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

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Published: 9 March 2023

Last updated: 10 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Sidorov, S. N. "Nonlocal problems for an equation of mixed parabolic-hyperbolic type with power degeneration." Russian Mathematics 59, no. 12 (November 7, 2015): 46–55. http://dx.doi.org/10.3103/s1066369x15120051.

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12

Berdyshev, A. S., E. T. Karimov, and N. Akhtaeva. "Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation." International Journal of Differential Equations 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/268465.

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Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.

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13

M Cotta, R., and J. E. V. Gerk. "MIXED FINITE-DIFFERENCE/INTEGRAL TRANSFORM APPROACH FOR PARABOLIC-HYPERBOLIC PROBLEMS IN TRANSIENT FORCED CONVECTION." Numerical Heat Transfer, Part B: Fundamentals 25, no. 4 (June 1994): 433–48. http://dx.doi.org/10.1080/10407799408955929.

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14

ELLING, VOLKER, and TAI-PING LIU. "THE ELLIPTICITY PRINCIPLE FOR SELF-SIMILAR POTENTIAL FLOWS." Journal of Hyperbolic Differential Equations 02, no. 04 (December 2005): 909–17. http://dx.doi.org/10.1142/s0219891605000646.

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We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear second-order PDE of mixed type which is hyperbolic at infinity (if the velocity is globally bounded). The type in each point is determined by the local pseudo-Mach number L, with L < 1 (respectively, L > 1) corresponding to elliptic (respectively, hyperbolic) regions. We prove an ellipticity principle: the interior of a parabolic-elliptic region of a sufficiently smooth solution must be elliptic; in fact L must be bounded above away from 1 by a domain-dependent function. In particular there are no open parabolic regions. We also discuss the case of slip boundary conditions at straight solid walls.

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15

Sabitov, K. B. "Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation." Mathematical Notes 102, no. 3-4 (September 2017): 378–95. http://dx.doi.org/10.1134/s0001434617090085.

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16

Yuldashev, T. K., B. I. Islomov, and U. Sh Ubaydullaev. "On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 101, no. 1 (March 30, 2021): 127–37. http://dx.doi.org/10.31489/2021m1/127-137.

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This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

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17

BOUCHUT, FRANÇOIS, and HERMANO FRID. "FINITE DIFFERENCE SCHEMES WITH CROSS DERIVATIVES CORRECTORS FOR MULTIDIMENSIONAL PARABOLIC SYSTEMS." Journal of Hyperbolic Differential Equations 03, no. 01 (March 2006): 27–52. http://dx.doi.org/10.1142/s0219891606000719.

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We propose finite difference schemes for multidimensional quasilinear parabolic systems whose main feature is the introduction of correctors which control the second-order terms with mixed derivatives. We show that with these correctors the schemes inherit physically relevant properties present at the continuous level, such as the existence of invariant domains and/or the nonincrease of the total amount of entropy. The analysis is performed with some general tools that could be used also in the analysis of finite volume methods based on flux vector splitting for first-order hyperbolic problems on unstructured meshes. Applications to the compressible Navier–Stokes system are given.

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18

Sidorov, Stanislav Nikolaevich. "Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides." Ufimskii Matematicheskii Zhurnal 11, no. 1 (2019): 75–89. http://dx.doi.org/10.13108/2019-11-1-75.

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19

Berdyshev, A. S. "The Volterra Property of Some Problems with the Bitsadze-Samarskii-Type Conditions for a Mixed Parabolic-Hyperbolic Equation." Siberian Mathematical Journal 46, no. 3 (May 2005): 386–95. http://dx.doi.org/10.1007/s11202-005-0041-y.

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20

Eleev, V. A., and S. K. Kumykova. "On some boundary-value problems with a shift on characteristics for a mixed equation of hyperbolic-parabolic type." Ukrainian Mathematical Journal 52, no. 5 (May 2000): 809–20. http://dx.doi.org/10.1007/bf02487291.

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21

Fortin, Michel, and Abdellatif Serghini Mounim. "Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case." Journal of Mathematics Research 9, no. 1 (January 9, 2017): 68. http://dx.doi.org/10.5539/jmr.v9n1p68.

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We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

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22

Alakashi, Abobaker Mohammed, and Bambang Basuno. "Comparison between Cell-Centered Schemes Computer Code and Fluent Software for a Transonic Flow Pass through an Array of Turbine Stator Blades." Applied Mechanics and Materials 437 (October 2013): 271–74. http://dx.doi.org/10.4028/www.scientific.net/amm.437.271.

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The Finite Volume Method (FVM) is a discretization method which is well suited for the numerical simulation of various types (elliptic, parabolic or hyperbolic, for instance) of conservation laws; it has been extensively used in several engineering fields. The Finite volume method uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations [. the developed computer code based Cell-centered scheme and Fluent software had been used to investigate the inviscid Transonic Flow Pass Through an array of Turbine Stator Blades. The governing equation of fluid motion of the flow problem in hand is assumed governed by the compressible Euler Equation. Basically this equation behave as a mixed type of partial differential equation elliptic and hyperbolic type of partial differential equation. If the local Mach number is less than one, the governing equation will behave as elliptic type of differential equation while if the Mach number is greater than one it will behave as hyperbolic type of differential equation. To eliminate the presence a mixed type behavior, the governing equation of fluid motion are treated as the governing equation of unsteady flow although the problem in hand is steady flow problems. Presenting the Euler equation in their unsteady form makes the equation becomes hyperbolic with respect to time. There are various Finite Volume Methods can used for solving hyperbolic type of equation, such as Cell-centered scheme [, Roe Upwind Scheme [ and TVD Scheme [. The present work use a cell centered scheme applied to the case of flow pass through an array of turbine stator blades. The comparison carried out with the result provided by Fluent Software for three different value of back pressure. The developed computer code shows the result close to the Fluent software although the Fluent software use a Time Averaged Navier stokes equation as its governing equation of fluid motion.

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23

GUSTAFSSON, BERTIL. "Analysis and Methods in Fluid Mechanics." International Journal of Modern Physics C 02, no. 01 (March 1991): 75–85. http://dx.doi.org/10.1142/s0129183191000093.

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When constructing numerical methods for partial differential equations, it is important to have a thorough understanding of the continuous model and the characteristic properties of its solutions. We shall present methods of analysis for determining well-posedness of hyperbolic and mixed hyperbolic-parabolic équations which are applicable to the time-dependent Euler and Navier-Stokes equations. We shall then discuss difference- and finite volume methods and the construction of grids. The geometry of realistic problems is usually such that it is almost impossible to construct one structured grid. One way to overcome this difficulty is to use overlapping grids, where each domain has a structured grid. We discuss stability and accuracy of difference methods applied on such grids. Many problems in physics and engineering are defined in boundary domains, and artificial boundaries are introduced for computational reasons. In some cases one can construct accurate boundary conditions at these open boundaries. We shall indicate how this can be achieved, but we will also point out certain cases where accurate solutions are impossible to be obtained on limited domains. Finally some comments will be given on the difficulties arising when almost incompressible flow is computed. This corresponds to small Mach-numbers, and extra care must be taken when designing numerical methods. The theory will be complemented by numerical experiments for various flow problems in two space dimensions.

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24

Shishkina, OA, and I. M. Indrupskiy. "Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs." Journal of Physics: Conference Series 2090, no. 1 (November 1, 2021): 012139. http://dx.doi.org/10.1088/1742-6596/2090/1/012139.

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Abstract Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

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25

GRAY, J. M. N. T., and C. ANCEY. "Multi-component particle-size segregation in shallow granular avalanches." Journal of Fluid Mechanics 678 (June 1, 2011): 535–88. http://dx.doi.org/10.1017/jfm.2011.138.

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A general continuum theory for particle-size segregation and diffusive remixing in polydisperse granular avalanches is formulated using mixture theory. Comparisons are drawn to existing segregation theories for bi-disperse mixtures and the case of a ternary mixture of large, medium and small particles is investigated. In this case, the general theory reduces to a system of two coupled parabolic segregation–remixing equations, which have a single diffusion coefficient and three parameters which control the segregation rates between each pair of constituents. Considerable insight into many problems where the effect of diffusive remixing is small is provided by the non-diffusive case. Here the equations reduce to a system of two first-order conservation laws, whose wave speeds are real for a very wide class of segregation parameters. In this regime, the system is guaranteed to be non-strictly hyperbolic for all admissible concentrations. If the segregation rates do not increase monotonically with the grain-size ratio, it is possible to enter another region of parameter space, where the equations may either be hyperbolic or elliptic, depending on the segregation rates and the local particle concentrations. Even if the solution is initially hyperbolic everywhere, regions of ellipticity may develop during the evolution of the problem. Such regions in a time-dependent problem necessarily lead to short wavelength Hadamard instability and ill-posedness. A linear stability analysis is used to show that the diffusive remixing terms are sufficient to regularize the theory and prevent unbounded growth rates at high wave numbers. Numerical solutions for the time-dependent segregation of an initially almost homogeneously mixed state are performed using a standard Galerkin finite element method. The diffuse solutions may be linearly stable or unstable, depending on the initial concentrations. In the linearly unstable region, ‘sawtooth’ concentration stripes form that trap and focus the medium-sized grains. The large and small particles still percolate through the avalanche and separate out at the surface and base of the flow due to the no-flux boundary conditions. As these regions grow, the unstable striped region is annihilated. The theory is used to investigate inverse distribution grading and reverse coarse-tail grading in multi-component mixtures. These terms are commonly used by geologists to describe particle-size distributions in which either the whole grain-size population coarsens upwards, or just the coarsest clasts are inversely graded and a fine-grained matrix is found everywhere. An exact solution is constructed for the steady segregation of a ternary mixture as it flows down an inclined slope from an initially homogeneously mixed inflow. It shows that for distribution grading, the particles segregate out into three inversely graded sharply segregated layers sufficiently far downstream, with the largest particles on top, the fines at the bottom and the medium-sized grains sandwiched in between. The heights of the layers are strongly influenced by the downstream velocity profile, with layers becoming thinner in the faster moving near-surface regions of the avalanche, and thicker in the slowly moving basal layers, for the same mass flux. Conditions for the existence of the solution are discussed and a simple and useful upper bound is derived for the distance at which all the particles completely segregate. When the effects of diffusive remixing are included, the sharp concentration discontinuities are smoothed out, but the simple shock solutions capture many features of the evolving size distribution for typical diffusive remixing rates. The theory is also used to construct a simple model for reverse coarse-tail grading, in which the fine-grained material does not segregate. The numerical method is used to calculate diffuse solutions for a ternary mixture and a sharply segregated shock solution is derived that looks similar to the segregation of a bi-disperse mixture of large and medium grains. The presence of the fine-grained material, however, prevents high concentrations of large or medium particles being achieved and there is a significant lengthening of the segregation distance.

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26

Feng, Pengbin, and Erkinjon T. Karimov. "Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations." Journal of Inverse and Ill-posed Problems 23, no. 4 (January 1, 2015). http://dx.doi.org/10.1515/jiip-2014-0022.

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AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

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27

Sabitov, Kamil Basirovich, and Stanislav Nikolaevich Sidorov. "Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time." Journal of Inverse and Ill-posed Problems, February 28, 2023. http://dx.doi.org/10.1515/jiip-2020-0154.

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Abstract For the equation of a mixed parabolic-hyperbolic type with a power degeneration on the type change line, the inverse problems to determine the time-dependent factors of right-hand sides are studied. Based on the formula for solving a direct problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Using the theory of integral equations, the corresponding theorems for the existence and uniqueness of the solutions of the stated inverse problems are proved and the explicit formulas for the solution have been given.

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28

"95/01007 Mixed finite-difference/integral transform approach for parabolic-hyperbolic problems in transient forced convection." Fuel and Energy Abstracts 36, no. 1 (January 1995): 59. http://dx.doi.org/10.1016/0140-6701(95)96264-6.

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29

Gassner, Gregor J., and Andrew R. Winters. "A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?" Frontiers in Physics 8 (January 29, 2021). http://dx.doi.org/10.3389/fphy.2020.500690.

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In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

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