- Bateman H., Some recent researches on the motion of fluids. Monthly Weather Review, Vol. 43, No. 4, pp. 163–170, 1915.
- Burgers J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, Vol. 1, pp. 171–199, 1948.
- San O., Analysis of low-pass filters for approximate deconvolution closure modelling in one-dimensional decaying Burgers turbulence, International Journal of Computational Fluid Dynamics, Vol. 30, No. 1, pp. 20-37, 2016.
- Rashidi M. M, Erfani E., New analytical method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM, Computer Physics Communications, Vol. 180, pp. 1539–1544, 2009.
- Yu L, Zhou B., The Burgers Equation for a New Continuum Model with Consideration of Driver’s Forecast Effect, Journal of Applied Mathematics, Vol. 2014, pp. 1-7, 2014.
- Kuo C. K, Lee S. Y., A New Exact Solution of Burgers’ Equation with Linearized Solution, Mathematical Problems in Engineering, Vol. 2015, Article ID 414808, 7 pages, 2015.
- Binatari N., A comparison between Cole-Hopf Tranformation and Homotopy Perturbation Method for Viscous Burger Equation in Traffic Flow, Journal of Physics: Conference Series, The 2nd International Seminar on Innovation in Mathematics and Mathemathics Education, Yogyakarta, Indonesia, 2018.
- EL-Kalaawy O. H., Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation, Computers & Mathematics with Applications, Vol. 72, No. 4, pp. 1031-1041, 2016.
- محمودی ع. رفعی ر., اثر هندسهی نازل بر عملکرد آن در حالت خارج از طرح در جریان دارای شوک و جدایش لایه مرزی. مهندسی مکانیک دانشگاه تبریز. د. 51، ش. 2، ص 205-213، 1400.
- Rahmani B, Moosaie A, Mansourian Tabaei A., Distributed control of nonlinear Burger’s equation, Modares Mechanical Engineering, Vol. 15, No. 4, pp. 214-220, 2015.
- حقیقی ا. ر. احمدی شالی ج. امامعلی پور ح. اصغری ن., مقایسه روشهای عددی تجزیه آدومیان و کرانک- نیکلسون بهبود یافته برای معادله برگرز دوبعدی. مهندسی مکانیک دانشگاه تبریز. د. 49، ش. 2، ص 61-67، 1398.
- Singh B. K, Gupta M., A new efficient fourth order collocation scheme for solving Burgers’ equation, Applied Mathematics and Computation, Vol. 399, 126011, 2021.
- Yang X, Ge Y, Zhang L., A class of high-order compact difference schemes for solving the Burgers’ equations. Applied Mathematics and Computation, Vol. 358, pp. 394-417, 2019.
- Breiten T, Damm T., Krylov subspace methods for model order reduction of bilinear control systems, Systems & Control Letters, 59, pp. 443-450, 2010.
- Al-Baiyat S. A, Bettayeb M, Al-Saggaf U. M., New model reduction scheme for bilinear systems, International Journal of Systems Science, Vol. 25, pp. 631–1642, 1994.
- Redmann M, Duff I. P., Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems, Applied Mathematics and Computation, Vol. 420, 126561, 2022.
- Benner P, Goyal P, Gugercin S., H2-quasi-optimal model order reduction for quadratic-bilinear control systems, SIAM Journal on Matrix Analysis and Application, Vol. 39, No. 2, pp. 983–1032, 2019.
- Ahmad M. I, Baur U, Benner P, Implicit volterra series interpolation for model reduction of bilinear systems, Journal of Computational and Applied Mathematics, Vol. 316, pp. 15-28, 2017.
- Goyal PK, Ahmad MI, Benner P. Model reduction of quadratic-bilinear descriptor systems via carleman bilinearization, European Control Conference, Linz, Austria, pp. 1177-1182, 2015.
- Nguyen V. B, Buffoni M, Willcox K, Khoo B. C., Model reduction for reacting flow applications, International Journal of Computational Fluid Dynamics, Vol. 28, No. 3-4, pp. 91-105, 2014.
- Hsu C. H, Desai U. B, Crawley C. A., Realization algorithms and approximation methods of bilinear systems, 22nd IEEE Conf. Decision. Control, San Antonio, Texas, pp. 783-788, 1983.
- Duff I. P, Goyal P, Benner P., Balanced truncation for a special class of bilinear descriptor systems, IEEE Control Systems Letters, Vol. 3, No. 3, pp. 535-540, 2019.
- Al-Baiyat S, Farag A. S, Bettayeb M., Transient approximation of a bilinear two-area Interconnected Power System, Electric Power Systems Research, Vol. 26, No. 1, pp. 11–19, 1993.
- Zhang L. Q, Lam J, Huang B, Yang G. H., On gramians and balanced truncation of discrete-time bilinear systems, International Journal of Control, Vol. 76, No. 4, pp. 414–427, 2003.
- Philips J. R., Projection frameworks for model reduction of weakly nonlinear systems, Proceedings 37th Design Automation Conference, Los Angeles, California, USA, pp. 184-189, 2000.
- Lin Y, Bao L, Wei Y., A model-order reduction method based on Krylov subspace for MIMO bilinear dynamical systems, Journal of Applied Mathematics and Computing, Vol. 25, pp. 293-304, 2007.
- Lin Y, Bao L, Wei Y., Order reduction of bilinear MIMO dynamical systems using new block krylov subspace, Computers & Mathematics with Applications, Vol. 58, pp. 1093-1102, 2009.
- Bai Z, Skoogh D., A projection method for model reduction of bilinear dynamical systems, Linear Algebra and its Applications, Vol. 415, pp. 406-425, 2006.
- Feng L, Benner P., A note on projection techniques for model order reduction of bilinear systems, International Conference on Numerical Analysis and Applied Mathematics, Vol. 936, No. 1, pp. 208-211, 2007.
- Benner P, Breiten T., Two-sided moment matching methods for nonlinear model reduction, SIAM Journal on Scientific Computing, Vol. 37, No. 2, pp. 239-260, 2015.
- Flagg G, Gugercin S., Multipoint volterra series interpolation and H2 optimal model reduction of bilinear systems, SIAM Journal on Numerical Analysis, Vol. 36, No. 2, pp. 549-579, 2015.
- Choudhary R, Ahuja K., Stability analysis of Bilinear Iterative Rational Krylov Algorithm. Linear Algebra and its Applications, Vol. 538, pp. 56-88, 2018.
- Choudhary R, Ahuja K., Inexact Linear Solves in Model Reduction of Bilinear Dynamical Systems, IEEE Access. Vol. 7, pp. 72297-72307, 2019.
- Zhang L, Lam J., On H2 model reduction of bilinear systems, Automatica. Vol. 38, No. 2, pp. 205-216, 2002.
- Benner P, Breiten T., Interpolation-based H2 model reduction of bilinear control systems, SIAM Journal on Matrix Analysis and Applications, Vol. 33, No. 3, pp. 859-885, 2012.
- Xu K. L, Jiang Y.L., An Approach to H2,ω model reduction on finite interval for bilinear systems, Journal of Franklin Institute, Vol. 354, No. 16, pp. 7429-7443, 2017.
- Benner P, Goyal P., Balanced truncation model order reduction for quadratic-bilinear control systems, Technical Report. https://arxiv.org/pdf/1705.00160.pdf April 2017.
- Penzl, T., Numerical solution of generalized Lyapunov equations, Advances in Computational Mathematics, Vol. 8, pp. 33–48, 1998.
- Yang P, Jiang Y. L, Xu K. L., A trust-region method for H2 model reduction of bilinear systems on the Stiefel manifold, Journal of the Franklin Institute, Vol. 356, No. 4, pp. 2258-2273, 2016.
- Choudhary R, Ahuja K., Stability analysis of Bilinear Iterative Rational Krylov Algorithm, Linear Algebra and its Applications, Vol. 538, pp. 56-88, 2018.
- Flagg G. M., Interpolation methods for the model reduction of bilinear systems, Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA 2012.
- Goyal P., System-theoretic Model Order Reduction for Bilinear and Quadratic-bilinear Systems, Doctoral thesis, Otto von Guericke University Library, Magdeburg, Germany, 2018.
- Kerschen K, Golinval J, Vakakis A. F, Bergman L. A., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An Overview, Nonlinear Dynamic, Vol. 41, pp. 147–169, 2005.
- Bruns, A., Benner, P., Parametric model order reduction of thermal models using the bilinear interpolatory rational Krylov algorithm, Mathematical and Computer Modelling of Dynamical Systems, Vol. 21, No. 2, pp. 103-129, 2015.
- Horne, B.G., Lower bounds for the spectral radius of a matrix, Linear Algebra Appl., Vol. 263, pp. 261-273, 1997.
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