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Готові списки джерел за темами / Differential equations, Parabolic / Статті в журналах

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Статті в журналах з теми "Differential equations, Parabolic"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 30 липня 2024

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1

Bonafede, Salvatore, and Salvatore A. Marano. "Implicit parabolic differential equations." Bulletin of the Australian Mathematical Society 51, no. 3 (June 1995): 501–9. http://dx.doi.org/10.1017/s0004972700014349.

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Анотація:

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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2

N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform." International Journal of Science and Research (IJSR) 12, no. 6 (June 5, 2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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3

Ishii, Katsuyuki, Michel Pierre, and Takashi Suzuki. "Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations." Mathematics 11, no. 3 (February 2, 2023): 758. http://dx.doi.org/10.3390/math11030758.

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Анотація:

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 (June 22, 2011): 1–35. http://dx.doi.org/10.1155/2011/469806.

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Анотація:

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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5

Ashyralyev, Allaberen, and Ülker Okur. "Stability of Stochastic Partial Differential Equations." Axioms 12, no. 7 (July 24, 2023): 718. http://dx.doi.org/10.3390/axioms12070718.

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Анотація:

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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6

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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Анотація:

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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7

Hrytchuk, M., and I. Klevchuk. "BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION." Bukovinian Mathematical Journal 11, no. 2 (2023): 100–103. http://dx.doi.org/10.31861/bmj2023.02.10.

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Анотація:

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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8

BOUFOUSSI, B., and N. MRHARDY. "MULTIVALUED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 08, no. 02 (June 2008): 271–94. http://dx.doi.org/10.1142/s0219493708002317.

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Анотація:

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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9

Simon, László, and Willi Jäger. "On non-uniformly parabolic functional differential equations." Studia Scientiarum Mathematicarum Hungarica 45, no. 2 (June 1, 2008): 285–300. http://dx.doi.org/10.1556/sscmath.2007.1036.

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Анотація:

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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10

Walter, Wolfgang. "Nonlinear parabolic differential equations and inequalities." Discrete & Continuous Dynamical Systems - A 8, no. 2 (2002): 451–68. http://dx.doi.org/10.3934/dcds.2002.8.451.

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11

Prato, Giuseppe Da, and Alessandra Lunardi. "Stabilizability of integro-differential parabolic equations." Journal of Integral Equations and Applications 2, no. 2 (June 1990): 281–304. http://dx.doi.org/10.1216/jie-1990-2-2-281.

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12

Dynkin, E. B. "Superdiffusions and Parabolic Nonlinear Differential Equations." Annals of Probability 20, no. 2 (April 1992): 942–62. http://dx.doi.org/10.1214/aop/1176989812.

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13

Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations." Stochastic Processes and their Applications 123, no. 12 (December 2013): 4294–336. http://dx.doi.org/10.1016/j.spa.2013.06.015.

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14

Blázquez, C. Miguel, and Elías Tuma. "Saddle connections in parabolic differential equations." Proyecciones (Antofagasta) 13, no. 1 (1994): 25–34. http://dx.doi.org/10.22199/s07160917.1994.0001.00005.

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15

Alicki, R., and D. Makowiec. "Functional integral for parabolic differential equations." Journal of Physics A: Mathematical and General 18, no. 17 (December 1, 1985): 3319–25. http://dx.doi.org/10.1088/0305-4470/18/17/012.

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16

Ibragimov, N. H., S. V. Meleshko, and E. Thailert. "Invariants of linear parabolic differential equations." Communications in Nonlinear Science and Numerical Simulation 13, no. 2 (March 2008): 277–84. http://dx.doi.org/10.1016/j.cnsns.2006.03.017.

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17

Chow, S. N., K. N. Lu, and J. Malletparet. "Floquet Theory for Parabolic Differential Equations." Journal of Differential Equations 109, no. 1 (April 1994): 147–200. http://dx.doi.org/10.1006/jdeq.1994.1047.

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18

Selitskii, Anton. "ON SOLVABILITY OF PARABOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES (II)." Eurasian Mathematical Journal 11, no. 2 (2020): 86–92. http://dx.doi.org/10.32523/2077-9879-2020-11-2-86-92.

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19

Wang, Hanxiao. "Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula." Stochastics and Dynamics 21, no. 01 (March 11, 2020): 2150004. http://dx.doi.org/10.1142/s0219493721500040.

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Анотація:

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

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20

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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Анотація:

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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21

Kehaili, Abdelkader, Ali Hakem, and Abdelkader Benali. "Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients." Global Journal of Pure and Applied Sciences 26, no. 1 (June 1, 2020): 35–55. http://dx.doi.org/10.4314/gjpas.v26i1.6.

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Анотація:

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

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22

Shakhmurov, Veli. "Regularity properties of nonlocal fractional differential equations and applications." Georgian Mathematical Journal 29, no. 2 (February 5, 2022): 275–84. http://dx.doi.org/10.1515/gmj-2021-2128.

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Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

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23

Bahuguna, D., and V. Raghavendra. "Rothe’s method to parabolic integra-differential equations via abstract integra-differential equations." Applicable Analysis 33, no. 3-4 (January 1989): 153–67. http://dx.doi.org/10.1080/00036818908839869.

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24

Turkyilmazoglu, M. "Parabolic Partial Differential Equations with Nonlocal Initial and Boundary Values." International Journal of Computational Methods 12, no. 05 (October 2015): 1550024. http://dx.doi.org/10.1142/s0219876215500243.

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Анотація:

Parabolic partial differential equations possessing nonlocal initial and boundary specifications are used to model some real-life applications. This paper focuses on constructing fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on a double power series representation of the solution via ordinary polynomials. Consequently, a parabolic partial differential equation is reduced to a system involving algebraic equations. Exact solutions are obtained when the solutions are themselves polynomials. Some parabolic partial differential equations are treated by the technique to judge its validity and also to measure its accuracy as compared to the existing methods.

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25

Netka, Milena. "Differential difference inequalities related to parabolic functional differential equations." Opuscula Mathematica 30, no. 1 (2010): 95. http://dx.doi.org/10.7494/opmath.2010.30.1.95.

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26

Matusik, Milena. "Iterative methods for parabolic functional differential equations." Applicationes Mathematicae 40, no. 2 (2013): 221–35. http://dx.doi.org/10.4064/am40-2-5.

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27

Anikushyn, A. V., and A. L. Hulianytskyi. "Generalized solvability of parabolic integro-differential equations." Differential Equations 50, no. 1 (January 2014): 98–109. http://dx.doi.org/10.1134/s0012266114010133.

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28

Lord, Gabriel J., and Tony Shardlow. "Postprocessing for Stochastic Parabolic Partial Differential Equations." SIAM Journal on Numerical Analysis 45, no. 2 (January 2007): 870–89. http://dx.doi.org/10.1137/050640138.

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29

Garroni, Maria Giovanna, and José Luis Menaldi. "Regularizing effect for integro-differential parabolic equations." Communications in Algebra 18, no. 12 (1993): 2023–50. http://dx.doi.org/10.1080/00927879308824122.

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30

Gear, C. W., and Ioannis G. Kevrekidis. "Telescopic projective methods for parabolic differential equations." Journal of Computational Physics 187, no. 1 (May 2003): 95–109. http://dx.doi.org/10.1016/s0021-9991(03)00082-2.

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31

Atkinson, Kendall, Olaf Hansen, and David Chien. "A spectral method for parabolic differential equations." Numerical Algorithms 63, no. 2 (July 27, 2012): 213–37. http://dx.doi.org/10.1007/s11075-012-9620-8.

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32

Al Horani, Mohammed, Angelo Favini, and Hiroki Tanabe. "Parabolic First and Second Order Differential Equations." Milan Journal of Mathematics 84, no. 2 (October 27, 2016): 299–315. http://dx.doi.org/10.1007/s00032-016-0260-7.

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33

Malek, St�phane. "Hypergeometric Functions and Parabolic Partial Differential Equations." Journal of Dynamical and Control Systems 11, no. 2 (April 2005): 253–62. http://dx.doi.org/10.1007/s10883-005-4173-y.

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34

Simon, L. "On higher order parabolic functional differential equations." Periodica Mathematica Hungarica 31, no. 1 (August 1995): 53–62. http://dx.doi.org/10.1007/bf01876354.

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35

Zuur, E. A. H. "Time-step sequences for parabolic differential equations." Applied Numerical Mathematics 17, no. 2 (May 1995): 173–86. http://dx.doi.org/10.1016/0168-9274(95)00012-j.

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36

Bayer, Christian, Denis Belomestny, Martin Redmann, Sebastian Riedel, and John Schoenmakers. "Solving linear parabolic rough partial differential equations." Journal of Mathematical Analysis and Applications 490, no. 1 (October 2020): 124236. http://dx.doi.org/10.1016/j.jmaa.2020.124236.

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37

Lefton, L. E., and V. L. Shapiro. "Resonance and Quasilinear Parabolic Partial Differential Equations." Journal of Differential Equations 101, no. 1 (January 1993): 148–77. http://dx.doi.org/10.1006/jdeq.1993.1009.

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38

Qahremani, E., T. Allahviranloo, S. Abbasbandy, and N. Ahmady. "A study on the fuzzy parabolic Volterra partial integro-differential equations." Journal of Intelligent & Fuzzy Systems 40, no. 1 (January 4, 2021): 1639–54. http://dx.doi.org/10.3233/jifs-201125.

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Анотація:

This paper is concerned with aspects of the analytical fuzzy solutions of the parabolic Volterra partial integro-differential equations under generalized Hukuhara partial differentiability and it consists of two parts. The first part of this paper deals with aspects of background knowledge in fuzzy mathematics, with emphasis on the generalized Hukuhara partial differentiability. The existence and uniqueness of the solutions of the fuzzy Volterra partial integro-differential equations by considering the type of [gH - p]-differentiability of solutions are proved in this part. The second part is concerned with the central themes of this paper, using the fuzzy Laplace transform method for solving the fuzzy parabolic Volterra partial integro-differential equations with emphasis on the type of [gH - p]-differentiability of solution. We test the effectiveness of method by solving some fuzzy Volterra partial integro-differential equations of parabolic type.

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39

M. El-Borai, Mahmoud, Hamed Kamal Awad, and Randa Hamdy M. Ali. "Method of Averaging for Some Parabolic Partial Differential Equations." Academic Journal of Applied Mathematical Sciences, no. 61 (January 25, 2020): 1–4. http://dx.doi.org/10.32861/ajams.61.1.4.

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Анотація:

Quantitative and qualitative analysis of the Averaging methods for the parabolic partial differential equation appears as an exciting field of the investigation. In this paper, we generalize some known results due to Krol on the averaging methods and use them to solve the parabolic partial differential equation.

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40

Barletta, Giuseppina. "Parabolic equations with discontinuous nonlinearities." Bulletin of the Australian Mathematical Society 63, no. 2 (April 2001): 219–28. http://dx.doi.org/10.1017/s0004972700019286.

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Анотація:

In this paper we deal with the homogeneous Cauchy-Dirichlet problem for a class of parabolic equations with either Carathéodory or discontinuous nonlinear terms. We then present an application and explicitly point out an existence result for a differential inclusion, which can be applied to the classical Stefan problem.

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41

R.Ramesh Rao, T. "Numerical Solution of Time Fractional Parabolic Differential Equations." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 790. http://dx.doi.org/10.14419/ijet.v7i4.10.26117.

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Анотація:

In this paper, we study the coupling of an approximate analytical technique called reduced differential transform (RDT) with fractional complex transform. The present method reduces the time fractional differential equations in to integer order differential equations. The fractional derivatives are defined in Jumaries modified Riemann-Liouville sense. Result shows that the present technique is effective and powerful for handling the fractional order differential equations.

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42

Hu, Zhanrong, and Zhen Jin. "Almost Automorphic Mild Solutions to Neutral Parabolic Nonautonomous Evolution Equations with Nondense Domain." Discrete Dynamics in Nature and Society 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/183420.

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Анотація:

Combining the exponential dichotomy of evolution family, composition theorems for almost automorphic functions with Banach fixed point theorem, we establish new existence and uniqueness theorems for almost automorphic mild solutions to neutral parabolic nonautonomous evolution equations with nondense domain. A unified framework is set up to investigate the existence and uniqueness of almost automorphic mild solutions to some classes of parabolic partial differential equations and neutral functional differential equations.

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43

Topolski, Krzysztof A. "On the existence of classical solutions for differential-functional IBVP." Abstract and Applied Analysis 3, no. 3-4 (1998): 363–75. http://dx.doi.org/10.1155/s1085337598000608.

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Анотація:

We consider the initial-boundary value problem for second order differential-functional equations of parabolic type. Functional dependence in the equation is of the Hale type. By using Leray-Schauder theorem we prove the existence of classical solutions. Our formulation and results cover a large class of parabolic problems both with a deviated argument and integro-differential equations.

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44

Hensel, Edward. "Inverse Problems for Multi-Dimensional Parabolic Partial Differential Equations." Applied Mechanics Reviews 41, no. 6 (June 1, 1988): 263–69. http://dx.doi.org/10.1115/1.3151898.

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Анотація:

The parabolic inverse boundary value problem is defined, and the major characteristics common to multi-dimensional parabolic inverse problems are discussed. These include sensitivity to measurement error and noise, the value of future time information, the nonuniqueness of estimates, and the effects of temperature dependent thermal properties. The importance of quantifying resolution degradation and estimate variances is discussed.

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45

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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Анотація:

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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46

Lee, Min Ho. "Mixed automorphic forms and differential equations." International Journal of Mathematics and Mathematical Sciences 13, no. 4 (1990): 661–68. http://dx.doi.org/10.1155/s0161171290000916.

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Анотація:

We construct mixed automorphic forms associated to a certain class of nonhomogeneous linear ordinary differential equations. We also establish an isomorphism between the space of mixed automorphic forms of the second kind modulo exact forms nd a certain parabolic cohomology explicitly in terms of the periods of mixed automorphic forms.

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Truman, Aubrey, FengYu Wang, JiangLun Wu, and Wei Yang. "A link of stochastic differential equations to nonlinear parabolic equations." Science China Mathematics 55, no. 10 (August 9, 2012): 1971–76. http://dx.doi.org/10.1007/s11425-012-4463-2.

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48

Ruaa Shawqi Ismael, Ali Al -Fayadh, and Saad M. Salman. "Solving Partial Differential Equations by using Efficient Hybrid Transform Iterative Method." Tikrit Journal of Pure Science 29, no. 3 (June 25, 2024): 75–83. http://dx.doi.org/10.25130/tjps.v29i3.1566.

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The aim of this article is to propose an efficient hybrid transform iteration method that combines the homotopy perturbation approach, the variational iteration method, and the Aboodh transform forsolving various partial differential equations. The Korteweg-de Vries (KdV), modified KdV, coupled KdV, and coupled pseudo-parabolic equations are given as examples to show how effective and practical the suggested method is. The obtained exact solitary solutions of the KdV equations as well as the exact solution of the coupled pseudo-parabolic equations are identified as a convergent series with easily calculable components. identified as a convergent series with easily calculable components. When used to solve KdV , Wave like and Pseudo – Parabolic equations , the proposed method helps to avoid Problems that frequently arise when determining the Lagrange Multiplier and the difficult integration usedin the variation iteration method , as well as the need to use the transform convolution theorem.

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Van Brunt, B., D. Pidgeon, M. Vlieg-Hulstman, and W. D. Halford. "Conservation laws for second-order parabolic partial differential equations." ANZIAM Journal 45, no. 3 (January 2004): 333–48. http://dx.doi.org/10.1017/s1446181100013407.

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AbstractConservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Fréchet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

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Ameer, Zainab Abdel, and Sameer Qasim Hasan. "Solution of Parabolic Differential Equations with Dirichlet Control Boundary problem." Journal of Physics: Conference Series 2322, no. 1 (August 1, 2022): 012047. http://dx.doi.org/10.1088/1742-6596/2322/1/012047.

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Abstract In this paper we study a system of two degenerate parabolic equations which defined in a bounded domain. Using concepts of parabolic for finding existence of the solutions of the Dirichlet control boundary problem and the existence of Periodic time-solutions were achieved with results presented in detail for the proposed system.

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