Relevant bibliographies by topics / Mixed hyperbolic-parabolic problems

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  2. Dissertations / Theses
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Academic literature on the topic 'Mixed hyperbolic-parabolic problems'

Author: Grafiati
Published: 9 March 2023
Last updated: 10 March 2023

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Journal articles on the topic "Mixed hyperbolic-parabolic problems"

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Aldashev, S. A. "TRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATION." Vestnik of Samara University. Natural Science Series 26, no. 4 (August 17, 2021): 7–14. http://dx.doi.org/10.18287/2541-7525-2020-26-4-7-14.

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It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the mediums conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.
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Aldashev, Serik. "The Tricomi Problem for a Class of Multidimensional Mixed Hyperbolic-Parabolic Equations." Mathematical Physics and Computer Simulation, no. 2 (August 2022): 5–16. http://dx.doi.org/10.15688/mpcm.jvolsu.2022.2.1.

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It is known that in the mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate multidimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) is reduced to degenerate multidimensional hyperbolic-parabolic equations. It is also known that the oscillations of elastic membranes in space can be modeled according to the Hamilton principle by degenerate multidimensional hyperbolic equations. The study of the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Therefore, by studying the mathematical modeling of the heat propagation process in oscillating elastic membranes, we also arrive at degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it becomes necessary to obtain an explicit representation of the solutions to the problems under study. Boundary value problems for hyperbolicparabolic equations on the plane are well studied, and their multidimensional analogues are little studied. The Tricomi problem for these equations was previously investigated. As far as we know, this problem has not been studied in space. In this paper, the Tricomi problem is shown to be ambiguously solvable for a class of multidimensional mixed hyperbolic-parabolic equations.
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Aldashev, C. A., and E. Kazez. "CORRECTNESS OF THE MIXED PROBLEM FOR ONE CLASS OF DEGENERATE MULTIDIMENSIONAL HYPERBOLO-PARABOLIC EQUATIONS." SERIES PHYSICO-MATHEMATICAL 6, no. 334 (December 15, 2020): 27–35. http://dx.doi.org/10.32014/2020.2518-1726.94.

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It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, we get degenerate multi-dimensional hyperbolic equations. If the medium has a high conductivity, then we go to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to degenerate multidimensional hyperbolic-parabolic equations. Also, it is known that the oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Studying the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Consequently, by studying the mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it is necessary to obtain an explicit representation of the solutions of the studied problems. The mixed problem for degenerate multidimensional hyperbolic equations was previously considered. As far as is known, these questions for degenerate multidimensional hyperbolic-parabolic equations have not been studied. In this paper, unique solvability is shown and an explicit form of the classical solution of the mixed problem for one class of degenerate multidimensional hyperbolic-parabolic equations is obtained.
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Sidorov, S. N. "Inverse problems for a mixed parabolic-hyperbolic equation with a degenerate parabolic part." Sibirskie Elektronnye Matematicheskie Izvestiya 16 (January 31, 2019): 144–57. http://dx.doi.org/10.33048/semi.2019.16.007.

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Abdumitalip uulu, K. "Boundary Value Problems for a Mixed Fourth-order Parabolic-Hyperbolic Equation With Discontinuous Gluing Conditions." Bulletin of Science and Practice, no. 11 (November 15, 2022): 12–23. http://dx.doi.org/10.33619/2414-2948/84/01.

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The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.
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Tarasenko, A. V. "ON SOME PROBLEMS FOR A LOADED PARABOLIC-HYPERBOLIC EQUATION." Vestnik of Samara University. Natural Science Series 19, no. 6 (June 2, 2017): 201–4. http://dx.doi.org/10.18287/2541-7525-2013-19-6-201-204.

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Some problems with various boundary conditions for the loaded mixed type equation in rectangular area are studied. The criterion of uniqueness is established and theorems of an existence of solutions to the problems are proved. The solutions are constructed as Fourier series with respect to eigenfunctions of a corresponding one-dimensional problem.
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Yunusova, G. R. "NONLOCAL PROBLEMS FOR THE EQUATION OF THE MIXED PARABOLIC-HYPERBOLIC TYPE." Vestnik of Samara University. Natural Science Series 17, no. 8 (June 14, 2017): 108–17. http://dx.doi.org/10.18287/2541-7525-2011-17-8-108-117.

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Boundary value problems with non-local conditions for partial differential equation are considered. In these problems, non-local conditions connect the values of a required solutions on the opposite sides of a rectangular domain. Criteria of uniqueness of each of the problems are obtained. Solutions to both problems are constructed as sums of Fourier series. The stability of solutions is proved.
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Milovanovic-Jeknic, Zorica. "Parabolic-hyperbolic transmission problem in disjoint domains." Filomat 32, no. 20 (2018): 6911–20. http://dx.doi.org/10.2298/fil1820911m.

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In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper we investigate a mixed parabolic-hyperbolic initial-boundary value problem in two nonadjacent rectangles with nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problem the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.
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Colli, Pierluigi, and Angelo Favini. "Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations." International Journal of Mathematics and Mathematical Sciences 19, no. 3 (1996): 481–94. http://dx.doi.org/10.1155/s0161171296000683.

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In this paper we deal with the equationL(d2u/dt2)+B(du/dt)+Au∋f, whereLandAare linear positive selfadjoint operators in a Hilbert spaceHand from a Hilbert spaceV⊂Hto its dual spaceV′, respectively, andBis a maximal monotone operator fromVtoV′. By assuming some coerciveness onL+BandA, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.
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Balkizov, Zh A., Z. Kh Guchaeva, and A. Kh Kodzokov. "Inner boundary value problem with displacement for a second order mixed parabolic-hyperbolic equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 106, no. 2 (June 30, 2022): 59–71. http://dx.doi.org/10.31489/2022m2/59-71.

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This paper investigates inner boundary value problems with a shift for a second-order mixed-hyperbolic equation consisting of a wave operator in one part of the domain and a degenerate hyperbolic operator of the first kind in the other part. We find sufficient conditions for the given functions to ensure the existence of a unique regular solution to the problems under study. In some special cases, solutions are obtained explicitly.
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Dissertations / Theses on the topic "Mixed hyperbolic-parabolic problems"

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ROSSI, ELENA. "Balance Laws: Non Local Mixed Systems and IBVPs." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2016. http://hdl.handle.net/10281/103090.

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Scalar hyperbolic balance laws in several space dimensions play a central role in this thesis. First, we deal with a new class of mixed parabolic-hyperbolic systems on all R^n: we obtain the basic well-posedness theorems, devise an ad hoc numerical algorithm, prove its convergence and investigate the qualitative properties of the solutions. The extension of these results to bounded domains requires a deep understanding of the initial boundary value problem (IBVP) for hyperbolic balance laws. The last part of the thesis provides rigorous estimates on the solution to this IBVP, under precise regularity assumptions. In Chapter 1 we introduce a predator-prey model. A non local and non linear balance law is coupled with a parabolic equation: the former describes the evolution of the predator density, the latter that of prey. The two equations are coupled both through the convective part of the balance law and the source terms. The drift term is a non local function of the prey density. This allows the movement of predators to be directed towards the regions where the concentration of prey is higher. We prove the well-posedness of the system, hence the existence and uniqueness of solution, the continuous dependence from the initial data and various stability estimates. In Chapter 2 we devise an algorithm to compute approximate solutions to the mixed system introduced above. The balance law is solved numerically by a Lax-Friedrichs type method via dimensional splitting, while the parabolic equation is approximated through explicit finite-differences. Both source terms are integrated by means of a second order Runge-Kutta scheme. The key result in Chapter 2 is the convergence of this algorithm. The proof relies on a careful tuning between the parabolic and the hyperbolic methods and exploits the non local nature of the convective part in the balance law. This algorithm has been implemented in a series of Python scripts. Using them, we obtain information about the possible order of convergence and we investigate the qualitative properties of the solutions. Moreover, we observe the formation of a striking pattern: while prey diffuse, predators accumulate on the vertices of a regular lattice. The analytic study of the system above is on all R^n. However, both possible biological applications and numerical integrations suggest that the boundary plays a relevant role. With the aim of studying the mixed hyperbolic-parabolic system in a bounded domain, we noticed that for balance laws known results lack some of the estimates necessary to deal with the coupling. In Chapter 3 we then focus on the IBVP for a general balance law in a bounded domain. We prove the well-posedness of this problem, first with homogeneous boundary condition, exploiting the vanishing viscosity technique and the doubling of variables method, then for the non homogeneous case, mainly thanks to elliptic techniques. We pay particular attention to the regularity assumptions and provide rigorous estimates on the solution.
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Gisclon, Marguerite. "Etude des conditions aux limites pour des systèmes strictement hyperboliques, via l'approximation parabolique." Lyon 1, 1994. http://www.theses.fr/1994LYO10294.

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On etudie les systemes hyperboliques de lois de conservation en dimension un d'espace, en particulier ce qu'il reste d'une condition aux limites de dirichlet, de neumann ou melee, posee pour une perturbation parabolique du systeme, lorsque le cfficient de diffusion tend vers zero. De telles perturbations ont en general un sens physique dans le probleme qu'on etudie, elles modelisent en effet les effets de dissipation. Dans un premier temps, on montre que les limites de deux problemes differents pour l'equation de burgers, que joseph et le floch avaient decrites par des formules complexes, sont en fait egales. Il s'agit d'un probleme scalaire. Pour des systemes, la couche limite qui se forme dans le cas ou le bord n'est pas caracteristique (le cas caracteristique serait analogue au probleme, toujours ouvert, de la convergence de navier-stokes vers euler dans un domaine borne) est decrite. Par une methode d'energie, on demontre la validite du developpement asymptotique sur un intervalle de temps fini, anterieur a la formation des chocs. Dans le cas du p-systeme notamment, la condition aux limites residuelle est explicitee
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Books on the topic "Mixed hyperbolic-parabolic problems"

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Elliptic, hyperbolic and mixed complex equations with parabolic degeneracy. Singapore: World Scientific, 2008.

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Volevich, L. R. Mixed problem for partial differential equations with quasihomogeneous principal part. Providence, R.I: American Mathematical Society, 1996.

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Book chapters on the topic "Mixed hyperbolic-parabolic problems"

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Maurer, Jochen. "A Genuinely Multi-dimensional Scheme for Mixed Hyperbolic-Parabolic Systems." In Hyperbolic Problems: Theory, Numerics, Applications, 713–22. Basel: Birkhäuser Basel, 1999. http://dx.doi.org/10.1007/978-3-0348-8724-3_22.

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"Mixed problem for q-parabolic and q-hyperbolic equation." In Translations of Mathematical Monographs, 199–225. Providence, Rhode Island: American Mathematical Society, 1995. http://dx.doi.org/10.1090/mmono/147/04.

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Conference papers on the topic "Mixed hyperbolic-parabolic problems"

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Chernikov, Dmitry, and Olesya I. Zhupanska. "Fully Coupled Dynamic Analysis of Electro-Magneto-Mechanical Problems in Electrically Conductive Composite Plates." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-37377.

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This paper will present a numerical method for solving fully coupled dynamic problems of the mechanical behavior of electrically conductive composite plates in the presence of an electromagnetic field. The mechanical behavior of electrically conductive materials in the presence of an electromagnetic field is described by the system of nonlinear partial differential equations (PDEs), including equations of motion and Maxwell’s equations that are coupled through the Lorentz ponderomotive force. In the case of thin plates, the system of governing equations is reduced to the two-dimensional (2D) time-dependent nonlinear mixed system of hyperbolic and parabolic PDEs. This paper discusses a numerical solution method for this system, which consists of a sequential application of the Newmark finite difference time integration scheme, spatial (with respect to one coordinate) integration scheme, method of lines (MOL), quasilinearization, and a finite difference spatial integration of the obtained two-point boundary-value problem. The final solution is obtained by the application of the superposition method followed by orthonormalization.
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Hagani, Fouad, M'hamed Boutaous, Ronnie Knikker, Shihe Xin, and Dennis Siginer. "Numerical Modeling of Non-Affine Viscoelastic Fluid Flow Including Viscous Dissipation Through a Square Cross-Section Duct: Heat Transfer Enhancement due to the Inertia and the Elastic Effects." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-23558.

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Abstract Non-isothermal laminar flow of a viscoelastic fluid including viscous dissipation through a square cross–section duct is analyzed. Viscoelastic stresses are described by Giesekus modele orthe Phan-Thien–Tanner model and the solvent shear stress is given by the linear Newtonian constitutive relationship. The flow through the tube is governed by the conservation equations of energy, mass, momentum associated with to one non–affine rheological model mentioned above. The mixed type of the governing system of equations (elliptic–parabolic–hyperbolic) requires coupling between discretisation methods designed for elliptic–type equations and techniques adapted to transport equations. To allow appropriate spatial discretisation of the convection terms, the system is rewritten in a quasi-linear first-order and homogeneous form without the continuity and energy equations. With the rheological models of the Giesekus type, the conformation tensor is by definition symmetrical and positive-definite, with the PTT model the hyperbolicity condition is subject to restrictions related to the rheological parameters. Based on this hyperbolicity condition, the contribution of the hyperbolic part is approximated by applying the characteristic method to extract pure advection terms which are then discretized by high ordre schemes WENO and HOUC. The algorithm thus developed makes it possible, to avoid the problems of instabilities related to the high Weissenberg number without the use of any stabilization method. Finally, a Nusselt number analysis is given as a function of inertia, elasticity, viscous dissipation, for constant solvent viscosity ratio and constant material and rheological parameters.
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Dildabek, Gulnar. "On a new nonlocal boundary value problem for an equation of the mixed parabolic-hyperbolic type." In APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’16): Proceedings of the 42nd International Conference on Applications of Mathematics in Engineering and Economics. Author(s), 2016. http://dx.doi.org/10.1063/1.4968471.

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Kalmenov, Tynysbek Sh, and Makhmud Sadybekov. "On a problem of the Frankl type for an equation of the mixed parabolic-hyperbolic type." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2016). Author(s), 2016. http://dx.doi.org/10.1063/1.4959615.

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"Initial boundary value problem for a three-dimensional homogeneous equation of mixed parabolic-hyperbolic type with power degeneration." In Уфимская осенняя математическая школа - 2022. 2 часть. Baskir State University, 2022. http://dx.doi.org/10.33184/mnkuomsh2t-2022-09-28.93.

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Dildabek, G., M. A. Sadybekov, and M. B. Saprygina. "On a Volterra property of an problem of the Frankl type for an equation of the mixed parabolic–hyperbolic type." In PROCEEDINGS OF THE 43RD INTERNATIONAL CONFERENCE APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS: (AMEE’17). Author(s), 2017. http://dx.doi.org/10.1063/1.5013971.

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