Journal articles: 'Systems of Parabolic Equations'

1

Simon, László. "On some singular systems of parabolic functional equations." Mathematica Bohemica 135, no. 2 (2010): 123–32. http://dx.doi.org/10.21136/mb.2010.140689.

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2

Amann, Herbert. "Quasilinear evolution equations and parabolic systems." Transactions of the American Mathematical Society 293, no. 1 (January 1, 1986): 191. http://dx.doi.org/10.1090/s0002-9947-1986-0814920-4.

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3

Kozhevnikov, A. "Multi–Weighted Parabolic Equations and Systems." Journal of Mathematical Sciences 193, no. 2 (July 31, 2013): 267–82. http://dx.doi.org/10.1007/s10958-013-1452-0.

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4

Klevchuk, I. I. "Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion." Carpathian Mathematical Publications 14, no. 2 (December 30, 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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5

Ishida, Sachiko, and Tomomi Yokota. "Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems." Archivum Mathematicum, no. 2 (2023): 181–89. http://dx.doi.org/10.5817/am2023-2-181.

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6

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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7

Kavian, Otared, and Luz de Teresa. "Unique continuation principle for systems of parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 16, no. 2 (February 10, 2009): 247–74. http://dx.doi.org/10.1051/cocv/2008077.

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8

Bensoussan, Alain, and Jens Frehse. "Smooth Solutions of systems of quasilinear parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 169–93. http://dx.doi.org/10.1051/cocv:2002059.

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9

Eden, A., B. Michaux, and J. M. Rakotoson. "Doubly nonlinear parabolic-type equations as dynamical systems." Journal of Dynamics and Differential Equations 3, no. 1 (January 1991): 87–131. http://dx.doi.org/10.1007/bf01049490.

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10

Rogovchenko, Yuri V. "Comparison principles for systems of impulsive parabolic equations." Annali di Matematica Pura ed Applicata 170, no. 1 (December 1996): 311–28. http://dx.doi.org/10.1007/bf01758993.

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11

Lyubanova, Anna Sh. "On nonlocal problems for systems of parabolic equations." Journal of Mathematical Analysis and Applications 421, no. 2 (January 2015): 1767–78. http://dx.doi.org/10.1016/j.jmaa.2014.08.027.

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12

Galaktionov, Victor A. "Dynamical systems of inequalities and nonlinear parabolic equations." Communications in Partial Differential Equations 24, no. 11-12 (January 1999): 2191–236. http://dx.doi.org/10.1080/03605309908821499.

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13

Nee, Janpou. "Almost periodic solutions to systems of parabolic equations." Journal of Applied Mathematics and Stochastic Analysis 7, no. 4 (January 1, 1994): 581–86. http://dx.doi.org/10.1155/s1048953394000456.

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In this paper we show that the second-order differential solution is 𝕃2-almost periodic, provided it is 𝕃2-bounded, and the growth of the components of a non-linear function of a system of parabolic equation is bounded by any pair of con-secutive eigenvalues of the associated Dirichlet boundary value problems.

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14

Ревина, С. В. "Diffusion Instability Region for Systems of Parabolic Equations." Владикавказский математический журнал, no. 4 (December 14, 2022): 117–26. http://dx.doi.org/10.46698/d6373-9335-7338-n.

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Рассматривается система двух уравнений реакции-диффузии в ограниченной области $m$-мерного пространства с краевыми условиями Неймана на границе, для которой слагаемые реакции $f(u,v)$ и $g(u,v)$ зависят от двух параметров $a$ и $b$. Предполагается, что система имеет пространственно-однородное решение $(u_0,v_0)$, причем $f_u(u_0,v_0)>0$, а $-g_v(u_0,v_0)=F( \mathrm{Det (\mathrm{J})})$, где $\mathrm{J}$~--- матрица Якоби соответствующей линеаризованной системы в бездиффузионном приближении, $F$~--- гладкая монотонно возрастающая функция. Предложен способ аналитического описания области необходимых и достаточных условий неустойчивости Тьюринга на плоскости параметров системы при фиксированном коэффициенте диффузии $d$. Показано, что область необходимых условий неустойчивости Тьюринга на плоскости $( \mathrm{Det (\mathrm{J})}, f_u)$ ограничена кривой нулевого следа, дискриминантной кривой и геометрическим местом точек $ \mathrm{Det (\mathrm{J})}=0$. Найдены явные выражения кривых достаточных условий и доказано, что дискриминантная кривая является огибающей семейства этих кривых. Показано, что одна из границ области неустойчивости Тьюринга состоит из фрагментов кривых достаточных условий, выражается через функцию F и собственные значения оператора Лапласа в рассматриваемой области. Найдены точки пересечения кривых достаточных условий и показано, что их абсциссы не зависят от вида функции $F$ и выражаются через коэффициент диффузии и собственные значения оператора Лапласа. Рассмотрен частный случай $F( \mathrm{Det (\mathrm{J})})= \mathrm{Det (\mathrm{J})}$. Для этого случая указан диапазон волновых чисел, при которых возникает неустойчивость Тьюринга. Получено разбиение полуоси $d>1$ на полуинтервалы, каждому из которых соответствует свое минимальное критическое волновое число. Точки пересечения кривых достаточных условий лежат на прямых, не зависящих от коэффициента диффузии $d$. В качестве примеров приложений доказанных утверждений рассматриваются система Шнакенберга и уравнения брюсселятора.

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15

Kahane, Charles S. "The Feynman-Kac formula for a system of parabolic equations." Czechoslovak Mathematical Journal 44, no. 4 (1994): 579–602. http://dx.doi.org/10.21136/cmj.1994.128499.

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16

Avalishvili, Gia, and Mariam Avalishvili. "On nonclassical problems for first-order evolution equations." gmj 18, no. 3 (July 14, 2011): 441–63. http://dx.doi.org/10.1515/gmj.2011.0028.

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Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

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17

Hong, Keum S., and Joseph Bentsman. "Stability Criterion For Linear Oscillatory Parabolic Systems." Journal of Dynamic Systems, Measurement, and Control 114, no. 1 (March 1, 1992): 175–78. http://dx.doi.org/10.1115/1.2896501.

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This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.

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18

Heinrichs, Wilhelm. "Enclosure statements for systems of semilinear parabolic differential equations." Applications of Mathematics 36, no. 2 (1991): 96–122. http://dx.doi.org/10.21136/am.1991.104448.

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19

Sivergina, Irina F., Michael P. Polis, and Ilya Kolmanovsky. "Source Identification for Parabolic Equations." Mathematics of Control, Signals, and Systems (MCSS) 16, no. 2-3 (September 1, 2003): 141–57. http://dx.doi.org/10.1007/s00498-003-0136-6.

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20

Bensoussan, Alain, Jens Frehse, and Sheung Chi Phillip Yam. "Systems of quasilinear parabolic equations in Rn and systems of quadratic backward stochastic differential equations." Journal de Mathématiques Pures et Appliquées 149 (May 2021): 135–85. http://dx.doi.org/10.1016/j.matpur.2021.01.006.

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21

Pao, C. V. "Numerical Analysis of Coupled Systems of Nonlinear Parabolic Equations." SIAM Journal on Numerical Analysis 36, no. 2 (January 1999): 393–416. http://dx.doi.org/10.1137/s0036142996313166.

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22

Litovchenko, V. A., and E. B. Nastasiĭ. "Degenerate parabolic systems of vector order Kolmogorov-type equations." Siberian Mathematical Journal 53, no. 1 (January 2012): 119–33. http://dx.doi.org/10.1134/s0037446612010107.

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23

Engler, Hans, Bernhard Kawohl, and Stephan Luckhaus. "Gradient estimates for solutions of parabolic equations and systems." Journal of Mathematical Analysis and Applications 147, no. 2 (April 1990): 309–29. http://dx.doi.org/10.1016/0022-247x(90)90350-o.

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24

Mierczyński, Janusz, and Wenxian Shen. "Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations." Journal of Dynamics and Differential Equations 23, no. 3 (July 20, 2010): 551–71. http://dx.doi.org/10.1007/s10884-010-9181-2.

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25

Simon, L. "On systems of strongly nonlinear parabolic functional differential equations." Periodica Mathematica Hungarica 33, no. 2 (October 1996): 135–51. http://dx.doi.org/10.1007/bf02093511.

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26

Belopolskaya, Yana. "Stochastic models for forward systems of nonlinear parabolic equations." Statistical Papers 59, no. 4 (August 29, 2018): 1505–19. http://dx.doi.org/10.1007/s00362-018-1033-x.

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27

Pao, C. V. "Systems of Parabolic Equations with Continuous and Discrete Delays." Journal of Mathematical Analysis and Applications 205, no. 1 (January 1997): 157–85. http://dx.doi.org/10.1006/jmaa.1996.5177.

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28

Cao, Yanzhao, Max Gunzburger, and James Turner. "The Controllability of Systems Governed by Parabolic Differential Equations." Journal of Mathematical Analysis and Applications 215, no. 1 (November 1997): 174–89. http://dx.doi.org/10.1006/jmaa.1997.5633.

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29

Li, Wei Nian, Bao Tong Cui, and Lokenath Debnath. "Oscillation of systems of certain neutral delay parabolic differential equations." Journal of Applied Mathematics and Stochastic Analysis 16, no. 1 (January 1, 2003): 81–94. http://dx.doi.org/10.1155/s1048953303000066.

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30

Biccari, Umberto, Dongnam Ko, and Enrique Zuazua. "Dynamics and control for multi-agent networked systems: A finite-difference approach." Mathematical Models and Methods in Applied Sciences 29, no. 04 (April 2019): 755–90. http://dx.doi.org/10.1142/s0218202519400050.

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We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time horizon and the size of the controls.

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Caussignac, Ph. "Incompletely Parabolic Systems from Friedrichs Theory Point of View." Mathematical Models and Methods in Applied Sciences 07, no. 08 (December 1997): 1141–52. http://dx.doi.org/10.1142/s0218202597000566.

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An incompletely parabolic system of partial differential equations consist of one parabolic subsystem coupled to a hyperbolic subsystem. For the initial-boundary value problem, it has been shown, by requiring that a solution remains bounded at any time by the data, that boundary conditions which make both subsystems well-posed render the global system well-posed too. In this paper, we establish the same type of result with the help of the notion of semi-admissible boundary conditions in the theory of Friedrichs positive systems of differential equations. Our aim is not to obtain the same results of existence and uniqueness as those for the Cauchy problem, but rather to find a way to establish boundary conditions on the subsystems which can be used for the global system too. The theory is illustrated by the two examples of the compressible Navier–Stokes equations and of the hydrodynamic model for semiconductor devices, both in two space dimensions.

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32

Gombao, Sophie, and Jean-Pierre Raymond. "Hamilton-Jacobi equations for control problems of parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 12, no. 2 (March 22, 2006): 311–49. http://dx.doi.org/10.1051/cocv:2006004.

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33

Boyer, Franck, and Víctor Hernández-Santamaría. "Carleman estimates for time-discrete parabolic equations and applications to controllability." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 12. http://dx.doi.org/10.1051/cocv/2019072.

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In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

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34

Papageorgiou, N. "Optimality conditions for systems driven by nonlinear evolution equations." Mathematical Problems in Engineering 1, no. 1 (1995): 27–36. http://dx.doi.org/10.1155/s1024123x95000044.

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Using the Dubovitskii-Milyutin theory we derive necessary and sufficient conditions for optimality for a class of Lagrange optimal control problems monitored by a nonlinear evolution equation and involving initial and/or terminal constraints. An example of a parabolic control system is also included.

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35

FAYYAD, D., and N. NASSIF. "APPROXIMATION OF BLOWING UP SOLUTIONS TO SEMILINEAR PARABOLIC EQUATIONS USING "MASS CONTROLLED" PARABOLIC SYSTEMS." Mathematical Models and Methods in Applied Sciences 09, no. 07 (October 1999): 1077–88. http://dx.doi.org/10.1142/s0218202599000488.

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This paper considers asymptotic approximations to the solutions of the semilinear parabolic equation: [Formula: see text] where the function f(u) is such that the solution to (0.1) blows up in a finite time Tb. In order to control the explosive behavior of this problem, we consider a "perturbation" to (0.1) defined by: [Formula: see text] where ε is a small positive number. The boundary and initial conditions on uε are those of u. For vε, the initial and boundary conditions are chosen to be 1. Note that system (0.2) belongs to a class of coupled semilinear parabolic equations, with positive solutions and "mass control" property, (see Ref. 10). The solution {uε, vε} of such systems is known to be global. As such, (0.2) appears to be a regular perturbation to a singular problem (0.1). In this work, our basic theorem is a convergence proof for uε and [Formula: see text] to u and ut, respectively, in the L∞ norm. These results constitute a framework for designing in subsequent work, numerical algorithms for the computation of blow-up times (see Ref. 6).

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36

Quittner, Pavol, and Philippe Souplet. "Admissible Lp norms for local existence and for continuation in semilinear parabolic systems are not the same." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (December 2001): 1435–56. http://dx.doi.org/10.1017/s0308210500001475.

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We say that a Banach space E is a continuation space for a given parabolic problem if the E-norm of any non-global solution has to become unbounded. We will prove that for large classes of parabolic systems of two equations, the space E = Lr1 × Lr2 can be a continuation space even though the problem is not locally well posed in E. This stands in contrast with classical results for analogous scalar equations.

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37

Bondarenko, Viktor, Anna Kravchenko, and Tetiana Sobko. "Generalization of the Trotter–Daletsky formula for systems of the "reaction–diffusion" type." System research and information technologies, no. 4 (December 22, 2021): 102–14. http://dx.doi.org/10.20535/srit.2308-8893.2021.4.08.

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An iterative method for constructing a solution to the Cauchy problem for a system of parabolic equations with a nonlinear potential has been proposed and substantiated. The method is based on the Trotter–Daletsky formula, generalized for a nonlinear perturbation of an elliptic operator. The idea of generalization is the construction of a composition of the semigroup generated by the Laplacian and the phase flow corresponding to a system of ordinary differential equations. A computational experiment performed for a two-dimensional system of semilinear parabolic equations of the “reaction–diffusion” type confirms estimates for the convergence of iterations established in the proof of this formula. Obtained results suggest the feasibility of an unconventional approach to modeling dynamic systems with distributed parameters.

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38

Cywiak-Códova, D., G. Gutiérrez-Juárez, and And M. Cywiak-Garbarcewicz. "Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients." Revista Mexicana de Física E 17, no. 1 Jan-Jun (January 28, 2020): 11. http://dx.doi.org/10.31349/revmexfise.17.11.

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A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated. In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differential equations, which can be seen as a generalization of d'Alambert solution. We found that the traditional classification, i.e., parabolic, hyperbolic and elliptic, is not necessary reducing the classification to only parabolic and non-parabolic cases. We put special attention for parabolic partial differential equations, analyzing the general 1D homogeneous solution of the Photoacoustic and Photothermal equations in the frequency and time domain. Finally, we also used the method to solve Helmholtz equation in cylindrical coordinates, showing that it can be used in other coordinates systems.

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39

Fattorini, H. O. "Robustness and convergence of suboptimal controls in distributed parameter systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 6 (1997): 1153–79. http://dx.doi.org/10.1017/s0308210500026998.

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A fundamental prerequisite for the numerical computation of optimal controls is to show that sequences of suboptimal (that is, close-to-optimal) controls converge. We show this in a version that applies to hyperbolic and parabolic distributed parameter systems, the latter including the Navier–Stokes equations. The optimal problems include control and state constraints; in the parabolic case, the constraints may be pointwise on the solution and the gradient.

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40

Ling, Rina. "Unstable periodic wave solutions of Nerve Axion diffusion equations." International Journal of Mathematics and Mathematical Sciences 10, no. 4 (1987): 787–96. http://dx.doi.org/10.1155/s0161171287000875.

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41

Skakauskas, V., and P. Katauskis. "Numerical solving of coupled systems of parabolic and ordinary differential equations." Nonlinear Analysis: Modelling and Control 15, no. 3 (July 25, 2010): 351–60. http://dx.doi.org/10.15388/na.15.3.14330.

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Two coupled systems of parabolic and nonlinear ordinary differential equations arising in kinetics of heterogeneous reactions are studied numerically by using computer calculations. Some numerical results are discussed.

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42

Kashchenko, Sergey. "Infinite Turing Bifurcations in Chains of Van der Pol Systems." Mathematics 10, no. 20 (October 13, 2022): 3769. http://dx.doi.org/10.3390/math10203769.

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A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. In a number of cases, normalized parabolic equations with two spatial variables arise while considering the most important diffusion type couplings. It has been established, for example, that for the considered systems with a large number of elements, the dynamics change significantly with a slight change in the number of such elements.

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43

Bokalo, M. M., and A. M. Tsebenko. "Optimal control problem for systems governed by nonlinear parabolic equations without initial conditions." Carpathian Mathematical Publications 8, no. 1 (June 30, 2016): 21–37. http://dx.doi.org/10.15330/cmp.8.1.21-37.

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An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

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44

Melnig, Elena-Alexandra. "Stability in 𝐿𝑞-norm for inverse source parabolic problems." Journal of Inverse and Ill-posed Problems 28, no. 6 (December 1, 2020): 797–814. http://dx.doi.org/10.1515/jiip-2019-0081.

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AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

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45

Tran, Hoang-Dung, Weiming Xiang, Stanley Bak, and Taylor T. Johnson. "Reachability Analysis for One Dimensional Linear Parabolic Equations." IFAC-PapersOnLine 51, no. 16 (2018): 133–38. http://dx.doi.org/10.1016/j.ifacol.2018.08.023.

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46

Mitra, Sourav. "Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 66. http://dx.doi.org/10.1051/cocv/2019036.

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In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

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47

Jaruszewska-Walczak, Danuta. "Comparison theorems for infinite systems of parabolic functional-differential equations." Annales Polonici Mathematici 77, no. 3 (2001): 261–70. http://dx.doi.org/10.4064/ap77-3-5.

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Baderko, E. A., and M. F. Cherepova. "Mixed Problems for Plane Parabolic Systems and Boundary Integral Equations." Journal of Mathematical Sciences 260, no. 4 (January 2022): 418–33. http://dx.doi.org/10.1007/s10958-022-05703-4.

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Kang, Jing, and Changzheng Qu. "Symmetry groups and fundamental solutions for systems of parabolic equations." Journal of Mathematical Physics 53, no. 2 (February 2012): 023509. http://dx.doi.org/10.1063/1.3684749.

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Lin Guo, Yung-Jen, and Masahiko Shimojo. "Blow-up for Parabolic Equations and Systems with Nonnegative Potential." Taiwanese Journal of Mathematics 15, no. 3 (June 2011): 995–1005. http://dx.doi.org/10.11650/twjm/1500406280.

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Journal articles on the topic 'Systems of Parabolic Equations'

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