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

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Published: 9 March 2023

Last updated: 10 March 2023

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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 (May 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 based upon a fixed point iteration along the characteristic curves. The solutions of the initial boundary value problem on arbitrary networks satisfy a maximum principle in terms of the Riemann invariants that states that the maximum of the absolute values is attained for the initial or the boundary data.

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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 (October 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 (January 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 (June 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 (January 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 (June 2021): 61–110. http://dx.doi.org/10.4208/aam.oa-2020-0004.

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9

Passarella, Francesca, and Vincenzo Tibullo. "Uniqueness of Solutions in Thermopiezoelectricity of Nonsimple Materials." Entropy 24, no. 9 (September 1, 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 uniqueness result is established. Next, the basic equations for the isotropic case are derived. Finally, a set of inequalities is obtained for the constant constitutive coefficients of the isotropic case that, on the basis on the previous theorem, ensure the uniqueness of the solution of the mixed initial-boundary value problem.

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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 (March 30, 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 solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.

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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 (November 22, 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 (August 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. It is shown that the initial-boundary value problem has only one solution which satisfies suitable boundedness estimates regarding calcium concentrations.

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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 (January 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, combined with mechanical balances and kinetic laws, yields an initial-boundary-value problem coupling the evolution of bulk and interfacial growth, on the one hand, and the evolution of growth and nutrient concentration on the other. The problem is solved, and discussed in detail, for two distinct examples: annual ring formation during tree growth and healing of cutaneous wounds in animals.

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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, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

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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 (December 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 laws defining the balance laws have limited regularity. Next, we establish a weak–strong uniqueness and stability theorem for the initial value problem associated with balance laws endowed with a strictly convex entropy field. We compare two solutions with limited regularity: on the one hand, a continuous solution that need not be Lipschitz continuous and, on the other hand, a weak solution that satisfies an "entropy inequality". Finally, we apply our theory to the Euler system for compressible fluid flows on a curved spacetime, and we exhibit an entropy field, which is strictly convex but fails to be uniformly convex as vacuum is approached. This leads us to a uniqueness theorem for (both the relativistic and non-relativistic versions of) the Euler system, which applies to a continuous solution with vacuum and an entropy weak solution with vacuum.

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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 (December 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 rigorously justified. Our proofs rely on the use of kinetic equations and the compensated compactness method for genuinely nonlinear balance laws.

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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 (February 15, 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 Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.

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18

Kan, Pui Tak, Marcelo M. Santos, and Zhouping Xin. "Initial Boundary Value Problem for Conservation Laws." Communications in Mathematical Physics 186, no. 3 (July 16, 1997): 701–30. http://dx.doi.org/10.1007/s002200050125.

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19

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 noted that some options of locally non-equilibrium processes lead to conflicting results. This study aims to develop a method for mathematical modeling of locally nonequilibrium heat conduction processes in solids, which allows determining their temperature with high accuracy during fast and high-intensity heat transfer processes. As applied to heat transfer processes in solids, a generalized heat equation that takes into account the relaxation properties of materials is formulated. The exact analytical solution is obtained using the Fourier method of separation of variables. The methodology for mathematical modeling of locally non-equilibrium transfer processes based on modified conservation laws has been developed. The generalized differential heat equation which allows performing N-fold relaxation of the heat flow and temperature in the modified heat balance equation has been formulated. For the first time, an exact analytical solution to the unsteady heat conduction problem for an infinite plate was obtained taking into account many-fold relaxation. The analysis of the solution to the boundary value problem of locally nonequilibrium heat conduction enabled to conclude that it is impossible to instantly has establish a boundary condition of the first kind. It has been demonstrated that each of the following terms in the relaxed heat equation has an ever smaller effect on the heat transfer process. The obtained results can be used by the scientific and technical personnel of organizations and higher educational institutions in the study of fuel ignition processes, the development of laser processing of materials, the design of highly efficient heat transfer equipment and the description of fast-flowing heat transfer processes.

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20

CHRISTOV, C. I., and M. G. VELARDE. "INELASTIC INTERACTION OF BOUSSINESQ SOLITONS." International Journal of Bifurcation and Chaos 04, no. 05 (October 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 (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

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21

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 (August 2008): 298–301. http://dx.doi.org/10.1007/s11741-008-0404-4.

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22

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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23

Xin, Zhouping, and Wen-Qing Xu. "Initial-boundary value problem to systems of conservation laws with relaxation." Quarterly of Applied Mathematics 60, no. 2 (June 1, 2002): 251–81. http://dx.doi.org/10.1090/qam/1900493.

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24

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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25

FRANKOWSKA, HÉLÈNE. "ON LeFLOCH'S SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR CONSERVATION LAWS." Journal of Hyperbolic Differential Equations 07, no. 03 (September 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 initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that Vx is continuous almost everywhere, and a result concerning the traces of the sign of f′(Vx(t, ⋅)) is proven.

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26

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

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27

Li, Lingxiao, Mingliang Wang, and Jinliang Zhang. "The Solutions of Initial (-Boundary) Value Problems for Sharma-Tasso-Olver Equation." Mathematics 10, no. 3 (January 29, 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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28

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 the energy system of equations that allow to investigate properties of the problem solution, its uniqueness, stability, dependence on initial data and more. Energy equation of motion of surface and groundwater flows are derived and investigated. It is shown that the total energy compatible flow depends on sources that are located inside the domain or on its border.

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29

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 (January 14, 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 volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is the posing of abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics.

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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 (August 2008): 306–10. http://dx.doi.org/10.1007/s11741-008-0406-3.

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31

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 (January 17, 2003): 1727–37. http://dx.doi.org/10.1090/s0002-9939-03-06961-2.

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32

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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33

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 (August 2002): 727–34. http://dx.doi.org/10.1016/s0893-9659(02)00034-4.

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34

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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35

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 (July 23, 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 and stability for a new class of conservation laws.

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36

Shao, Zhi-Qiang. "Blow-up of solutions to the initial–boundary value problem for quasilinear hyperbolic systems of conservation laws." Nonlinear Analysis: Theory, Methods & Applications 68, no. 4 (February 2008): 716–40. http://dx.doi.org/10.1016/j.na.2006.11.029.

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37

Donadello, Carlotta, and Andrea Marson. "Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws." Nonlinear Differential Equations and Applications NoDEA 14, no. 5-6 (December 2007): 569–92. http://dx.doi.org/10.1007/s00030-007-5010-7.

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38

Saxena, Prashant. "On the General Governing Equations of Electromagnetic Acoustic Transducers." Archive of Mechanical Engineering 60, no. 2 (June 1, 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 wave propagation problems of boundary value type.

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39

Maugin, Ge´rard A. "Material Forces: Concepts and Applications." Applied Mechanics Reviews 48, no. 5 (May 1, 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 expression of relevant material forces and devising criteria of progress for defects, in a general way. The advances made along this line, which may be referred to as Eshelbian mechanics, are assessed and perpectives are drawn.

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40

Zhang, Guizhou. "Convergence rate of the solution toward boundary layer solution for initial-boundary value problem of the 2-D viscous conservation laws." Applied Mathematics and Computation 217, no. 19 (June 2011): 7799–805. http://dx.doi.org/10.1016/j.amc.2011.02.087.

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41

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 (November 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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42

Kassab, Ghassan S. "Biomechanics of the cardiovascular system: the aorta as an illustratory example." Journal of The Royal Society Interface 3, no. 11 (July 5, 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 appropriate boundary conditions to formulate and solve several well-posed boundary value problems. Among other issues, we shall consider the effect of longitudinal pre-stretch and surrounding tissue on the mechanical status of the vessel wall. The solutions of the boundary value problems predict the presence of mechanical homeostasis in the vessel wall. The implications of mechanical homeostasis on growth, remodelling and postnatal development of the aorta are considered.

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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 (March 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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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 (April 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 limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions, and on some further properties of both ε-approximate front tracking solutions and limit solutions.

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Karakas, Mehmet. "Numerical solution of initial-boundary value problems with integral conditional for third-order-differential equations." Thermal Science 22, Suppl. 1 (2018): 211–19. http://dx.doi.org/10.2298/tsci170614288k.

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Water quality differential equation based on the theoretical bases of change is a multiparameter mathematical. When we compared with water quality measurement valves, it is determined that the concentration valve rate is not balanced and the two parameters, change solution is current and unique. When change conditions only one solution will not be the determinant of Jacobi matrix linear connection. Therefore, this research will help the availability in theory and uniqueness of the solution to the problem of water quality parameters. This method provides compatibility between real data to issue water quality parameter change obtained using the equation of the estimated value of the third row and differantive. The numerical solution of start-border value problem which is integral conditioned for third-order-differential balance and the analytical property of problem is analyzed. The application phases are shown, contribution is given theorem, some remarks about the results produced and made in the light of their theorems.

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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 properties of the solution and nonlinear coefficients of the differential problem. Here we use restrictions only on input data (initial and boundary conditions and the right-hand side). The sufficient conditions of the shock wave generation is formulated for input data. For the Riemann problem two exact and stable difference schemes are analyzed.

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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 (September 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 analysis applies to both the characteristic and the non-characteristic case. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields.

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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 (June 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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Bauzet, Caroline, Julia Charrier, and Thierry Gallouët. "Numerical approximation of stochastic conservation laws on bounded domains." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 1 (December 2, 2016): 225–78. http://dx.doi.org/10.1051/m2an/2016020.

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This paper is devoted to the study of finite volume methods for the discretization of scalar conservation laws with a multiplicative stochastic force defined on a bounded domain D of Rd with Dirichlet boundary conditions and a given initial data in L∞(D). We introduce a notion of stochastic entropy process solution which generalizes the concept of weak entropy solution introduced by F.Otto for such kind of hyperbolic bounded value problems in the deterministic case. Using a uniqueness result on this solution, we prove that the numerical solution converges to the unique stochastic entropy weak solution of the continuous problem under a stability condition on the time and space steps.

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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 (December 1, 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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