Separation of Variables for Partial Differential Equations

Cain, George D.; Meyer, Gunter H.

doi:10.4324/9780203498781

Cited by 16 publications

(23 citation statements)

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“…It is readily seen that this algorithm yields F Q (H a ) m = 0 and that P Q = I m , which is precisely (8). Moreover, x = [I n , 0 n×m ] Qy and u = [0 m×n , I m ] Qy identically satisfy (7).…”

Section: Fractional Flatnessmentioning

confidence: 91%

“…The following algorithm to compute a fractionally flat output is immediately deduced: Output: Defining matrices P and Q, i.e. satisfying (8).…”

Section: Fractional Flatnessmentioning

confidence: 99%

“…We follow a separation of variables approach developed, e.g. in [8] and generalizing the one followed by [22,21,40]. We recall that, for a function f of several variables ξ and t ∈ R, its (partial) Laplace transform is given bŷ f (ξ, s) = +∞ 0 f (ξ, t)e −ts dt for all ξ and s ∈ C where this integral is finite.…”

Section: Fractional 0-flatnessmentioning

confidence: 99%

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Flatness for linear fractional systems with application to a thermal system

et al. 2015

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This paper is devoted to the study of the flatness property of linear time-invariant fractional systems. In the framework of polynomial matrices of the fractional derivative operator, we give a characterization of fractionally flat outputs and a simple algorithm to compute them. We also obtain a characterization of the so-called fractionnally 0-flat outputs. We then present an application to a two dimensional heated metallic sheet, whose dynamics may be approximated by a fractional model of order 1/2. The trajectory planning of the temperature at a given point of the metallic sheet is obtained thanks to the fractional flatness property, without integrating the system equations. The pertinence of this approach is discussed on simulations.

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Section: Fractional Flatnessmentioning

confidence: 91%

“…The following algorithm to compute a fractionally flat output is immediately deduced: Output: Defining matrices P and Q, i.e. satisfying (8).…”

Section: Fractional Flatnessmentioning

confidence: 99%

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Flatness for linear fractional systems with application to a thermal system

et al. 2015

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“…This is just a consequence of the Stone-Weierstrass theorem [20]. Discussion of the most general case (5) would require us to go into integral operators, which we will not do as in the present framework we are interested in applications that rely on the inverse problems corresponding to (6) and (7). Nonetheless (5) is expected to be useful and we state it here for completeness.…”

Section: Sparse Separable Decompositionsmentioning

confidence: 99%

“…Nonetheless (5) is expected to be useful and we state it here for completeness. Henceforth, we will simply refer to (6) or (7) as a multilinear decomposition, by which we mean a decomposition into a linear combination of separable functions. We note here that the finite-dimensional discrete version has been studied under several different names -see Section IX.…”

Section: Sparse Separable Decompositionsmentioning

confidence: 99%

Blind Multilinear Identification

Lim

Comon

2014

IEEE Trans. Inform. Theory

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International audienceWe discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible: (i) closely located or highly correlated sources in antenna array processing, (ii) highly correlated spreading codes in CDMA radio communication, (iii) nearly dependent spectra in fluorescent spectroscopy. This has important implications --- in the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescent spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that specialize to the usual ones for matrices and operators but applies to also hypermatrices and tensors

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The analytical solution of the transient radial diffusion equation with a nonuniform loss term

2017

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Many works have been done during the past 40 years to perform the analytical solution of the radial diffusion equation that models the transport and loss of electrons in the magnetosphere, considering a diffusion coefficient proportional to a power law in L shell and a constant loss term. In this paper, we propose an original analytical method to address this challenge with a nonuniform loss term. The strategy is to match any L‐dependent electron losses with a piecewise constant function on M subintervals, i.e., dealing with a constant lifetime on each subinterval. Applying an eigenfunction expansion method, the eigenvalue problem becomes presently a Sturm‐Liouville problem with M interfaces. Assuming the continuity of both the distribution function and its first spatial derivatives, we are able to deal with a well‐posed problem and to find the full analytical solution. We further show an excellent agreement between both the analytical solutions and the solutions obtained directly from numerical simulations for different loss terms of various shapes and with a diffusion coefficient DLL∼L6. We also give two expressions for the required number of eigenmodes N to get an accurate snapshot of the analytical solution, highlighting that N is proportional to 1/t0, where t0 is a time of interest, and that N increases with the diffusion power. Finally, the equilibrium time, defined as the time to nearly reach the steady solution, is estimated by a closed‐form expression and discussed. Applications to Earth and also Jupiter and Saturn are discussed.

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Separation of Variables for Partial Differential Equations

Cited by 16 publications

References 0 publications

Flatness for linear fractional systems with application to a thermal system

Flatness for linear fractional systems with application to a thermal system

Blind Multilinear Identification

The analytical solution of the transient radial diffusion equation with a nonuniform loss term

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