly to the DAEs system and expand the nonlinear terms using the Adomian polynomials. Then, taking account of the index of the DAE, we derive a nonsingular algebraic recursion system for the expansion components of the solution. Finally, by solving this algebraic system we obtain the solution of the DAE.

To solve DAEs initial-value problem (36)–(37) by the procedure developed in the previous section, we let \(Lu=\ddot{u}\) and have \(L^{-1}u=\int _{0}^{t}\int _{0}^{t}udtdt.\)

*v*of (36) to have the form

Now, we solve system (44) recursively to obtain the solution of DAEs system (36)–(37).

## Discussion

System of higher-index differential–algebraic equations (DAEs) still require new numerical and analytical methods to solve them efficiently. Such problems are known to be difficult to solve. In this paper, we developed a novel technique that applies the Adomian decomposition method (ADM) directly to solve a class of nonlinear higher-index Hessenberg DAEs. Our technique has successfully handled this class of DAEs without the need for complex transformations like index-reductions. The proposed method transforms these DAEs into easily solvable algebraic systems for the expansion components of the solution. To demonstrate the effectiveness of our technique, we solve one nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints is solved. It is important to note that nonlinear algebraic constraints make the DAE more difficult to solve. In particular some transformations like those in Hosseini (2006b) cannot not be used. This example shows that the direct application of the ADM is a simple powerful technique to obtain the exact or approximate solutions of nonlinear higher-index Hessenberg DAEs. In the case we want to solve a DAE with an unknown solution, one way to measure the error for the approximate solution is to use the mean square residual (MSR) as in Benhammouda and Vazquez-Leal (2015) since the convergence of the method is still not shown.

## Conclusion

This work presents the analytical solution of a class of nonlinear higher-index Hessenberg DAEs using the Adomian decomposition method (ADM). A procedure for solving this class of DAEs is presented. For this class, the technique was tested on a nonlinear higher-index Hessenberg DAEs system with nonlinear algebraic constraints. The results obtained show that the method can be applied to nonlinear higher-index Hessenberg DAEs efficiently to obtain the exact or an approximate solution. On the one hand, it is important to note that these types of DAEs are difficult to solve and on the other, the direct application of the ADM was able to solve this class of nonlinear higher-index Hessenberg DAEs. Also, it is important to note that unlike previous works (Duan and Sun 2014; Çelika et al. 2006; Hosseini 2006a, b), our procedure does not make transformations to DAEs before applying the ADM to them. Our technique is based on a straightforward procedure that can be programmed in Maple or Mathematica to simulate real application problems. Finally, further work is needed to show the convergence of the proposed method, apply it with its modifications (for example a multistage ADM) to solve nonlinear higher-index Hessenberg partial differential–algebraic equations and other nonlinear higher-index DAEs.

## Declarations

### Competing interests

The author declare that he has no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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