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Relevant bibliographies by topics / $a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators

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Academic literature on the topic '$a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators'

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Published: 4 June 2021

Last updated: 4 February 2022

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Journal articles on the topic "$a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators"

1

Meyer, Y. "Pseudodifferential Operators and Nonlinear Partial Differential Equations (M. E. Taylor)." SIAM Review 35, no. 4 (December 1993): 671–72. http://dx.doi.org/10.1137/1035163.

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Garetto, Claudia, and Michael Oberguggenberger. "Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities." Proceedings of the Edinburgh Mathematical Society 57, no. 2 (September 19, 2013): 426–63. http://dx.doi.org/10.1017/s0013091513000424.

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AbstractThis paper addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalized functions. We employ the recently developed theory of generalized Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalized solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalized Fourier integral operators are obtained.

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Garello, Gianluca. "Pseudodifferential Operators with Symbols in Weighted Sobolev Spaces and Regularity for non Linear Partial Differential Equations." Mathematische Nachrichten 239-240, no. 1 (June 2002): 62–79. http://dx.doi.org/10.1002/1522-2616(200206)239:1<62::aid-mana62>3.0.co;2-w.

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Tervo, J., M. T. Nihtilä, and P. Kokkonen. "On controllability, parametrization, and output tracking of a linearized bioreactor model." Journal of Applied Mathematics 2003, no. 5 (2003): 243–76. http://dx.doi.org/10.1155/s1110757x03204046.

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The paper deals with a distributed parameter system related to the so-called fixed-bed bioreactor. The original nonlinear partial differential system is linearized around the steady state. We find that the linearized system is not exactly controllable but it is approximatively controllable when certain algebraic equations hold. We apply frequency-domain methods (transfer function analysis) to consider a related output tracking problem. The input-output system can be formulated as a translation invariant pseudodifferential equation. A simulation shows that the calculation scheme is stable. An idea to use frequency-domain methods and certain pseudodifferential operators for parametrization of control systems of more general systems is pointed out.

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5

Previato, Emma. "Multivariable Burchnall–Chaundy theory." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (June 22, 2007): 1155–77. http://dx.doi.org/10.1098/rsta.2007.2064.

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Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A 118 , 557–583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

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Xu, Sheng, and Hongbo Zhou. "Accurate simulations of pure quasi-P-waves in complex anisotropic media." GEOPHYSICS 79, no. 6 (November 1, 2014): T341—T348. http://dx.doi.org/10.1190/geo2014-0242.1.

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Reverse time migration (RTM) in complex anisotropic media requires calculation of the propagation of a single-mode wave, the quasi-P-wave. This was conventionally realized by solving a [Formula: see text] system of second-order partial differential equations. The implementation of this [Formula: see text] system required at least twice the computational resources as compared with the acoustic wave equation. The S-waves, an introduced auxiliary function in this system, were treated as artifacts in the RTM. Furthermore, the [Formula: see text] system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist, which brings more challenges to real data implementation. On the other hand, the Alkhalifah’s equation, which governs the pure quasi-P-wave propagation, was hard to solve because it was a pseudodifferential equation. We proposed a pure quasi-P-wave equation that can be easily implemented within current imaging framework. Our new equation was obtained by decomposing the original pseudodifferential operator into two numerical solvable operators: a differential operator and a scalar operator. The combination of these two operators yielded an accurate phase of quasi-P-wave propagation. Our solution was S-wave free and numerically stable for very complicated models. Because only one equation was required to resolve numerically, the new proposed scheme was more efficient than those conventional schemes that solve the [Formula: see text] second-order differential equations system. For tilted transverse isotropy (TTI) RTM implementation, the required increase of numerical cost was minimal, and we could expect at least a factor of two of improvement of efficiency. We showed the effectiveness and robustness of our method with numerical examples with simple and very complicated TTI models, the SEG Advanced Modeling (SEAM) model. Further extension of our approach to orthorhombic anisotropy or tilted orthorhombic anisotropy was straightforward.

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Anh, Vo V., Nikolai N. Leonenko, and María D. Ruiz-Medina. "Fractional-in-time and multifractional-in-space stochastic partial differential equations." Fractional Calculus and Applied Analysis 19, no. 6 (January 1, 2016). http://dx.doi.org/10.1515/fca-2016-0074.

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AbstractThis paper derives the weak-sense Gaussian solution to a family of fractional-in-time and multifractional-in-space stochastic partial differential equations, driven by fractional-integrated-in-time spatiotemporal white noise. Some fundamental results on the theory of pseudodifferential operators of variable order, and on the Mittag-Leffler function are applied to obtain the temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the derived solution.

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8

"Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer." V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, no. 89 (2019). http://dx.doi.org/10.26565//2221-5646-2019-89-03.

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A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

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"Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer." V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, no. 89 (2019). http://dx.doi.org/10.26565/2221-5646-2019-89-03.

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Abstract:

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

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Dissertations / Theses on the topic "$a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators"

1

Wang, Luqi. "Pseudo-differential operators with rough coefficients." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ30120.pdf.

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Krüger, Matthias. "On the Cauchy problem for a class of degenerate hyperbolic equations." Doctoral thesis, 2018. http://hdl.handle.net/11858/00-1735-0000-002E-E491-C.

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Jain, Rahul. "Regularity And Propagation Phenomena In Some Linear And Non-Linear Partial Differential Equations With Particular Reference To Microlocal Analysis." Thesis, 2005. http://etd.iisc.ernet.in/handle/2005/1447.

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Books on the topic "$a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators"

1

Lectures on linear partial differential equations. Providence, R.I: American Mathematical Society, 2011.

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2

Semiclassical analysis. Providence, R.I: American Mathematical Society, 2012.

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3

Spectral theory and geometric analysis: An international conference in honor of Mikhail Shubin's 65th birthday, July 29 - August 2, 2009, Northeastern University, Boston, Massachusetts. Providence, R.I: American Mathematical Society, 2010.

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4

service), SpringerLink (Online, ed. Pseudodifferential Operators with Applications. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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5

Singular ordinary differential operators and pseudodifferential equations. Berlin: Springer-Verlag, 1985.

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1930-, Treves Francois, and American Mathematical Society, eds. Pseudodifferential operators and applications. Providence, R.I: American Mathematical Society, 1985.

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Southeast Geometry Seminar (15th 2009 University of Alabama at Birmingham). Geometric analysis, mathematical relativity, and nonlinear partial differential equations: Southeast Geometry Seminars Emory University, Georgia Institute of Technology, University of Alabama, Birmingham, and the University of Tennessee, 2009-2011. Edited by Ghomi Mohammad 1969-. Providence, Rhode Island: American Mathematical Society, 2013.

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1951-, Wong Man Wah, Zhu Hongmei, and SpringerLink (Online service), eds. Pseudo-Differential Operators: Analysis, Applications and Computations. Basel: Springer Basel AG, 2011.

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Ville, Turunen, and SpringerLink (Online service), eds. Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics. Basel: Birkhäuser Basel, 2010.

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Luigi, Rodino, and SpringerLink (Online service), eds. Global Pseudo-Differential Calculus on Euclidean Spaces. Basel: Birkhäuser Basel, 2010.

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Book chapters on the topic "$a Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators"

1

Taylor, Michael E. "Pseudodifferential Operators." In Partial Differential Equations II, 1–90. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7052-7_1.

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Taylor, Michael E. "Pseudodifferential Operators." In Partial Differential Equations II, 1–73. New York, NY: Springer New York, 1996. http://dx.doi.org/10.1007/978-1-4757-4187-2_1.

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Brenner, A. V., and E. M. Shargorodsky. "Boundary Value Problems for Elliptic Pseudodifferential Operators." In Partial Differential Equations IX, 145–215. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-662-06721-5_2.

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Melrose, Richard B. "Fibrations, Compactifications and Algebras of Pseudodifferential Operators." In Partial Differential Equations and Mathematical Physics, 246–61. Boston, MA: Birkhäuser Boston, 1996. http://dx.doi.org/10.1007/978-1-4612-0775-7_16.

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Egorov, Yu V., and M. A. Shubin. "Asymptotics of Eigenvalues of Self-adjoint Differential and Pseudodifferential Operators." In Partial Differential Equations II, 96–120. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57876-2_8.

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6

McKeag, Peter, and Yuri Safarov. "Pseudodifferential Operators on Manifolds: A Coordinate-free Approach." In Partial Differential Equations and Spectral Theory, 321–41. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0024-2_6.

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Garello, Gianluca. "Pseudodifferential Operators with Symbols in Weighted Function Spaces of Quasi-Subadditive Type." In Partial Differential Equations and Spectral Theory, 133–38. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8231-6_15.

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Dencker, Nils. "The Solvability and Subellipticity of Systems of Pseudodifferential Operators." In Advances in Phase Space Analysis of Partial Differential Equations, 73–94. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4861-9_5.

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Budylin, A. M., and V. S. Buslaev. "Semiclassical Pseudodifferential Operators with Double Discontinuous Symbols and their Application to Problems of Quantum Statistical Physics." In Partial Differential Equations and Spectral Theory, 41–51. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8231-6_6.

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Rozenblum, Grigori, and Eugene Shargorodsky. "Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case." In Partial Differential Equations, Spectral Theory, and Mathematical Physics, 331–54. Zuerich, Switzerland: European Mathematical Society Publishing House, 2021. http://dx.doi.org/10.4171/ecr/18-1/20.

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