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Journal articles on the topic 'Elliptic/parabolic problems'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Kim, Inwon C., and Norbert Požár. "Nonlinear Elliptic–Parabolic Problems." Archive for Rational Mechanics and Analysis 210, no. 3 (September 13, 2013): 975–1020. http://dx.doi.org/10.1007/s00205-013-0663-3.

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2

Mannucci, Paola, and Juan Luis Vazquez. "Viscosity solutions for elliptic-parabolic problems." Nonlinear Differential Equations and Applications NoDEA 14, no. 1-2 (October 2007): 75–90. http://dx.doi.org/10.1007/s00030-007-4044-1.

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3

Goldstein, C. I. "Preconditioning Singularity Perturbed Elliptic and Parabolic Problems." SIAM Journal on Numerical Analysis 28, no. 5 (October 1991): 1386–418. http://dx.doi.org/10.1137/0728072.

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4

Guidotti, Patrick. "Elliptic and parabolic problems in unbounded domains." Mathematische Nachrichten 272, no. 1 (August 2004): 32–45. http://dx.doi.org/10.1002/mana.200310187.

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5

Kenmochi, Nobuyuki, and Masahiro Kubo. "Periodic solutions of parabolic-elliptic obstacle problems." Journal of Differential Equations 88, no. 2 (December 1990): 213–37. http://dx.doi.org/10.1016/0022-0396(90)90096-8.

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6

Lee, Ki-ahm, and J. L. Vázquez. "Parabolic approach to nonlinear elliptic eigenvalue problems." Advances in Mathematics 219, no. 6 (December 2008): 2006–28. http://dx.doi.org/10.1016/j.aim.2008.07.012.

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7

Jiang, Daijun, Hui Feng, and Jun Zou. "Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 3 (May 2018): 1085–107. http://dx.doi.org/10.1051/m2an/2018016.

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We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods.

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8

Dancer, E. N., and Yihong Du. "The generalized Conley index and multiple solutions of semilinear elliptic problems." Abstract and Applied Analysis 1, no. 1 (1996): 103–35. http://dx.doi.org/10.1155/s108533759600005x.

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We establish some framework so that the generalized Conley index can be easily used to study the multiple solution problem of semilinear elliptic boundary value problems. Both the parabolic flow and the gradient flow are used. Some examples are given to compare our approach here with other well-known methods. Our abstract results with parabolic flows may have applications to parabolic problems as well.

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9

SHAKHMUROV, VELI B., and AIDA SAHMUROVA. "Mixed problems for degenerate abstract parabolic equations and applications." Carpathian Journal of Mathematics 34, no. 2 (2018): 247–54. http://dx.doi.org/10.37193/cjm.2018.02.13.

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Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

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10

Aiki, Toyohiko. "Two-phase Stefan problems for parabolic-elliptic equations." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 10 (1988): 377–80. http://dx.doi.org/10.3792/pjaa.64.377.

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11

Lieberman, Gary. "Oblique derivative problems for elliptic and parabolic equations." Communications on Pure & Applied Analysis 12, no. 6 (2013): 2409–44. http://dx.doi.org/10.3934/cpaa.2013.12.2409.

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12

Stiemer, Marcus. "Finite element discretization of parabolic-elliptic interface problems." PAMM 7, no. 1 (December 2007): 2020117–18. http://dx.doi.org/10.1002/pamm.200700859.

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13

Byun, Sun-Sig, Ki-Ahm Lee, Jehan Oh, and Jinwan Park. "Nondivergence elliptic and parabolic problems with irregular obstacles." Mathematische Zeitschrift 290, no. 3-4 (March 1, 2018): 973–90. http://dx.doi.org/10.1007/s00209-018-2048-7.

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14

Ženíšek, Alexander. "Finite element variational crimes in parabolic-elliptic problems." Numerische Mathematik 55, no. 3 (May 1989): 343–76. http://dx.doi.org/10.1007/bf01390058.

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15

Zhang, Tie, and Lixin Tang. "A Discontinuous Finite Volume Element Method Based on Bilinear Trial Functions." International Journal of Computational Methods 14, no. 03 (April 13, 2017): 1750025. http://dx.doi.org/10.1142/s0219876217500256.

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We propose a discontinuous finite volume element (DFVE) method for second order elliptic and parabolic problems. Discontinuous bilinear functions are used as the trial functions. We give the stability analysis of this DFVE method and derive the optimal error estimates in the broken [Formula: see text]-norm. Specifically, the optimal [Formula: see text]-error is obtained for the first time for the bilinear DFVE methods solving elliptic and parabolic problems.

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16

KUMAR, VIVEK, and MANI MEHRA. "CUBIC SPLINE ADAPTIVE WAVELET SCHEME TO SOLVE SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 02 (March 2007): 317–31. http://dx.doi.org/10.1142/s021969130700177x.

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In this paper, the collocation method proposed by Cai and Wang1 has been reviewed in detail to solve singularly perturbed reaction diffusion equation of elliptic and parabolic types. The method is based on an interpolating wavelet transform using cubic spline on dyadic points. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples are presented for elliptic and parabolic problems. The purposed method comes up as a powerful tool for studying singular perturbation problems in term of effective grid generation and CPU time.

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17

Ammar, Kaouther, and Petra Wittbold. "Existence of renormalized solutions of degenerate elliptic-parabolic problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 3 (June 2003): 477–96. http://dx.doi.org/10.1017/s0308210500002493.

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We consider a general class of degenerate elliptic-parabolic problems associated with the equation b(υ)t = div a(υ, Dυ) + f. Existence of renormalized solutions is established for general L1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

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18

Ashyralyev, Allaberen, and Okan Gercek. "On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/230190.

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We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

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19

Hamamuki, Nao, and Qing Liu. "A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 13. http://dx.doi.org/10.1051/cocv/2019076.

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This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.

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20

Barrios, Begoña, and Maria Medina. "Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 1 (January 26, 2019): 475–95. http://dx.doi.org/10.1017/prm.2018.77.

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AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

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21

Gavrilyuk, I. P. "Approximation of the Operator Exponential and Applications." Computational Methods in Applied Mathematics 7, no. 4 (2007): 294–320. http://dx.doi.org/10.2478/cmam-2007-0019.

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AbstractA review of the exponentially convergent approximations to the operator exponential is given. The applications to inhomogeneous parabolic and elliptic equations, nonlinear parabolic equations, tensor-product approximations of multidimensional solution operators as well as to parabolic problems with time dependent coefficients and boundary conditions are discussed.

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22

Grinfeld, M., P. Benilan, M. Chipot, L. C. Evans, and M. Pierre. "Recent Advances in Non-Linear Elliptic and Parabolic Problems." Mathematical Gazette 74, no. 470 (December 1990): 414. http://dx.doi.org/10.2307/3618183.

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23

Chipot, M., and B. Lovat. "Some remarks on non local elliptic and parabolic problems." Nonlinear Analysis: Theory, Methods & Applications 30, no. 7 (December 1997): 4619–27. http://dx.doi.org/10.1016/s0362-546x(97)00169-7.

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24

Hulshof, J., and J. R. King. "Asymptotic analysis of some elliptic-parabolic moving boundary problems." Applied Mathematics Letters 12, no. 2 (March 1999): 87–94. http://dx.doi.org/10.1016/s0893-9659(98)00154-2.

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25

Garcı́a Azorero, J. P., and I. Peral Alonso. "Hardy Inequalities and Some Critical Elliptic and Parabolic Problems." Journal of Differential Equations 144, no. 2 (April 1998): 441–76. http://dx.doi.org/10.1006/jdeq.1997.3375.

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26

Carrillo, José, and Petra Wittbold. "Uniqueness of Renormalized Solutions of Degenerate Elliptic–Parabolic Problems." Journal of Differential Equations 156, no. 1 (July 1999): 93–121. http://dx.doi.org/10.1006/jdeq.1998.3597.

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27

Bernis, Francisco. "Elliptic and parabolic semilinear problems without conditions at infinity." Archive for Rational Mechanics and Analysis 106, no. 3 (September 1989): 217–41. http://dx.doi.org/10.1007/bf00281214.

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28

Nittka, Robin. "Quasilinear elliptic and parabolic Robin problems on Lipschitz domains." Nonlinear Differential Equations and Applications NoDEA 20, no. 3 (September 23, 2012): 1125–55. http://dx.doi.org/10.1007/s00030-012-0201-2.

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29

Ashyralyev, Allaberen, and Okan Gercek. "Finite difference method for multipoint nonlocal elliptic–parabolic problems." Computers & Mathematics with Applications 60, no. 7 (October 2010): 2043–52. http://dx.doi.org/10.1016/j.camwa.2010.07.044.

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30

Kenmochi, Nobuyuki, and Irena Pawlow. "Parabolic-elliptic free boundary problems with time-dependent obstacles." Japan Journal of Applied Mathematics 5, no. 1 (February 1988): 87–121. http://dx.doi.org/10.1007/bf03167902.

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31

Al-Sultani, Mohamed Saleh Mehdi, and Igor Boglaev. "Block monotone iterations for solving coupled systems of nonlinear parabolic equations." ANZIAM Journal 61 (July 28, 2020): C166—C180. http://dx.doi.org/10.21914/anziamj.v61i0.15144.

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The article deals with numerical methods for solving a coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing. We employ block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of a solution are discussed. Numerical experiments are presented. References Al-Sultani, M. and Boglaev, I. ''Numerical solution of nonlinear elliptic systems by block monotone iterations''. ANZIAM J. 60:C79–C94, 2019. doi:10.21914/anziamj.v60i0.13986 Al-Sultani, M. ''Numerical solution of nonlinear parabolic systems by block monotone iterations''. Tech. Report, 2019. https://arxiv.org/abs/1905.03599 Boglaev, I. ''Inexact block monotone methods for solving nonlinear elliptic problems'' J. Comput. Appl. Math. 269:109–117, 2014. doi:10.1016/j.cam.2014.03.029 Lui, S. H. ''On monotone iteration and Schwarz methods for nonlinear parabolic PDEs''. J. Comput. Appl. Math. 161:449–468, 2003. doi:doi.org/10.1016/j.cam.2003.06.001 Pao, C. V. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992. doi:10.1007/s002110050168 Pao C. V. ''Numerical analysis of coupled systems of nonlinear parabolic equations''. SIAM J. Numer. Anal. 36:393–416, 1999. doi:10.1137/S0036142996313166 Varga, R. S. Matrix iterative analysis. Springer, Berlin, 2000. 10.1007/978-3-642-05156-2 Zhao, Y. Numerical solutions of nonlinear parabolic problems using combined-block iterative methods. Masters Thesis, University of North Carolina, 2003. http://dl.uncw.edu/Etd/2003/zhaoy/yaxizhao.pdf

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32

GODOY, T., and U. KAUFMANN. "INHOMOGENEOUS PERIODIC PARABOLIC PROBLEMS WITH INDEFINITE DATA." Bulletin of the Australian Mathematical Society 84, no. 3 (September 6, 2011): 516–24. http://dx.doi.org/10.1017/s0004972711002553.

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AbstractLet Ω⊂ℝN be a smooth bounded domain and let f⁄≡0 be a possibly discontinuous and unbounded function. We give a necessary and sufficient condition on f for the existence of positive solutions for all λ>0 of Dirichlet periodic parabolic problems of the form Lu=h(x,t,u)+λf(x,t), where h is a nonnegative Carathéodory function that is sublinear at infinity. When this condition is not fulfilled, under some additional assumptions on h we characterize the set of λs for which the aforementioned problem possesses some positive solution. All results remain true for the corresponding elliptic problems.

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33

Guidotti, Patrick. "Singular elliptic and parabolic problems and a class of free boundary problems." PAMM 7, no. 1 (December 2007): 2040049–50. http://dx.doi.org/10.1002/pamm.200700658.

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34

Tamburrino, A. "Monotonicity based imaging methods for elliptic and parabolic inverse problems." Journal of Inverse and Ill-posed Problems 14, no. 6 (September 2006): 633–42. http://dx.doi.org/10.1515/156939406778474578.

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35

Pozio, Maria Assunta, Fabio Punzo, and Alberto Tesei. "Criteria for well-posedness of degenerate elliptic and parabolic problems." Journal de Mathématiques Pures et Appliquées 90, no. 4 (October 2008): 353–86. http://dx.doi.org/10.1016/j.matpur.2008.06.001.

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36

Kostin, A. B. "Counterexamples in inverse problems for parabolic, elliptic, and hyperbolic equations." Computational Mathematics and Mathematical Physics 54, no. 5 (May 2014): 797–810. http://dx.doi.org/10.1134/s0965542514020092.

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37

KENMOCHI, Nobuyuki, and Masahiro KUBO. "Periodic behavior of solutions to parabolic-elliptic free boundary problems." Journal of the Mathematical Society of Japan 41, no. 4 (October 1989): 625–40. http://dx.doi.org/10.2969/jmsj/04140625.

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38

Makridakis, Charalambos, and Ricardo H. Nochetto. "Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems." SIAM Journal on Numerical Analysis 41, no. 4 (January 2003): 1585–94. http://dx.doi.org/10.1137/s0036142902406314.

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39

Chow, S. N., D. R. Dunninger, and M. Miklavčlč. "Galerkin approximations for singular linear elliptic and semilinear parabolic problems." Applicable Analysis 40, no. 1 (January 1991): 41–52. http://dx.doi.org/10.1080/00036819008839991.

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40

Egger, Herbert, Christoph Erath, and Robert Schorr. "On the Nonsymmetric Coupling Method for Parabolic-Elliptic Interface Problems." SIAM Journal on Numerical Analysis 56, no. 6 (January 2018): 3510–33. http://dx.doi.org/10.1137/17m1158276.

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41

Kannai, Yakar. "A hyperbolic approach to elliptic and parabolic singular perturbation problems." Journal d'Analyse Mathématique 59, no. 1 (December 1992): 75–87. http://dx.doi.org/10.1007/bf02790218.

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42

Bazaliy, B. V., and S. P. Degtyarev. "Classical solutions of many-dimensional elliptic–parabolic free boundary problems." Nonlinear Differential Equations and Applications NoDEA 16, no. 4 (May 20, 2009): 421–43. http://dx.doi.org/10.1007/s00030-009-0020-2.

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43

Sinha, Rajen K., and Bhupen Deka. "Finite element methods for semilinear elliptic and parabolic interface problems." Applied Numerical Mathematics 59, no. 8 (August 2009): 1870–83. http://dx.doi.org/10.1016/j.apnum.2009.02.001.

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44

Patel, Ajit, Sanjib Kumar Acharya, and Amiya Kumar Pani. "Stabilized Lagrange multiplier method for elliptic and parabolic interface problems." Applied Numerical Mathematics 120 (October 2017): 287–304. http://dx.doi.org/10.1016/j.apnum.2017.05.011.

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45

Dutt, P., P. Biswas, and G. Naga Raju. "Preconditioners for spectral element methods for elliptic and parabolic problems." Journal of Computational and Applied Mathematics 215, no. 1 (May 2008): 152–66. http://dx.doi.org/10.1016/j.cam.2007.03.030.

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46

Burman, Erik, and Benjamin Stamm. "Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems." Numerische Mathematik 116, no. 2 (May 12, 2010): 213–41. http://dx.doi.org/10.1007/s00211-010-0304-9.

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47

Chan, W. L., and S. P. Yung. "Error estimate for optimality of distributed parameter control problems via duality." Journal of Applied Mathematics and Stochastic Analysis 8, no. 2 (January 1, 1995): 177–88. http://dx.doi.org/10.1155/s1048953395000165.

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Sharp error estimates for optimality are established for a class of distributed parameter control problems that include elliptic, parabolic, hyperbolic systems with impulsive control and boundary control. The estimates are obtained by constructing manageable dual problems via the extremum principle.

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48

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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49

Chipot, Michel, and José-Francisco Rodrigues. "Comparison and stability of solutions to a class of quasilinear parabolic problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 3-4 (1988): 275–85. http://dx.doi.org/10.1017/s0308210500022265.

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SynopsisThis paper presents new comparison and uniqueness results for the solutions of parabolic quasilinear boundary value problems with (and without) obstacles. A stability result in L1(Ω) yields the asymptotic stabilisation in this space, when t → ∞) towards the corresponding elliptic problem.

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50

Calvo-Jurado, Carmen, and Juan Casado-Díaz. "Homogenization of Dirichlet parabolic systems with variable monotone operators in general perforated domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 6 (December 2003): 1231–48. http://dx.doi.org/10.1017/s0308210500002894.

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We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

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