Modulating functions-based method for parameters and source estimation in one-dimensional partial differential equations

References

• Robinson EA. Predictive decomposition of time series with application to seismic exploration. Geophysics. 1967;32:418–484.
   Web of Science ®Google Scholar
• Cameron M, Fomel S, Sethian J. Inverse problem in seismic imaging. PAMM. 2007;7:1024803–1024804.
   Google Scholar
• Fear E, Stuchly M. Microwave detection of breast cancer. IEEE Trans. Microwave Theory Tech. 2000;48:1854–1863.
   Web of Science ®Google Scholar
• Kirsch A. Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Prob. 1998;14:1489–1512.
   Web of Science ®Google Scholar
• Muniz W, Ramos F, de Campos Velho H. Entropy-and tikhonov-based regularization techniques applied to the backwards heat equation. Comput. Math. Appl. 2000;40:1071–1084.
   Web of Science ®Google Scholar
• Clason C, Klibanov MV. The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 2007;30:1–23.
   Web of Science ®Google Scholar
• Han H, Ling L, Takeuchi T. An energy regularization for cauchy problems of laplace equation in annulus domain. Commun. Comput. Phys. 2011;9:878–896.
   Web of Science ®Google Scholar
• Poli L, Oliveri G, Rocca P, et al. Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under te illuminations. IEEE Trans. Geosci. Remote Sens. 2013;51:2920–2936.
   Web of Science ®Google Scholar
• Moireau P, Chapelle D, Le Tallec P. Joint state and parameter estimation for distributed mechanical systems. Comput. Methods Appl. Mech. Eng. 2008;197:659–677.
   Web of Science ®Google Scholar
• Ramdani K, Tucsnak M, Weiss G. Recovering the initial state of an infinite-dimensional system using observers. Automatica. 2010;46:1616–1625.
   Web of Science ®Google Scholar
• Asiri S, Laleg-Kirati T, Zayane C. Inverse source problem for a one-dimensional wave equation using observers. In: 11th International Conference on Mathematical and Numerical Aspects of Waves; Tunisia; 2013. p. 149–150.
   Google Scholar
• Tucsnak M, Weiss G. From exact observability to identification of singular sources. Math. Control Signals Syst. 2014;27:1–21.
   Web of Science ®Google Scholar
• Zuazua E. Propagation, observation, control and numerical approximation of waves. SIAM Rev. 2005;47:197–243.
   Web of Science ®Google Scholar
• Shinbrot M. On the analysis of linear and nonlinear dynamical systems from transient-response data. National Advisory Committee for Aeronautics NACA; 1954.
   Google Scholar
• Shinbrot M. On the analysis of linear and nonlinear systems. Trans. ASME. 1957;79:547–552.
   Google Scholar
• Perdreauville FJ, Goodson R. Identification of systems described by partial differential equations. J. Fluids Eng. 1966;88:463–468.
   Web of Science ®Google Scholar
• Fairman F, Shen D. Parameter identification for a class of distributed systems. Int. J. Control. 1970;11:929–940.
   Web of Science ®Google Scholar
• Co TB, Ungarala S. Batch scheme recursive parameter estimation of continuous-time systems using the modulating functions method. Automatica. 1997;33:1185–1191.
   Web of Science ®Google Scholar
• Liu DY, Laleg-Kirati TM, Perruquetti W, et al. Non-asymptotic state estimation for a class of linear time-varying systems with unknown inputs. IFAC P. Vol. 2014;47:3732–3738.
   Google Scholar
• Liu DY, Laleg-Kirati TM. Robust fractional order differentiators using generalized modulating functions method. Signal Process. 2015;107:395–406.
   Web of Science ®Google Scholar
• Liu DY, Tian Y, Boutat D, et al. An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation. Signal Process. 2015;116:78–90.
   Web of Science ®Google Scholar
• Aldoghaither A, Liu D, Laleg-Kirati T. A novel approach for parameter and differentiation order estimation for a space fractional advection dispersion equation. SIAM J. Sci. Comput. 2015;37:A2813–A2839.
   Web of Science ®Google Scholar
• Takaya K. The use of hermite functions for system identification. IEEE Trans. Autom. Control. 1968;13:446–447.
   Web of Science ®Google Scholar
• Preisig H, Rippin D. Theory and application of the modulating function method–i. review and theory of the method and theory of the spline-type modulating functions. Comput. Chem. Eng. 1993;17:1–16.
   Web of Science ®Google Scholar
• Saha DC, Rao BP, Rao GP. Structure and parameter identification in linear continuous lumped systems: the poisson moment functional approach. Int. J. Control. 1982;36:477–491.
   Web of Science ®Google Scholar
• Patra A, Unbehauen H. Identification of a class of nonlinear continuous-time systems using hartley modulating functions. Int. J. Control. 1995;62:1431–1451.
   Web of Science ®Google Scholar
• Pearson A, Lee F. Parameter identification of linear differential systems via fourier based modulating functions. Control-Theory Adv. Technol. 1985;1:239–266.
   Google Scholar
• Fliess M. Analyse non standard du bruit. C.R. Math. 2006;342:797–802.
   Web of Science ®Google Scholar
• Fliess M. Critique du rapport signal à bruit en communications numériques-questioning the signal to noise ratio in digital communications. ARIMA (Revue africaine d’informatique et de Mathématiques appliquées). 2008;9:419–429.
   Google Scholar
• Liberzon D. Calculus of variations and optimal control theory: a concise introduction. New Jersey: Princeton University Press; 2012.
   Google Scholar
• Liu DY, Laleg-Kirati TM, Gibaru O, et al. Identification of fractional order systems using modulating functions method. In: American Control Conference (ACC), 2013. IEEE, Washington, DC; 2013. p. 1679–1684.
   Google Scholar
• Rabiner LR. Multirate digital signal processing. New Jersey: Prentice Hall PTR; 1996.
   Google Scholar
• Benney D. General theory for interactions between short and long waves. Stud. Appl. Math. 1977;56:81–94. cited By 110.
   Web of Science ®Google Scholar
• Gorshkov K, Ostrovsky L. Interactions of solitons in nonintegrable systems: direct perturbation method and applications. Phys. D: Nonlinear Phenom. 1981;3:428–438.
   Web of Science ®Google Scholar
• Bridges T, Derks G. Linear instability of solitary wave solutions of the kawahara equation and its generalizations. SIAM J. Math. Anal. 2002;33:1356–1378.
   Web of Science ®Google Scholar
• Janno J, Šeletski A. Reconstruction of coefficients of higher order nonlinear wave equations by measuring solitary waves. Wave Motion. 2014;52:15–25.
   Web of Science ®Google Scholar
• Kaya D. An explicit and numerical solutions of some fifth-order kdv equation by decomposition method. Appl. Math. Comput. 2003;144:353–363.
   Web of Science ®Google Scholar