Discrete Dynamics in Nature and Society Volume, Article ID 474, pages doi:.55//474 Research Article On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Inde- Melike Karta and Ercan Çelik Department of Mathematics, Faculty of Art and Science, Ağrı Ibrahim Ceçen University, 4 Agrı, Turkey Department of Mathematics, Atatürk University Faculty of Science, 54 Erzurum, Turkey Correspondence should be addressed to Ercan Çelik, ercelik@atauni.edu.tr Received April ; Revised 8 October ; Accepted 9 November Academic Editor: Antonia Vecchio Copyright q M. Karta and E. Çelik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Numerical solution of differential-algebraic equations with Hessenberg inde- is considered by variational iteration method. We applied this method to two eamples, and solutions have been compared with those obtained by eact solutions.. Introduction Many important mathematical models can be epressed in terms of differential-algebraic equations DAEs. Many physical problems are most easily initially modeled as a system of differential-algebraic equations DAEs. Some numerical methods have been developed, using both BDF and implicit Runge-Kutta methods,padé and Chebysev approimations method 4 6. These methods are only directly suitable for low-inde problems and often require that the problem, have special structure. Although many important applications can be solved by these methods, there is a need for more general approaches. There are many new publication in the field of analytical sueveys such as 7. The variational iteration method VIM was developed by He in. The method is used by many researchers in a variety of scientific fields. The method has been proved by many authors 6 to be reliable and efficient for a variety of scientific applications, linear and nonlinear as well. The most general form of a DAE is given by F ( t,, ),.
Discrete Dynamics in Nature and Society where F/ may be singular. The rank and structure of this Jacobian matri may depend, in general, on the solution t, and for simplicity we will always assume that it is independent of t. The important special case is of a semieplicit DAE or an ODE with constraints: f t,, z,.a g t,, z..b This is a special case of.. The inde is if g/ z is nonsingular, because then one differentiation of.b yields z in principle. For the semi-eplicit inde- DAE we can distinguish between differential variables t and algebraic variables z t. The algebraic variables may be less smooth than the differential variables by one derivative. In the general case, each component of may contain a mi of differential and algebraic components, which makes the numerical solution of such high-inde problems much harder and riskier.. Special Differential-Algebraic Equations (DAEs) Forms Most of the higher-inde problems encountered in practice can be epressed as a combination of more restrictive structures of ODEs coupled with constraints. In such systems the algebraic and differential variables are eplicitly identified for higher-inde DAEs as well, and the algebraic variables may all be eliminated using the same number of differentiations. These are called Hessenberg forms of the DAE and are given below. In this paper, the variational iteration method has been proposed for solving differential-algebraic equations with Hessenberg inde-... Hessenberg Inde- One has f t,, z, g t,, z.. Here the Jacobian matri function g z is assumed to be nonsingular for all t. This is also often referred to as a semi-eplicit inde- system. Semi-eplicit inde- DAEs are very closely related to implicit ODEs... Hessenberg Inde- One has f t,, z,.a g t,..b Here the product of Jacobians g f z is nonsingular for all t. Note the absence of the algebraic variables z from the constraints.b. This is a pure inde- DAE, and all algebraic variables play the role of inde- variables.
Discrete Dynamics in Nature and Society.. Hessenberg Inde- One has f ( t,, y, z ), y g ( t,, y ),. h ( t, y ). Here the product of three matri functions h y g f z is nonsingular. The inde of a Hessenberg DAE is found, as in the general case, by differentiation. However, here only algebraic constraints must be differentiated.. He s Variational Iteration Method (VIM) Consider the differential equation Lu Nu g,. where L and N are linear and nonlinear operators, respectively, and g is the source inhomogeneous term. In, He proposed the variational iteration method where a correction functional for. can be written as u n u n λ t ( Lu n t Nũ n t g t ) dt,. where λ is a general Lagrange s multiplier, which can be identified optimally via the variational theory and ũ n as a restricted variation which means δũ n. It is to be noted that the Lagrange multiplier λ can be a constant or a function. The variational iteration method should be employed by following two essential steps. It is required first to determine the Lagrange multiplier λ that can be identified optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approimations u n, n, of the solution u. The zeroth approimation u can be any selective function. However, using the initial values u, u,andu are preferably used for the selective zeroth approimation u as will be seen later. Consequently, the solution is given by u n lim n u n.... First-Order ODEs We first start our analysis by studying the first-order linear ODE of a standard form u p u q, u α..4
4 Discrete Dynamics in Nature and Society The VIM admits the use of the correction functional for this equation by u n u n λ t ( u n t p t ũ n t g t ) dt,.5 where λ is the lagrange multiplier, that in this method may be a constant or a function, and ũ n is a restricted value where δũ n. Taking the variation of both sides of.5 with respect to the independent variable u n we have ( δu n δu n δ λ t ( u n t p t ũ n t q t dt )).6 that gives ( δu n δu n t δ ) λ t u n t dt.7 obtained upon using δũ n andδq t. Integrating the integral of.6 by parts we obtain δu n δu n δλu n δ λ u n dt.8 or equivalently δu n δ λ t u n δ λ u n dt..9 The etremum condition of u n requires that u n. This means that the left hand side of.9 is, and as a result the right hand side should be as well. This yields the stationary conditions λ t, λ t.. This in turn gives λ.. Substituting this value of the lagrange multiplier into the functional.5 gives the iteration formula u n u n ( u n t p t u n t q t ) dt,.
Discrete Dynamics in Nature and Society 5 obtained upon deleting the restriction on u n that was used for the determination of λ. Considering the given condition u α, we can select the zeroth approimation u o α. Using the selection into. we obtain the following successive approimations: u t α, u α u u u u ( u t p t u t q t ) dt, ( u t p t u t q t ) dt, ( u t p t u t q t ) dt,. u n u n. ( u n t p t u n t q t ) dt. Recall that u lim n u n,.4 that may give the eact solution if a closed form solution eists, or we can use the n th approimation for numerical purposes. 4. Applications Eample 4.. We first considered the following differential-algebraic equations with Hessenberg inde- form:,, 4. e with initial conditions. 4.
6 Discrete Dynamics in Nature and Society The eact solutions are e, e, e, 4. where, represent the differential variables and represents the algebraic variables. After three times of differentiation of 4. we have the following ODE system: e, e, 4.4. Differential-algebraic equation DAE is a Hessenberg inde- form. To solve system 4.4, we can construct following correction functionals: ( n n λ t n t e t) dt, ( n n λ t n t e t) dt, 4.5 ( ) n n λ t n t t n n t dt, where λ t, λ t, and,λ t are general Lagrange multipliers and n, n denote restricted variations, that is, δ n δ n. Making the above correct functional stationary, ( δ n δ n δ λ t n t e t) dt, ( δ n δ n δ λ t n t e t) dt, ( δ n δ n δ λ t n t t n t n )dt, t t δ n δ n δλ t n t δ n δ n δλ t n t δ n δ n δλ t n t λ t δ n t dt, λ t δ n t dt, λ t δ n t dt. 4.6
Discrete Dynamics in Nature and Society 7 Its stationary conditions can be obtained as follows: λ t λ t λ t, λ t t λ t t λ t t. 4.7 The Lagrange multipliers can be identified as follows: λ t λ t λ t, 4.8 and the following multipliers can be obtained as ( n n n t e t) dt, ( n n n t e t) dt, 4.9 ( ) n n n t t n n t dt. Beginning with,, by the iteration formula 4.9, we have 6 4 4 5 7 6 54 7 6 4 4 5 7 6 54 7 4 8 4 8 688 9, 688 9, 5 4 4 5 7 7 6 6 7 448 8 688 9. 4. Eample 4.. One has v v v e v v v 4. with initial conditions. 4.
8 Discrete Dynamics in Nature and Society Table : Numerical solution of....5798.5798. 9..4758.4758. 9..4985888.49858876.4 9.4.4984698.49846977. 9.5.64877.648775.5 9.6.888.887987. 8.7.7577.75699.8 8.8.55498.554897. 7.9.4596.45966.5 6..78888.78857. 6 4 6 4 4 6 () () Figure : Values of and its variational iteration. The eact solutions are v e, v, v : v v v v, v e v v, 4. v,
Discrete Dynamics in Nature and Society 9 Table : Numerical solution of....95798.95798. 9..84758.84758. 9..74985888.749858876.4 9.4.6984698.69846977. 9.5.64877.648775.5 9.6.688.687987. 8.7.67577.675699.78 8.8.655498.65548967. 7.9.6596.659666.44 6..78888.788556.4 6 4 5 6 6 4 4 6 () () Figure : Values of and its variational iteration. where v, v represent the differential variables and v represents the algebraic variables. After three times of differentiation of 4. we have the following ODE system: v v v v, v e v v, 4.4 v. Differential-algebraic equation DAE is a Hessenberg inde- form.
Discrete Dynamics in Nature and Society Table : Numerical solution of....95688.95688..8568.85689. 8..5748645.5748645. 8.4.768554577.768554576. 8.5.97896.9789. 8.6.9598.9596.9 7.7.44796.44795.9 7.8.759767.75977.5 6.9.5459.54475.6 5..4656656.465695.6 5 8 6 4 6 4 4 6 () () Figure : Values of and its variational iteration. To solve system 4.4, we can construct the following correction functionals: ( v n v n λ t v n t ṽ n t ṽ n t tṽ )dt, t t ( v n v n λ t v n t e t v n ) t t ṽ n t t t dt, 4.5 v n v n λ t v n t dt.
Discrete Dynamics in Nature and Society Table 4: Numerical solution of v. v v v v...948748.948748. 9..88775.88775. 9..74887.74886. 9.4.6746.6746. 9.5.6656597.6656595. 9.6.548866.5488645.6 8.7.4965858.496585966.7 8.8.4998964.4998965.76 7.9.465696597.4656957.887 7..6787944.67879888.54 6 8 6 4 6 4 4 6 v () v () Figure 4: Values of v and its v variational iteration. By using the basic definition of the variational iteration method can obtain that v 6 4 4 5 7 6 54 7 v, 4 8 688 9, v. 4.6
Discrete Dynamics in Nature and Society 5. Conclusion The method has been proposed for solving differential-algebraic equations with Hessenberg inde-. Results show the advantages of the method. Tables 4 and Figures 4 show that the numerical solution approimates the eact solution very well in accordance with the above method. References K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Elsevier, New York, NY, USA, 989. U. M. Ascher, On symmetric schemes and differential-algebraic equations, SIAM Journal on Scientific Computing, vol., no. 5, pp. 97 949, 989. C. W. Gear and L. R. Petzold, ODE methods for the solution of differential/algebraic systems, SIAM Journal on Numerical Analysis, vol., no. 4, pp. 76 78, 984. 4 E. Çelik, E. Karaduman, and M. Bayram, Numerical method to solve chemical differential-algebraic equations, International Quantum Chemistry, vol. 89, pp. 447 45,. 5 E. Çelik and M. Bayram, On the numerical solution of differential-algebraic equations by Padé series, Applied Mathematics and Computation, vol. 7, no., pp. 5 6,. 6 M. Bayram and E. Çelik, Chebysev approimation for numerical solution of differential-algebraic equations DAEs, International Applied Mathematics & Statistics, pp. 9 9, 4. 7 Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, Analytical solution of time-fractional Navier- Stokes equation in polar coordinate by homotopy perturbation method, Numerical Methods for Partial Differential Equations, vol. 6, no., pp. 7 4,. 8 M. Shateri and D. D. Ganji, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique, International Differential Equations, vol., Article ID 954674, pages,. 9 S. R. Seyed Alizadeh, G. G. Domairry, and S. Karimpour, An approimation of the analytical solution of the linear and nonlinear integro-differential equations by homotopy perturbation method, Acta Applicandae Mathematicae, vol. 4, no., pp. 55 66, 8. A. R. Sohouli, M. Famouri, A. Kimiaeifar, and G. Domairry, Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed surface heat flu, Communications in Nonlinear Science and Numerical Simulation, vol. 5, no. 7, pp. 69 699,. J. H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, vol. 4, no. -, pp. 5,. J. Biazar and H. Ghazvini, He s variational iteration method for solving linear and non-linear systems of ordinary differential equations, Applied Mathematics and Computation, vol. 9, no., pp. 87 97, 7. A. M. Wazwaz, The variational iteration method for analytic treatment of linear and nonlinear ODEs, Applied Mathematics and Computation, vol., no., pp. 4, 9. 4 D. D. Ganji, E. M. M. Sadeghi, and M. Safari, Application of He s variational iteration method and adomian s decom- position method method to Prochhammer Chree equation, International Modern Physics B, vol., no., pp. 45 446, 9. 5 M. Safari, D. D. Ganji, and M. Moslemi, Application of He s variational iteration method and Adomian s decomposition method to the fractional KdV-Burgers-Kuramoto equation, Computers & Mathematics with Applications, vol. 58, no. -, pp. 9 97, 9. 6 D. D. Ganji, M. Safari, and R. Ghayor, Application of He s variational iteration method and Adomian s decomposition method to Sawada-Kotera-Ito seventh-order equation, Numerical Methods for Partial Differential Equations, vol. 7, no. 4, pp. 887 897,.
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