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Relevant bibliographies by topics / Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations

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Academic literature on the topic 'Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations"

1

Tokibetov, Zh A., N. E. Bashar, and А. К. Pirmanova. "THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (December 29, 2020): 68–72. http://dx.doi.org/10.51889/2020-4.1728-7901.10.

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For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

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Song, Mingliang, and Shuyuan Mei. "Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations." Journal of Function Spaces 2021 (January 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/6668037.

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The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

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3

Wei, Lin. "Some second-order systems of partial differential equations of composite type." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 106, no. 1-2 (1987): 73–88. http://dx.doi.org/10.1017/s0308210500018217.

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SynopsisThe Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.

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Li, Xiaoou, Jingchen Liu, and Shun Xu. "A multilevel approach towards unbiased sampling of random elliptic partial differential equations." Advances in Applied Probability 50, no. 4 (November 29, 2018): 1007–31. http://dx.doi.org/10.1017/apr.2018.49.

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Abstract Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

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5

Cosner, Chris, and Philip W. Schaefer. "Sign-definite solutions in some linear elliptic systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 3-4 (1989): 347–58. http://dx.doi.org/10.1017/s030821050001862x.

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SynopsisWe consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.

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Sowa, Artur. "Modeling a Quantum Hall System via Elliptic Equations." Advances in Mathematical Physics 2009 (2009): 1–9. http://dx.doi.org/10.1155/2009/514081.

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Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework to search for universal principles relevant to nanosystem modeling and nanosystem-specific signal processing. Recently, we have been able to construct a partial differential equations-based model of a quantum Hall system, which consists of the Schrödinger equation supplemented with a special-type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. Here we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first-order elliptic equations in the plane.

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Ogiwara, Toshiko, and Hiroshi Matano. "Stability analysis in order-preserving systems in the presence of symmetry." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 2 (1999): 395–438. http://dx.doi.org/10.1017/s0308210500021429.

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Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.

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8

Shakhmurov, Veli B. "Maximal regular boundary value problems in Banach-valued function spaces and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–26. http://dx.doi.org/10.1155/ijmms/2006/92134.

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The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

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TSAI, LONG-YI, and S. T. WU. "EXISTENCE OF SOLUTIONS FOR ELLIPTIC INTEGRO-DIFFERENTIAL SYSTEMS." Tamkang Journal of Mathematics 25, no. 1 (March 1, 1995): 61–70. http://dx.doi.org/10.5556/j.tkjm.25.1994.4426.

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In this paper the existence of the solution for elliptic integro-differential systems are discussed. Those systems are motivated by certain physical processes such as in epidemics, predator-prey dynamics and the others. We extend the method of mixed monotony to second order elliptic partial integro-differential equations. By assuming the existence of a satellite $f$ of the give function $\Phi$, we prove the existence of solutions by using fixed point theory. Moreover, we provide the modified method of mixed monotony to construct two monotone sequences which converge uniformly to the solution. We also give sufficient conditions for the existence of $f$ and obtain the construction of upper and lower solutions in some applications.

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10

Köster, M., and S. Turek. "The Influence of Higher Order FEM Discretisations on Multigrid Convergence." Computational Methods in Applied Mathematics 6, no. 2 (2006): 221–32. http://dx.doi.org/10.2478/cmam-2006-0011.

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AbstractQuadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.

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Dissertations / Theses on the topic "Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations"

1

Sassone, Edoardo [Verfasser]. "Existence, multiplicity and behaviour of solutions of some elliptic partial differential equations of higher order / von Edoardo Sassone." 2009. http://d-nb.info/1007862092/34.

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Books on the topic "Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations"

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Nonlinear elliptic equations of the second order. Providence, Rhode Island: American Mathematical Society, 2016.

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

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Fokker-Planck-Kolmogorov equations. Providence, Rhode Island: American Mathematical Society, 2015.

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1943-, Gossez J. P., and Bonheure Denis, eds. Nonlinear elliptic partial differential equations: Workshop in celebration of Jean-Pierre Gossez's 65th birthday, September 2-4, 2009, Université libre de Bruxelles, Belgium. Providence, R.I: American Mathematical Society, 2011.

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1966-, Pérez Joaquín, and Galvez José A. 1972-, eds. Geometric analysis: Partial differential equations and surfaces : UIMP-RSME Santaló Summer School geometric analysis, June 28-July 2, 2010, University of Granada, Granada, Spain. Providence, R.I: American Mathematical Society, 2012.

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6

Habib, Ammari, Capdeboscq Yves 1971-, and Kang Hyeonbae, eds. Multi-scale and high-contrast PDE: From modelling, to mathematical analysis, to inversion : Conference on Multi-scale and High-contrast PDE:from Modelling, to Mathematical Analysis, to Inversion, June 28-July 1, 2011, University of Oxford, United Kingdom. Providence, R.I: American Mathematical Society, 2010.

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Imaging, multi-scale, and high-contrast partial differential equations: Seoul ICM 2014 Satellite Conference, August 7-9, 2014, Daejeon, Korea. Providence, Rhode Island: American Mathematical Society, 2016.

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Sequeira, A., H. Beirão da Veiga, and V. A. Solonnikov. Recent advances in partial differential equations and applications: International conference in honor of Hugo Beirao de Veiga's 70th birthday, February 17-214, 2014, Levico Terme (Trento), Italy. Edited by Rădulescu, Vicenţiu D., 1958- editor. Providence, Rhode Island: American Mathematical Society, 2016.

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Nahmod, Andrea R. Recent advances in harmonic analysis and partial differential equations: AMS special sessions, March 12-13, 2011, Statesboro, Georgia : the JAMI Conference, March 21-25, 2011, Baltimore, Maryland. Edited by American Mathematical Society and JAMI Conference (2011 : Baltimore, Md.). Providence, Rhode Island: American Mathematical Society, 2012.

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P, Minicozzi William, ed. A course in minimal surfaces. Providence, R.I: American Mathematical Society, 2011.

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Book chapters on the topic "Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations"

1

Dzhuraev, A. "On a class of Second Order Elliptic Overdetermined Systems." In Complex Methods for Partial Differential Equations, 189–204. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4613-3291-6_12.

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Sprekels, J., and D. Tiba. "On the Approximation and Optimization of Fourth Order Elliptic Systems." In Optimal Control of Partial Differential Equations, 277–86. Basel: Birkhäuser Basel, 1999. http://dx.doi.org/10.1007/978-3-0348-8691-8_24.

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3

Dai, D. Q. "On some Problems for First Order Elliptic Systems in the Plane." In Complex Methods for Partial Differential Equations, 21–27. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4613-3291-6_2.

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Dzhuraev, Abduhamid. "Some Boundary Value Problems for Second Order Overdetermined Elliptic Systems in the Unit Ball Of ℂ N." In Partial Differential and Integral Equations, 37–57. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4613-3276-3_3.

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Akal, M. "On a Generalized Riemann-Hilbert Boundary Value Problem for Second Order Elliptic Systems in the Plane." In Complex Methods for Partial Differential Equations, 41–55. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4613-3291-6_4.

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Kenig, Carlos, and Jenn-Nan Wang. "Unique Continuation for the Elasticity System and a Counterexample for Second-Order Elliptic Systems." In Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1), 159–78. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30961-3_10.

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Barton, Ariel, and Svitlana Mayboroda. "Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey." In Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1), 55–121. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30961-3_4.

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8

"Chapter 8. Solving General Nonlinear First-Order Elliptic Systems." In Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48), 235–58. Princeton University Press, 2008. http://dx.doi.org/10.1515/9781400830114.235.

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Conference papers on the topic "Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations"

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Dalton, Charles, and Wu Zheng. "Numerical Solutions of a Viscous Uniform Approach Flow Past Square and Diamond Cylinders." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32287.

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Numerical results are presented for a uniform approach flow past square and diamond cylinders, with and without rounded corners, at Reynolds numbers of 250 and 1000. This unsteady viscous flow problem is formulated by the 2-D Navier-Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. A grid generation technique is applied to provide an efficient mesh system for the flow. The elliptic partial differential equation for stream-function and vorticity in the transformed plane is solved by the multigrid iteration method. The Strouhal number and the average in-line force coefficients agree very well with the experimental and previous numerical values. The vortex structures and the evolution of vortex shedding are illustrated by vorticity contours. Rounding the corners of the square and diamond cylinders produced a noticeable decrease on the calculated drag and lift coefficients.

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Aguirre-Rivas, Donovan A., and Karim H. Muci-Küchler. "Formulation of a Higher Order Finite Element for Two-Dimensional Heat Conduction Problems." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-71201.

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Recent research has identified Adini’s rectangular element as an efficient higher order element for solving second order elliptic partial differential equations such as Poisson’s equation, which governs the steady state heat conduction problem. This type of element includes both the primary field variable and its spatial derivatives as nodal degrees of freedom. Compared to the conventional cubic elements of the Serendipity and Lagrange families, Adini’s element includes the minimum number of nodes per element and has the advantage that the nodal values of the spatial derivatives of the temperature field are directly retrieved from the FEM solution. As a result, the differentiation and averaging procedures that are typically used to obtain the nodal values of the temperature gradients are avoided. In this paper a generalized version of Adini’s element for solving two-dimensional steady state heat transfer problems in non-rectangular geometries is presented. Also, the traditional finite element formulation is modified to allow the application of essential boundary conditions without having to constrain the nodal values of the tangential derivative of the temperature. The resulting higher order element and modified FEM formulation are used to solve an example problem and the accuracy of the solution is compared with solutions obtained using the traditional linear, quadratic, and cubic Serendipity elements to show the efficiency, in terms of accuracy per number of degrees of freedom, of the proposed approach for finding the nodal values of the temperature gradients, which are required to compute the nodal values of the heat flux vector.

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