A Novel Numerical Solution Method for Semi-explicit Differential-Algebraic Equations

Year 2023, Volume: 15 Issue: 1, 184 - 191, 30.06.2023

Nurettin Doğan , Hasan Hüseyin Sayan

Abstract

Generally, DAEs do not have a closed form solution, so these equations have to be solved numerically. In this work, an approximate analytic series solution of the semi-explicit DAEs is obtained by using Laplace Adomian Decomposition Method (LADM). Before directly solving the high-index semi-explicit DAEs, we apply the index reduction method to high-index semi-explicit DAEs since solving high-index semi-explicit DAEs is difficult. Then, we use the LADM obtaining the numerical solution. To show computational capability and efficiency of the LADM for the solution of semi-explicit DAEs, a couple of numerical examples are given. It has been shown that the intoduced algorithm has a very good accuricy compared with exact solution for the semi-explicit DAEs. So it can be applied to other DAEs.

Keywords

Differential algebraic equations, index reduction, Laplace transform, Adomian decomposition method, approximation solution

References

• Adomian,G., Nonlinear Stochastic Operator Equations, Academic Press, San Diego, 1986.
• Adomian,G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135(1988), 501–544.
• Adomian,G., Rach,R., Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition, J. Math. Anal. Appl., 174(1993) , 118–137.
• Adomian,G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994.
• Ali, G., Bartel, A., Rotundo, N., Index-2 elliptic partial differential-algebraic models for circuits and devices, Journal of Mathemtical Analysis and Applications, 423(2015), 1348-1369.
• Ascher, U.M., Lin, P., Sequential regularization methods for higher index differential-algebraic equations with constraint singularities: the linear index-2 case, SIAM J Anal, 33(1996),1921–1940.
• Babolian, E., Hosseini, M.M., Reducing index, and pseudospectral methods for differential-algebraic equations, Appl Math Comput, 140(2003),77–90.
• Babolian, E., Biazar,J., Vahidi, A.R., A new computational method for Laplace transforms by decomposition method, Applied Mathematics and Computation, 150(2004), 841–846.
• Bai, Z.Z., Yang, X., On convergence conditions of waveform relaxation methods for linear differential-algebraic equations, Journal of Computational and Applied Mathematics, 235(2011), 2790–2804.
• Beykal, B., Onel, M., Onel, O., Pistikopoulos, E.N., A data-driven optimization algorithm for differential algebraic equations with numerical infeasibilities, AIChE J., 66(2020), e16657.
• Brenan, K.E., Campbell, S.L., Petzold, L.R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia, 1996.
• Bujakiewicz, P., Maximum Weighted Matching for High Index Dierential Algebraic Equations. Doctor’s dissertation, Delft University of Technology, 1994.
• Dehghan,M., Shakeri,F., The numerical solution of the second Painleve equation, Numer. Methods PDEs, 25(2009),1238–1259.
• Doğan, N., Solution of the system Of ordinary differential equations by combined Laplace transform-Adomian decomposition method, Mathematical and Computational Applications An International Journal, 17(2012), 203-211.
• Doğan, N., Akin, Ö., Series solution of epidemic model, TWMS Journal of Applied and Engineering Mathematics, 2(2)(2012), 238-244.
• Doğan, N., Numerical treatment of the model for HIV infection of CD4+T cells by using multi-step Laplace Adomian decomposition method, Discrete Dynamics in Nature and Society, 2012(2012), Article ID 976352.
• DoĞan, N., Numerical solution of chaotic Genesio system with multi-step Laplace Adomian decomposition method, Kuwait Journal of Science, 40(1) (2013), 109–121.
• Hanke, M., M¨arz, R., Convergence analysis of least-squares collocation methods for nonlinear higher-index differential–algebraic equations, Journal of Computational and Applied Mathematics, 387(2021), 112514.
• Hosseini, M.M., An index reduction method for linear Hessenberg systems, J Appl Math Comput, 171(2005), 596–603.
• Khuri, S.A., A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, 1(4)(2001), 141–155.
• Peng, H., Li, F., Liu, J., Ju,Z., A symplectic instantaneous optimal control for robot trajectory tracking ith differential-algebraic equation models, in IEEE Transactions on Industrial Electronics, 67(5)(2020), 3819-3829.
• Petzold, L.R., Differential/algebraic equations are not ODE’s, SIAM Journal of Science and Statistical Computing, 3(3)(1982), 367-384.
• Pöll, C., Hafner, I., Index reduction and regularisation methods for multibody systems, IFAC-Papers OnLine, 48(2015), 306-311.
• Pulch, R., Narayan, A., Stykel, T., Sensitivity analysis of random linear dierential–algebraic equations using system norms, Journal of Computational and Applied Mathematics, 397(2021), 113666.
• Rach, R., On the Adomian decomposition method and comparisons with Picards method, J. Math. Anal. Appl., 128(1987), 480–483.
• Schwarz, D.E., Tischendorf, C., Structural analysis of electric circuits and consequences for MNA, Int. J. Circ. Theory Appl., 28(2000), 131–162.
• Schulz, S., Four Lectures on Differential-Algebraic Equations. Technical Report 497, The University of Auckland, New Zealand, 2003.
• Soltanian, F., Karbassi, S.M., Hosseini, M.M., Application of He’s variational iteration method for solution of differential-algebraic equations, Chaos, Solitons and Fractals, 41(2009), 436–445.
• Tang, J., Rao, Y., A new block structural index reduction approach for large-scale differential algebraic equations, Mathematics, 8()2020), 2057.
• Wazwaz, A.M., A comparison between Adomian decomposition method and Taylor series method in the series solutions, Appl. Math. Comput., 79(1998), 37–44.
• Wazwaz, A.M., The numerical solution of fifth-order boundary value problems by the decomposition method, J. Comput. Appl. Math., 136(2001), 259–270.
• Wazwaz, A.M., The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. Math. Comput., 118(2001), 311–325.
• Yan, X., Qian, X., Zhang, H., Song, S., Numerical approximation to nonlinear delay-differential–algebraic equations with proportional delay using block boundary value methods, Journal of Computational and Applied Mathematics, 404(2022), 113867.
• Zolfaghari, R., Taylor, J., Spiteri, R. J., Structural analysis of integro-dierential–algebraic equations, Journal of Computational and Applied Mathematics, 394(2021), 113568.