Bibliografias temáticas / Differential equations, Parabolic

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Literatura científica selecionada sobre o tema "Differential equations, Parabolic"

Autor: Grafiati
Publicado: 4 de junho de 2021
Última modificação: 30 de julho de 2024

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Artigos de revistas sobre o assunto "Differential equations, Parabolic"

1

Bonafede, Salvatore, e Salvatore A. Marano. "Implicit parabolic differential equations". Bulletin of the Australian Mathematical Society 51, n.º 3 (junho de 1995): 501–9. http://dx.doi.org/10.1017/s0004972700014349.

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Let QT = ω x (0, T), where ω is a bounded domain in ℝn (n ≥ 3) having the cone property and T is a positive real number; let Y be a nonempty, closed connected and locally connected subset of ℝh; let f be a real-valued function defined in QT × ℝh × ℝnh × Y; let ℒ be a linear, second order, parabolic operator. In this paper we establish the existence of strong solutions (n + 2 ≤ p < + ∞) to the implicit parabolic differential equationwith the homogeneus Cauchy-Dirichlet conditions where u = (u1, u2, …, uh), Dxu = (Dxu1, Dxu2, …, Dxuh), Lu = (ℒu1, ℒu2, … ℒuh).
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N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform". International Journal of Science and Research (IJSR) 12, n.º 6 (5 de junho de 2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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Ishii, Katsuyuki, Michel Pierre e Takashi Suzuki. "Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations". Mathematics 11, n.º 3 (2 de fevereiro de 2023): 758. http://dx.doi.org/10.3390/math11030758.

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We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution.
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4

Rubio, Gerardo. "The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain". International Journal of Stochastic Analysis 2011 (22 de junho de 2011): 1–35. http://dx.doi.org/10.1155/2011/469806.

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We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.
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Ashyralyev, Allaberen, e Ülker Okur. "Stability of Stochastic Partial Differential Equations". Axioms 12, n.º 7 (24 de julho de 2023): 718. http://dx.doi.org/10.3390/axioms12070718.

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In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.
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Klevchuk, I. I. "Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion". Carpathian Mathematical Publications 14, n.º 2 (30 de dezembro de 2022): 493–503. http://dx.doi.org/10.15330/cmp.14.2.493-503.

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The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.
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Hrytchuk, M., e I. Klevchuk. "BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION". Bukovinian Mathematical Journal 11, n.º 2 (2023): 100–103. http://dx.doi.org/10.31861/bmj2023.02.10.

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The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diffusion. The quasiperiodic solution of parabolic system is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce the norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with weak diffusion. We use bifurcation theory for ordinary differential equations and quasilinear parabolic equations. The existence of quasiperiodic solutions in an autonomous parabolic system of differential equations on the circle with small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with weak diffusion are investigated.
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BOUFOUSSI, B., e N. MRHARDY. "MULTIVALUED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS". Stochastics and Dynamics 08, n.º 02 (junho de 2008): 271–94. http://dx.doi.org/10.1142/s0219493708002317.

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In this paper, we establish by means of Yosida approximation, the existence and uniqueness of the solution of a backward doubly stochastic differential equation whose coefficient contains the subdifferential of a convex function. We will use this result to prove the existence of stochastic viscosity solution for some multivalued parabolic stochastic partial differential equation.
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Simon, László, e Willi Jäger. "On non-uniformly parabolic functional differential equations". Studia Scientiarum Mathematicarum Hungarica 45, n.º 2 (1 de junho de 2008): 285–300. http://dx.doi.org/10.1556/sscmath.2007.1036.

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We consider initial boundary value problems for second order quasilinear parabolic equations where also the main part contains functional dependence on the unknown function and the equations are not uniformly parabolic. The results are generalizations of that of [10]
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Walter, Wolfgang. "Nonlinear parabolic differential equations and inequalities". Discrete & Continuous Dynamical Systems - A 8, n.º 2 (2002): 451–68. http://dx.doi.org/10.3934/dcds.2002.8.451.

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Teses / dissertações sobre o assunto "Differential equations, Parabolic"

1

Yung, Tamara. "Traffic Modelling Using Parabolic Differential Equations". Thesis, Linköpings universitet, Kommunikations- och transportsystem, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102745.

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The need of a working infrastructure in a city also requires an understanding of how the traffic flows. It is known that increasing number of drivers prolong the travel time and has an environmental effect in larger cities. It also makes it more difficult for commuters and delivery firms to estimate their travel time. To estimate the traffic flow the traffic department can arrange cameras along popular roads and redirect the traffic, but this is a costly method and difficult to implement. Another approach is to apply theories from physics wave theory and mathematics to model the traffic flow; in this way it is less costly and possible to predict the traffic flow as well. This report studies the application of wave theory and expresses the traffic flow as a modified linear differential equation. First is an analytical solution derived to find a feasible solution. Then a numerical approach is done with Taylor expansions and Crank-Nicolson’s method. All is performed in Matlab and compared against measured values of speed and flow retrieved from Swedish traffic department over a 24 hours traffic day. The analysis is performed on a highway stretch outside Stockholm with no entries, exits or curves. By dividing the interval of the highway into shorter equal distances the modified linear traffic model is expressed in a system of equations. The comparison between actual values and calculated values of the traffic density is done with a nominal average difference. The results reveal that the numbers of intervals don’t improve the average difference. As for the small constant that is applied to make the linear model stable is higher than initially considered.
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.

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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Baysal, Arzu. "Inverse Problems For Parabolic Equations". Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605623/index.pdf.

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In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Problem III and IV) or on a time interval (in Problem I and II). If solution of inverse problem exists, then the temperature can be defined everywhere on the domain at all instants. The thesis consists of six chapters. In the first chapter, there is the introduction where the definition and applications of inverse problems are given and definition of the four inverse problems, that we will analyze in this thesis, are stated. In the second chapter, some definitions and theorems which we will use to obtain some conclusions about the corresponding direct problem of our four inverse problems are stated, and the conclusions about direct problem are obtained. In the third, fourth, fifth and sixth chapters we have the analysis of inverse problems I, II, III and IV, respectively.
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Keras, Sigitas. "Numerical methods for parabolic partial differential equations". Thesis, University of Cambridge, 1997. https://www.repository.cam.ac.uk/handle/1810/251611.

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Ascencio, Pedro. "Adaptive observer design for parabolic partial differential equations". Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/49454.

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This thesis addresses the observer design problem, for a class of linear one-dimensional parabolic Partial Differential Equations, considering the simultaneous estimation of states and parameters from boundary measurements. The design is based on the Backstepping methodology for Partial Differential Equations and extends its central idea, the Volterra transformation, to compensate for the parameters uncertainties. The design steps seek to reject time-varying parameter uncertainties setting forth a type of differential boundary value problems (Kernel-PDE/ODEs) to accomplish its objective, the solution of which is computed at every fixed sampling time and constitutes the observer gains for states and parameters. The design does not include any pre-transformation to some canonical form and/or a finite-dimensional formulation, and performs a direct parameter estimation from the original model. The observer design problem considers two cases of parameter uncertainty, at the boundary: control gain coefficient, and in-domain: diffusivity and reactivity parameters, respectively. For a Luenberger-type observer structure, the problems associated to one and two points of measurement at the boundary are studied through the application of an intuitive modification of the Volterra-type and Fredholm-type transformations. The resulting Kernel-PDE/ODEs are addressed by means of a novel methodology based on polynomial optimization and Sum-of-Squares decomposition. This approach allows recasting these coupled differential equations as convex optimization problems readily implementable resorting to semidefinite programming, with no restrictions to the spectral characteristics of some integral operators or system's coefficients. Additionally, for polynomials Kernels, uniqueness and invertibility of the Fredholm-type transformation are proved in the space of real analytic and continuous functions. The direct and inverse Kernels are approximated as the optimal polynomial solution of a Sum-of-Squares and Moment problem with theoretically arbitrary precision. Numerical simulations illustrate the effectiveness and potentialities of the methodology proposed to manage a variety of problems with different structures and objectives.
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Williams, J. F. "Scaling and singularities in higher-order nonlinear differential equations". Thesis, University of Bath, 2003. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275878.

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Tsang, Siu Chung. "Preconditioners for linear parabolic optimal control problems". HKBU Institutional Repository, 2017. https://repository.hkbu.edu.hk/etd_oa/464.

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In this thesis, we consider the computational methods for linear parabolic optimal control problems. We wish to minimize the cost functional while fulfilling the parabolic partial differential equations (PDE) constraint. This type of problems arises in many fields of science and engineering. Since solving such parabolic PDE optimal control problems often lead to a demanding computational cost and time, an effective algorithm is desired. In this research, we focus on the distributed control problems. Three types of cost functional are considered: Target States problems, Tracking problems, and All-time problems. Our major contribution in this research is that we developed a preconditioner for each kind of problems, so our iterative method is accelerated. In chapter 1, we gave a brief introduction to our problems with a literature review. In chapter 2, we demonstrated how to derive the first-order optimality conditions from the parabolic optimal control problems. Afterwards, we showed how to use the shooting method along with the flexible generalized minimal residual to find the solution. In chapter 3, we offered three preconditioners to enhance our shooting method for the problems with symmetric differential operator. Next, in chapter 4, we proposed another three preconditioners to speed up our scheme for the problems with non-symmetric differential operator. Lastly, we have the conclusion and the future development in chapter 5.
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Rivera, Noriega Jorge. "Some remarks on certain parabolic differential operators over non-cylindrical domains /". free to MU campus, to others for purchase, 2001. http://wwwlib.umi.com/cr/mo/fullcit?p3025649.

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Hammer, Patricia W. "Parameter identification in parabolic partial differential equations using quasilinearization". Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/37226.

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We develop a technique for identifying unknown coefficients in parabolic partial differential equations. The identification scheme is based on quasilinearization and is applied to both linear and nonlinear equations where the unknown coefficients may be spatially varying. Our investigation includes derivation, convergence, and numerical testing of the quasilinearization based identification scheme
Ph. D.
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Prinja, Gaurav Kant. "Adaptive solvers for elliptic and parabolic partial differential equations". Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/adaptive-solvers-for-elliptic-and-parabolic-partial-differential-equations(f0894eb2-9e06-41ff-82fd-a7bde36c816c).html.

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In this thesis our primary interest is in developing adaptive solution methods for parabolic and elliptic partial differential equations. The convection-diffusion equation is used as a representative test problem. Investigations are made into adaptive temporal solvers implementing only a few changes to existing software. This includes a comparison of commercial code against some more academic releases. A novel way to select step sizes for an adaptive BDF2 code is introduced. A chapter is included introducing some functional analysis that is required to understand aspects of the finite element method and error estimation. Two error estimators are derived and proofs of their error bounds are covered. A new finite element package is written, implementing a rather interesting error estimator in one dimension to drive a rather standard refinement/coarsening type of adaptivity. This is compared to a commercially available partial differential equation solver and an investigation into the properties of the two inspires the development of a new method designed to very quickly and directly equidistribute the errors between elements. This new method is not really a refinement technique but doesn't quite fit the traditional description of a moving mesh either. We show that this method is far more effective at equidistribution of errors than a simple moving mesh method and the original simple adaptive method. A simple extension of the new method is proposed that would be a mesh reconstruction method. Finally the new code is extended to solve steady-state problems in two dimensions. The mesh refinement method from one dimension does not offer a simple extension, so the error estimator is used to supply an impression of the local topology of the error on each element. This in turn allows us to develop a new anisotropic refinement algorithm, which is more in tune with the nature of the error on the parent element. Whilst the benefits observed in one dimension are not directly transferred into the two-dimensional case, the obtained meshes seem to better capture the topology of the solution.
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Livros sobre o assunto "Differential equations, Parabolic"

1

DiBenedetto, Emmanuele. Degenerate parabolic equations. New York: Springer-Verlag, 1993.

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2

Zheng, Songmu. Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Harlow, Essex, England: Longman, 1995.

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3

Watson, N. A. Parabolic equations on an infinite strip. New York: M. Dekker, 1989.

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4

Zheng, S. Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Harlow, Essex, England: Longman, 1995.

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5

Pao, C. V. Nonlinear parabolic and elliptic equations. New York: Plenum Press, 1992.

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6

1943-, Bandle Catherine, ed. Elliptic and parabolic problems. Harlow: Longman Scientific & Technical, 1995.

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7

Zeleni͡ak, T. I. Qualitative theory of parabolic equations. Utrecht: VSP, 1997.

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Zeleni︠a︡k, Tadeĭ Ivanovich. Qualitative theory of parabolic equations. Utrecht: VSP, 1997.

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9

Pao, C. V. Nonlinear parabolic and elliptic equations. New York: Plenum Press, 1992.

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10

A, Samarskiĭ A., ed. Blow-up in quasilinear parabolic equations. Berlin: De Gruyter, 1995.

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Capítulos de livros sobre o assunto "Differential equations, Parabolic"

1

Quarteroni, Alfio. "Parabolic equations". In Numerical Models for Differential Problems, 121–40. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-49316-9_5.

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Quarteroni, Alfio. "Parabolic equations". In Numerical Models for Differential Problems, 121–40. Milano: Springer Milan, 2014. http://dx.doi.org/10.1007/978-88-470-5522-3_5.

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Marin, Marin, e Andreas Öchsner. "Parabolic Equations". In Essentials of Partial Differential Equations, 169–99. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-90647-8_6.

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4

Bellman, Richard, e George Adomian. "Nonlinear Parabolic Equations". In Partial Differential Equations, 120–28. Dordrecht: Springer Netherlands, 1985. http://dx.doi.org/10.1007/978-94-009-5209-6_11.

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DiBenedetto, Emmanuele, e Ugo Gianazza. "PARABOLIC DEGIORGI CLASSES". In Partial Differential Equations, 451–508. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-46618-2_13.

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Pap, Endre, Arpad Takači e Djurdjica Takači. "Parabolic Equations". In Partial Differential Equations through Examples and Exercises, 183–226. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5574-8_6.

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Kavdia, Mahendra. "Parabolic Differential Equations, Diffusion Equation". In Encyclopedia of Systems Biology, 1621–24. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_273.

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Taylor, Michael E. "Nonlinear Parabolic Equations". In Partial Differential Equations III, 271–358. New York, NY: Springer New York, 1996. http://dx.doi.org/10.1007/978-1-4757-4190-2_3.

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Taylor, Michael E. "Nonlinear Parabolic Equations". In Partial Differential Equations III, 335–433. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-33928-8_3.

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Taylor, Michael E. "Nonlinear Parabolic Equations". In Partial Differential Equations I, 313–411. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7049-7_3.

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Trabalhos de conferências sobre o assunto "Differential equations, Parabolic"

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LITVAK-HINENZON, ANNA. "THE MECHANISM OF PARABOLIC RESONANCE ORBITS". In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0123.

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POLÁČIK, P. "ASYMPTOTIC SYMMETRY OF POSITIVE SOLUTIONS OF PARABOLIC EQUATIONS". In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0009.

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ROCHA, CARLOS. "TRANSVERSALITY IN SEMILINEAR PARABOLIC EQUATIONS ON THE CIRCLE". In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0112.

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HORSTMANN, D. "FORWARD-BACKWARD PARABOLIC EQUATIONS AND THEIR TIME DELAY APPROXIMATIONS". In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0188.

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Simon, László. "On contact problems for nonlinear parabolic functional differential equations". In The 7'th Colloquium on the Qualitative Theory of Differential Equations. Szeged: Bolyai Institute, SZTE, 2003. http://dx.doi.org/10.14232/ejqtde.2003.6.22.

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MORDUKHOVICH, BORIS S., e THOMAS I. SEIDMAN. "FEEDBACK CONTROL OF CONSTRAINED PARABOLIC SYSTEMS IN UNCERTAINTY CONDITIONS VIA ASYMMETRIC GAMES". In Applied Analysis and Differential Equations - The International Conference. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812708229_0020.

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DIAZ, J. I., e L. TELLO. "ON A PARABOLIC PROBLEM WITH DIFFUSION ON THE BOUNDARY ARISING IN CLIMATOLOGY". In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0179.

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Viglialoro, Giuseppe, Stella Vernier Piro e Monica Marras. "Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system". In The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid, Spain). American Institute of Mathematical Sciences, 2015. http://dx.doi.org/10.3934/proc.2015.0809.

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Faragó, István, e Róbert Horváth. "Qualitative properties of monotone linear parabolic operators". In The 8'th Colloquium on the Qualitative Theory of Differential Equations. Szeged: Bolyai Institute, SZTE, 2007. http://dx.doi.org/10.14232/ejqtde.2007.7.8.

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Ashyralyev, Allaberen, Yasar Sozen e Fatih Hezenci. "A note on parabolic differential equations on manifold". In FOURTH INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2020). AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0042762.

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Relatórios de organizações sobre o assunto "Differential equations, Parabolic"

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Dalang, Robert C., e N. Frangos. Stochastic Hyperbolic and Parabolic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, julho de 1994. http://dx.doi.org/10.21236/ada290372.

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Ostashev, Vladimir, Michael Muhlestein e D. Wilson. Extra-wide-angle parabolic equations in motionless and moving media. Engineer Research and Development Center (U.S.), setembro de 2021. http://dx.doi.org/10.21079/11681/42043.

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Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.
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Urban, Karsten, e Anthony T. Patera. A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, janeiro de 2012. http://dx.doi.org/10.21236/ada557547.

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