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Journal articles on the topic 'Initial-boundary value problem for balance laws'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Gugat, Martin, and Stefan Ulbrich. "Lipschitz solutions of initial boundary value problems for balance laws." Mathematical Models and Methods in Applied Sciences 28, no. 05 (2018): 921–51. http://dx.doi.org/10.1142/s0218202518500240.

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The flow of gas through networks of pipes can be modeled by the isothermal Euler equations and algebraic node conditions that model the flow through the vertices of the network graph. This motivates our analysis of the well-posedness of a coupled system of [Formula: see text] conservation laws on a network. We consider initial data and control functions that are Lipschitz continuous and compatible with the node and boundary conditions. We show the existence of semi-global Lipschitz continuous solutions of the initial boundary value problem on a network. The construction of the solution is base

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2

Chou, Shih-Wei, John M. Hong, and Ying-Chin Su. "The initial-boundary value problem of hyperbolic integro-differential systems of nonlinear balance laws." Nonlinear Analysis: Theory, Methods & Applications 75, no. 15 (2012): 5933–60. http://dx.doi.org/10.1016/j.na.2012.06.006.

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3

Hong, John M., and Ying-Chin Su. "Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws." Nonlinear Analysis: Theory, Methods & Applications 72, no. 2 (2010): 635–50. http://dx.doi.org/10.1016/j.na.2009.07.003.

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4

Rossi, Elena. "Well-posedness of general 1D initial boundary value problems for scalar balance laws." Discrete & Continuous Dynamical Systems - A 39, no. 6 (2019): 3577–608. http://dx.doi.org/10.3934/dcds.2019147.

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5

Rossi, Elena. "Definitions of solutions to the IBVP for multi-dimensional scalar balance laws." Journal of Hyperbolic Differential Equations 15, no. 02 (2018): 349–74. http://dx.doi.org/10.1142/s0219891618500133.

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We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.

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6

Li, Huicong, and Kun Zhao. "Initial–boundary value problems for a system of hyperbolic balance laws arising from chemotaxis." Journal of Differential Equations 258, no. 2 (2015): 302–38. http://dx.doi.org/10.1016/j.jde.2014.09.014.

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7

Galeş, C. "A Mixture Theory for Micropolar Thermoelastic Solids." Mathematical Problems in Engineering 2007 (2007): 1–21. http://dx.doi.org/10.1155/2007/90672.

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We derive a nonlinear theory of heat-conducting micropolar mixtures in Lagrangian description. The kinematics, balance laws, and constitutive equations are examined and utilized to develop a nonlinear theory for binary mixtures of micropolar thermoelastic solids. The initial boundary value problem is formulated. Then, the theory is linearized and a uniqueness result is established.

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8

Feng, Zefu. "Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis: Global Dynamics and Diffusivity Limit." Annals of Applied Mathematics 37, no. 1 (2021): 61–110. http://dx.doi.org/10.4208/aam.oa-2020-0004.

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Passarella, Francesca, and Vincenzo Tibullo. "Uniqueness of Solutions in Thermopiezoelectricity of Nonsimple Materials." Entropy 24, no. 9 (2022): 1229. http://dx.doi.org/10.3390/e24091229.

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This article presents the theory of thermopiezoelectricity in which the second displacement gradient and the second electric potential gradient are included in the set of independent constitutive variables. This is achieved by using the entropy production inequality proposed by Green and Laws. At first, appropriate thermodynamic restrictions and constitutive equations are obtained, using the well-established Coleman and Noll procedure. Then, the balance equations and the constitutive equations of linear theory are derived, and the mixed initial-boundary value problem is set. For this problem a

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10

Hong, John Meng-Kai, and Reyna Marsya Quita. "Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times." Tamkang Journal of Mathematics 48, no. 1 (2017): 73–94. http://dx.doi.org/10.5556/j.tkjm.48.2017.2274.

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In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solu

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11

Wei, Jinlong, Bin Liu, Rongrong Tian, and Liang Ding. "Stochastic Entropy Solutions for Stochastic Scalar Balance Laws." Entropy 21, no. 12 (2019): 1142. http://dx.doi.org/10.3390/e21121142.

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We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.

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12

COLLI, PIERLUIGI. "GLOBAL SOLUTION TO A MODEL FOR CELL MORPHOGENESIS BY CALCIUM-REGULATED STRAIN FIELDS." Mathematical Models and Methods in Applied Sciences 03, no. 04 (1993): 497–512. http://dx.doi.org/10.1142/s0218202593000266.

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In this paper we prove an existence and uniqueness result for a time-dependent problem related to a mechanochemical model of cellular morphogenesis, based on calcium ion regulation of the viscoelastic properties of the cellular cortex. Calcium input and output processes, as well as diffusive effects, are accounted in the description of the cell dynamics. The resulting system of nonlinear partial differential equations consists of the balance laws for the concentrations of free calcium and bound (to specific macro-molecules) calcium, coupled with the equilibrium equations for the displacements.

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13

Swain, Digendranath, and Anurag Gupta. "Biological growth in bodies with incoherent interfaces." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2209 (2018): 20170716. http://dx.doi.org/10.1098/rspa.2017.0716.

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A general theory of thermodynamically consistent biomechanical–biochemical growth in a body, considering mass addition in the bulk and at an incoherent interface, is developed. The incoherency arises due to incompatibility of growth and elastic distortion tensors at the interface. The incoherent interface therefore acts as an additional source of internal stress besides allowing for rich growth kinematics. All the biochemicals in the model are essentially represented in terms of nutrient concentration fields, in the bulk and at the interface. A nutrient balance law is postulated which, combine

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14

Hayes, Brian, and Michael Shearer. "Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 4 (1999): 733–54. http://dx.doi.org/10.1017/s0308210500013111.

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The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters,

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15

LEFLOCH, PHILIPPE G. "HYPERBOLIC BALANCE LAWS WITH ENTROPY ON A CURVED SPACETIME: THE WEAK–STRONG UNIQUENESS THEORY." Journal of Hyperbolic Differential Equations 10, no. 04 (2013): 773–98. http://dx.doi.org/10.1142/s0219891613500288.

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We are interested in nonlinear partial differential equations of hyperbolic type, written as first-order balance laws posed on a curved spacetime, whose unknown is typically a section of the bundle of future-oriented, timelike vectors. We are especially interested in dealing with systems and solutions with low regularity and, to this end, we introduce here the notion of "strictly convex entropy field". We then prove that a system of balance laws endowed with such an entropy admits a symmetrization which, however, may be "degenerate" if the entropy is not uniformly convex or if the constitutive

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16

Kuznetsov, Ivan, and Sergey Sazhenkov. "Singular limits of the quasi-linear Kolmogorov-type equation with a source term." Journal of Hyperbolic Differential Equations 18, no. 04 (2021): 789–856. http://dx.doi.org/10.1142/s0219891621500247.

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Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic–parabolic (ultra-parabolic) equation with a smooth source term is established. In addition, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is investigated and the formal limit is rigorous

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17

Meyer, Fabian, Christian Rohde, and Jan Giesselmann. "A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method." IMA Journal of Numerical Analysis 40, no. 2 (2019): 1094–121. http://dx.doi.org/10.1093/imanum/drz004.

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Abstract In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization of the stochastic Galerkin system a Runge–Kutta discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016, Hyperbolic Conservation Laws in Continuum Phy

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18

Belov, P. A., and S. A. Lurie. "VARIATIONAL FORMULATION OF GRADIENT IRREVERSIBLE THERMODYNAMICS." PNRPU Mechanics Bulletin, no. 5 (December 6, 2023): 36–44. http://dx.doi.org/10.15593/perm.mech/2023.5.04.

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This work proposes the elaboration of the variational principle of L.I. Sedov for modeling dissipative processes. The formulated variational principle makes it possible to propose the dissipative models using the known model of a reversible process (the known Lagrangian), adding the required number of dissipation channels. Dissipation channels are non-integrable variational forms that are linear in the variations of the ar-guments. The arguments of the dissipation channels are the generalized variables of the corresponding bilinear terms of the Lagrangian. Variational models of heat transfer p

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19

Kan, Pui Tak, Marcelo M. Santos, and Zhouping Xin. "Initial Boundary Value Problem for Conservation Laws." Communications in Mathematical Physics 186, no. 3 (1997): 701–30. http://dx.doi.org/10.1007/s002200050125.

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20

Eremin, A. V. "Modeling methodology of locally non-equilibrium heat conductivity processes." Vestnik IGEU, no. 2 (2020): 65–71. http://dx.doi.org/10.17588/2072-2672.2020.2.065-071.

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With the development of laser technologies and the ability to carry out processing steps under extreme conditions (ul-trahigh temperatures, pressures and their gradients), the interest in studying the processes that occur under locally non-equilibrium conditions has grown significantly. The key directions for the description of locally non-equilibrium pro-cesses include thermodynamic, kinetic and phenomenological ones. The locally non-equilibrium transfer equations can also be derived from the Boltzmann equation by using the theory of random walks and molecular-kinetic methods. It should be no

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21

CHRISTOV, C. I., and M. G. VELARDE. "INELASTIC INTERACTION OF BOUSSINESQ SOLITONS." International Journal of Bifurcation and Chaos 04, no. 05 (1994): 1095–112. http://dx.doi.org/10.1142/s0218127494000800.

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Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussines

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22

FRANKOWSKA, HÉLÈNE. "ON LeFLOCH'S SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR CONSERVATION LAWS." Journal of Hyperbolic Differential Equations 07, no. 03 (2010): 503–43. http://dx.doi.org/10.1142/s0219891610002219.

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We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients Vx satisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the init

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23

Lin, Gui-cheng, and Wan-cheng Sheng. "Godunov’s method for initial-boundary value problem of scalar conservation laws." Journal of Shanghai University (English Edition) 12, no. 4 (2008): 298–301. http://dx.doi.org/10.1007/s11741-008-0404-4.

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24

Mitrović, Darko, and Andrej Novak. "Transport-collapse scheme for scalar conservation laws: initial-boundary value problem." Communications in Mathematical Sciences 15, no. 4 (2017): 1055–71. http://dx.doi.org/10.4310/cms.2017.v15.n4.a7.

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25

Xin, Zhouping, and Wen-Qing Xu. "Initial-boundary value problem to systems of conservation laws with relaxation." Quarterly of Applied Mathematics 60, no. 2 (2002): 251–81. http://dx.doi.org/10.1090/qam/1900493.

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26

Guarguaglini, F. R., and A. Terracina. "A BGK approximation to nonlinear parabolic initial‐boundary value problems." Asymptotic Analysis 28, no. 1 (2001): 75–89. https://doi.org/10.3233/asy-2001-463.

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We present a relaxation semilinear system of conservation laws with source which approximates nonlinear parabolic equations with initial and boundary conditions. The system can be interpreted as a BGK (Bhatnagar, Gross, Krook) model with a finite number of velocities. We prove the well‐posedness of the model, a priori estimates and we obtain the convergence towards the solution of the parabolic problem. Moreover we prove a similar result for a weakly degenerate problem.

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27

Chen, Huazhou, and Tao Pan. "Two-Dimension Riemann Initial-Boundary Value Problem of Scalar Conservation Laws with Curved Boundary." Boundary Value Problems 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/138396.

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28

Teng, Zhen-huan. "Exact boundary conditions for the initial value problem of convex conservation laws." Journal of Computational Physics 229, no. 10 (2010): 3792–801. http://dx.doi.org/10.1016/j.jcp.2010.01.028.

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29

Li, Lingxiao, Mingliang Wang, and Jinliang Zhang. "The Solutions of Initial (-Boundary) Value Problems for Sharma-Tasso-Olver Equation." Mathematics 10, no. 3 (2022): 441. http://dx.doi.org/10.3390/math10030441.

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A nonlinear transformation from the solution of linear KdV equation to the solution of Sharma-Tasso-Olver (STO) equation is derived out by using simplified homogeneous balance (SHB) method. According to the nonlinear transformation derived here, the exact explicit solution of initial (-boundary) value problem for STO equation can be constructed in terms of the solution of initial (-boundary) value problem for the linear KdV equation. The exact solution of the latter problem is obtained by using Fourier transformation.

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30

Venherskyi, Petro. "CONSTRUCTION AND RESEARCH OF FULL BALANCE ENERGY OF VARIATIONAL PROBLEM MOTION SURFACE AND GROUNDWATER FLOWS." EUREKA: Physics and Engineering 1 (January 31, 2017): 45–52. http://dx.doi.org/10.21303/2461-4262.2017.00270.

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Based on the laws of conservation of mass and momentum the basic equations of motion with unknown quantities velocity and piezometric pressure are written. These equations are supplemented with boundary and initial conditions describing the motion of compatible flows. Based on the laws of motion continuum, received conditions contact on the common border interaction of surface and groundwater flows. Variational problems formulated compatible flow. Energy norms of basic components of variational problem are analyzed. Correctness of constructing variational problem arising from construction of t

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31

DU, QIANG, MAX GUNZBURGER, R. B. LEHOUCQ, and KUN ZHOU. "A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS." Mathematical Models and Methods in Applied Sciences 23, no. 03 (2013): 493–540. http://dx.doi.org/10.1142/s0218202512500546.

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A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The operators of the nonlocal calculus are used to define volu

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32

Ton, Bui An. "Time-dependent Stokes equations with measure data." Abstract and Applied Analysis 2003, no. 17 (2003): 953–73. http://dx.doi.org/10.1155/s1085337503308012.

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We establish the existence of a unique solution of an initial boundary value problem for the nonstationary Stokes equations in a bounded fixed cylindrical domain with measure data. Feedback laws yield the source and its intensity from the partial measurements of the solution in a subdomain.

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33

Dong, Shijie, and Philippe G. LeFloch. "Convergence of the finite volume method on a Schwarzschild background." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (2019): 1459–76. http://dx.doi.org/10.1051/m2an/2019037.

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We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence

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34

Yao, Ai-di, and Wan-cheng Sheng. "Initial-boundary value problem of nonlinear hyperbolic system for conservation laws with delta-shock waves." Journal of Shanghai University (English Edition) 12, no. 4 (2008): 306–10. http://dx.doi.org/10.1007/s11741-008-0406-3.

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35

A. V., Sedelnikov, Nikolaeva A. S., Serdakova V. V., and Evtushenko M. A. "Approximate Solution of Initial Boundary Value Problem of One-Dimensional Heat Conduction for the Thermal Shock of Thin Plate." WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS 19 (December 31, 2024): 200–205. https://doi.org/10.37394/232011.2024.19.22.

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The article represents a new approximate analytical solution of the initial boundary value problem of one-dimensional heat conduction with boundary conditions of the third kind. It describes the temperature field of a homogeneous thin plate under thermal shock more accurately than previous solutions. The obtained solution was determined by the ANSYS mathematical package. The results of the analysis can be used to construct control laws for a small spacecraft taking into account the thermal shock of its solar panels.

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36

Ryazhskih, A. V., А. А. Khvostov, Е. А. Soboleva, and V. I. Ryazhskih. "THE TEMPERATURE PATTERN OF A HOMOGENEOUS SQUARE AREA WITH ADJACENT SIDES MOVING WITHOUT ACCELERATION UNDER BOUNDARY CONDITIONS OF THE FIRST KIND." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 15, no. 1 (2023): 55–62. http://dx.doi.org/10.14529/mmph230106.

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A square area with homogeneous thermal and physical characteristics, deformed preserving 2-D similarity, is investigated. At the initial moment of time, two adjacent sides start moving respectively towards the abscissa and ordinate axes with constant speed while remaining equidistant to the other two adjacent sides (the fixed and moving sides are kept at different constant temperatures). A nonlinear initial boundary value problem with boundary conditions of the first kind and special coordinates immobilizes the moving boundary of the area into a fixed one with the corresponding transformation

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37

Saxena, Prashant. "On the General Governing Equations of Electromagnetic Acoustic Transducers." Archive of Mechanical Engineering 60, no. 2 (2013): 231–46. http://dx.doi.org/10.2478/meceng-2013-0015.

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In this paper, we present the general governing equations of electrodynamics and continuum mechanics that need to be considered while mathematically modelling the behaviour of electromagnetic acoustic transducers (EMATs). We consider the existence of finite deformations for soft materials and the possibility of electric currents, temperature gradients, and internal heat generation due to dissipation. Starting with Maxwell’s equations of electromagnetism and balance laws of nonlinear elasticity, we present the governing equations and boundary conditions in incremental form in order to solve wav

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38

BONETTI, ELENA, PIERLUIGI COLLI, and MICHEL FREMOND. "A PHASE FIELD MODEL WITH THERMAL MEMORY GOVERNED BY THE ENTROPY BALANCE." Mathematical Models and Methods in Applied Sciences 13, no. 11 (2003): 1565–88. http://dx.doi.org/10.1142/s0218202503003033.

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We introduce a thermomechanical model describing dissipative phase transitions with thermal memory in terms of the entropy balance and the principle of virtual power written for microscopic movements. The thermodynamical consistence of this model is verified and existence of solutions is proved for a related initial and boundary value problem.

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39

Beybalaev, Vetlugin Dzhabrailovich, Abutrab Aleksandrovich Aliverdiev, and Jordan Hristov. "Transient Heat Conduction in a Semi-Infinite Domain with a Memory Effect: Analytical Solutions with a Robin Boundary Condition." Fractal and Fractional 7, no. 10 (2023): 770. http://dx.doi.org/10.3390/fractalfract7100770.

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The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton’s law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically.

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40

Li, Tatsien, and Lei Yu. "Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 2 (2018): 793–810. http://dx.doi.org/10.1051/cocv/2017072.

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In this paper, we study the local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws with characteristics of constant multiplicity. We prove the two-sided boundary controllability, the one-sided boundary controllability and the two-sided boundary controllability with fewer controls, by applying the strategy used in [T. Li and L. Yu, J. Math. Pures et Appl. 107 (2017) 1–40; L. Yu, Chinese Ann. Math., Ser. B (To appear)]. Our constructive method is based on the well-posedness of semi-global solutions constructed by the l

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41

Maugin, Ge´rard A. "Material Forces: Concepts and Applications." Applied Mechanics Reviews 48, no. 5 (1995): 213–45. http://dx.doi.org/10.1115/1.3005101.

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The unifying notion of material force which gathers under one vision all types of driving “forces” on defects and smooth or abrupt inhomogeneities in fracture, defect mechanics, elastodynamics (localized solutions) and allied theories such as in electroelasticity, magnetoelasticity, and the propagation of phase transition fronts, is reviewed together with its many faceted applications. The presentation clearly distinguishes between the role played by local physical balance laws in the solution of boundary-value problems and that played by global material balance laws in obtaining the expressio

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42

Kassab, Ghassan S. "Biomechanics of the cardiovascular system: the aorta as an illustratory example." Journal of The Royal Society Interface 3, no. 11 (2006): 719–40. http://dx.doi.org/10.1098/rsif.2006.0138.

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Biomechanics relates the function of a physiological system to its structure. The objective of biomechanics is to deduce the function of a system from its geometry, material properties and boundary conditions based on the balance laws of mechanics (e.g. conservation of mass, momentum and energy). In the present review, we shall outline the general approach of biomechanics. As this is an enormously broad field, we shall consider a detailed biomechanical analysis of the aorta as an illustration. Specifically, we will consider the geometry and material properties of the aorta in conjunction with

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BANK, MIRIAM, and MATANIA BEN-ARTZI. "SCALAR CONSERVATION LAWS ON A HALF-LINE: A PARABOLIC APPROACH." Journal of Hyperbolic Differential Equations 07, no. 01 (2010): 165–89. http://dx.doi.org/10.1142/s0219891610002086.

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The initial-boundary value problem for the (viscous) nonlinear scalar conservation law is considered, [Formula: see text] The flux f(ξ) ∈ C2(ℝ) is assumed to be convex (but not strictly convex, i.e. f″(ξ)≥ 0). It is shown that a unique limit u = lim ∊ → 0 u∊ exists. The classical duality method is used to prove uniqueness. To this end parabolic estimates for both the direct and dual solutions are obtained. In particular, no use is made of the Kružkov entropy considerations.

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44

Milišić, Vuk. "Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws." Proceedings of the American Mathematical Society 131, no. 6 (2003): 1727–37. http://dx.doi.org/10.1090/s0002-9939-03-06961-2.

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45

De Filippis, Cristiana, and Paola Goatin. "The initial–boundary value problem for general non-local scalar conservation laws in one space dimension." Nonlinear Analysis 161 (September 2017): 131–56. http://dx.doi.org/10.1016/j.na.2017.05.017.

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46

Pan, Tao, and Hongxoa Liu. "Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws." Applied Mathematics Letters 15, no. 6 (2002): 727–34. http://dx.doi.org/10.1016/s0893-9659(02)00034-4.

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Matus, P., and S. Lemeshevsky. "Stability and Monotonicity of Difference Schemes for Nonlinear Scalar Conservation Laws and Multidimensional Quasi-linear Parabolic Equations." Computational Methods in Applied Mathematics 9, no. 3 (2009): 253–80. http://dx.doi.org/10.2478/cmam-2009-0016.

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Abstract We have proved the difference analogue of a Bihari-type inequality. Using this inequality, we study the stability in C and monotonicity of the difference schemes approximating initial-boundary value problems for nonlinear conservation laws and multi-dimensional parabolic equations. It has been shown that in the nonlinear case the stability and monotonicity are determined not only by the behavior of the approximate solution but also by its difference derivatives appearing in the nonlinear terms of the equation. The stability estimates are obtained without any assumptions about the prop

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BOURDARIAS, C., M. GISCLON, and S. JUNCA. "BLOW UP AT THE HYPERBOLIC BOUNDARY FOR A 2 × 2 SYSTEM ARISING FROM CHEMICAL ENGINEERING." Journal of Hyperbolic Differential Equations 07, no. 02 (2010): 297–316. http://dx.doi.org/10.1142/s0219891610002116.

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We consider an initial boundary value problem for a 2 × 2 system of conservation laws modeling heatless adsorption of a gaseous mixture with two species and instantaneous exchange kinetics, close to the system of chromatography. In this model the velocity is not constant because the sorption effect is taken into account. Exchanging the roles of the x, t variables we obtain a strictly hyperbolic system with a zero eigenvalue. Our aim is to construct a solution with a velocity which blows up at the corresponding characteristic "hyperbolic boundary" {t = 0}.

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CHRISTOFOROU, CLEOPATRA, and LAURA V. SPINOLO. "A UNIQUENESS CRITERION FOR VISCOUS LIMITS OF BOUNDARY RIEMANN PROBLEMS." Journal of Hyperbolic Differential Equations 08, no. 03 (2011): 507–44. http://dx.doi.org/10.1142/s0219891611002482.

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We deal with the initial boundary value problem for systems of conservation laws in one space dimension and we focus on the boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. In this paper, we establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that, under reasonable assumptions, the self-similar second-order approximation [Formula: see text] and the classical viscous approximation [Formula: see text] provide the same limit as ε → 0+. Our a

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Kolenda, Z. S., and J. S. Szmyd. "Entropy generation minimization in transient heat conduction processes PART II – Transient heat conduction in solids." Bulletin of the Polish Academy of Sciences Technical Sciences 62, no. 4 (2014): 883–87. http://dx.doi.org/10.2478/bpasts-2014-0097.

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Abstract Formulation and solution of the initial boundary-value problem of heat conduction in solids have been presented when an entropy generation minimization principle is imposed as the arbitrary constraint. Using an entropy balance equation and the Euler-Lagrange variational approach a new form of the heat conduction equation (non-linear partial difference equation) is derived.

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