Showing papers on "Hyperbolic partial differential equation published in 2002"

Finite Volume Methods for Hyperbolic Problems

Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

Inverse Problems for Partial Differential Equations

Abstract: Auxiliary information from functional analysis and theory of differential equations the basic notions and notations inequalities some concepts and theorems of functional analysis linear partial differential equation of the first order the maximum principle for parabolic equations of second order the weak approximation method examples reducing to the concept of the weak approximation method general formulation of the weak approximation method two theorems - an example the linear partial differential equation identification problems for parabolic equations with Cauchy data the unknown source function the unknown lowest coefficient the unknown coefficient to the first order derivative an unknown coefficient to the time derivative inverse problem for a semilinear parabolic equation equations of Burgers type the splitting of one many-dimensional inverse problem into problems of lower dimension the identification of the source function for a system of composite type and parabolic equation the behaviour of the problem's solution under t-> the statement of the problem theorems of existence and uniqueness "on the whole" the behaviour of solution by t-> stationary problem convergence to the stationary problem solution unique solvability of the problem of identifying the source function for a parabolic equation the stabilization of the solution to the inverse problem the problem of determining the coefficient in a parabolic equation and some properties of its solution statement of the problem theorems of existence and uniqueness "on the whole" properties of solution under t-> two unknown coefficients of a parabolic type equation uniform boundary conditions inhomogeneous conditions of over-determination input data of the special form some inverse boundary problems unknown source function nonlinear heat equation hyperbolic equation with small parameter.

Functional equations and inequalities in several variables

Abstract: Functional Equations and Inequalities in Linear Spaces: Linear Spaces and Semilinear Topology Convex Functions Cauchy's Exponential Equation Polynomial Functions and Their Extensions Quadratic Mappings Quadratic Equation on an Interval Ulam-Hyers-Rassias Stability of Functional Equations: Additive Cauchy Equation Multiplicative Cauchy Equation Jensen's Functional Equation Gamma Functional Equation Stability of Homogeneous Mappings Stability of Functional Equations in Function Spaces Stability in the Lipschitz Norms Round-off Stability of Iterations Functional Equations in Set-Valued Functions: Cauchy's Set-Valued Functional Equation Pexider's Functional Equation Subadditive Set-Valued Functions Hahn-Banach Type Theorem and Applications Subquadratic Set-Valued Functions Iteration Semigroups of Set-Valued Functions.

Handbook of linear partial differential equations for engineers and scientists

Abstract: INTRODUCTION: SOME DEFINITIONS, FORMULAS, METHODS, AND SOLUTIONS Classification of Second Order Partial Differential Equations Basic Problems of Mathematical Physics Properties and Particular Solutions of Linear Equations Separation of Variables Method Integral Transforms Method Representation of the Solution of the Cauchy Problem via the Fundamental Solution Nonhomogeneous Boundary Value Problems with One Space Variable Nonhomogeneous Boundary Value Problems with Many Space Variables Construction of the Green's Functions: General Formulas and Relations Duhamel's Principles in Nonstationary Problems Transformation Simplifying Initial and boundary Conditions EQUATIONS OF PARABOLIC TYPE WITH ONE SPACE VARIABLE Constant Coefficient Equations Heat Equation with Axial or Central Symmetry and Related Equations Equations Containing Power Functions and Arbitrary Parameters Equations Containing Exponential Functions and Arbitrary Parameters Equations Containing Hyperbolic Functions and Arbitrary Parameters Equations Containing Logarithmic Functions and Arbitrary Parameters Equations Containing Trigonometric Functions and Arbitrary Parameters Equations Containing Arbitrary Functions Equations of Special Form PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations with Three Space Variables Equations with n Space Variables HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE Constant Coefficient Equations Wave Equation with Axial or Central Symmetry Equations Containing Power Functions and Arbitrary Parameters Equations Containing the First Time Derivative Equations Containing Arbitrary Functions HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equation with Two Space Variables HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equations with Three Space Variables Equations with n Space Variables ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations Equations with n Space Variables HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS Third Order Partial Differential Equations Fourth Order One-Dimensional Nonstationary Equations Two-Dimensional Nonstationary Fourth Order Equations Fourth Order Stationary Equations Higher Order Linear Equations with Constant Coefficients Higher Order Linear Equations with Variable Coefficients SUPPLEMENT A: Special Functions and Their Properties SUPPLEMENT B: Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics REFERENCES INDEX

Front Tracking for Hyperbolic Conservation Laws

Abstract: In this edition we have added the following new material: In Chapt. 1 we have added a section on linear equations, which allows us to present some of the material in the book in the simpler linear setting. In Chapt. 2 we have made some changes in the presentation of Kružkov’s fundamental doubling of variables method. In Chapt. 3 on finite difference methods the focus has been changed to finite volume methods. A section on higher-order schemes has been added. The section on measure-valued solutions has been rewritten. The main existence theorem in Chapt. 4, Theorem 4.3, now resembles the one-dimensional case. The presentation of the solution of the Riemann problem for systems in Chapt. 5 has been supplemented by the complete solution of the Riemann problem for the 3 3 Euler equations of gas dynamics. The solution of the Cauchy problem for systems in Chapt. 6 has been rewritten and simplified. We have added a new chapter, Chapt. 8, on one-dimensional scalar conservation laws where the flux function depends explicitly on space in a discontinuous manner

Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves

Abstract: I. Fundamental concepts and examples.- 1. Hyperbolicity, genuine nonlinearity, and entropies.- 2. Shock formation and weak solutions.- 3. Singular limits and the entropy inequality.- 4. Examples of diffusive-dispersive models.- 5. Kinetic relations and traveling waves.- 1. Scalar Conservation Laws.- II. The Riemann problem.- 1. Entropy conditions.- 2. Classical Riemann solver.- 3. Entropy dissipation function.- 4. Nonclassical Riemann solver for concave-convex flux.- 5. Nonclassical Riemann solver for convex-concave flux.- III. Diffusive-dispersive traveling waves.- 1. Diffusive traveling waves.- 2. Kinetic functions for the cubic flux.- 3. Kinetic functions for general flux.- 4. Traveling waves for a given speed.- 5. Traveling waves for a given diffusion-dispersion ratio.- IV. Existence theory for the Cauchy problem.- 1. Classical entropy solutions for convex flux.- 2. Classical entropy solutions for general flux.- 3. Nonclassical entropy solutions.- 4. Refined estimates.- V. Continuous dependence of solutions.- 1. A class of linear hyperbolic equations.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- 1. Shock and rarefaction waves.- 2. Classical Riemann solver.- 3. Entropy dissipation and wave sets.- 4. Kinetic relation and nonclassical Riemann solver.- VII. Classical entropy solutions of the Cauchy problem.- 1. Glimm interaction estimates.- 2. Existence theory.- 3. Uniform estimates.- 4. Pointwise regularity properties.- VIII. Nonclassical entropy solutions of the Cauchy problem.- 1. A generalized total variation functional.- 2. A generalized weighted interaction potential.- 3. Existence theory.- 4. Pointwise regularity properties.- IX. Continuous dependence of solutions.- 1. A class of linear hyperbolic systems.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- X. Uniqueness of entropy solutions.- 1. Admissible entropy solutions.- 2. Tangency property.- 3. Uniqueness theory.- 4. Applications.

Direct construction method for conservation laws of partial differential equations. Part II: General treatment

Abstract: This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

Riemann wave description of erosional dam-break flows

Abstract: This work examines the sudden erosional flow initiated by the release of a dam-break wave over a loose sediment bed. Extended shallow-water equations are formulated to describe the development of the surge. Accounting for bed material inertia, a transport layer of finite thickness is introduced, and a sharp interface view of the morphodynamic boundary is adopted. Approximations are sought for an intermediate range of wave evolution, in which equilibration of the sediment load can be assumed instantaneous but momentum loss due to bed friction has not yet been felt. The resulting homogeneous hyperbolic equations are mathematically tractable using the Riemann techniques of gas dynamics. Dam-break initial conditions give rise to self-similar flow profiles. The wave structure features piecewise constant states, two smoothly varied simple waves, and a special type of shock: an erosional bore forming at the forefront of the wave. Profiles are constructed through a semi-analytical procedure, yielding a geomorphic generalization of the Stoker solution for dam-break waves over rigid bed. For most flow properties, the predictions of the theoretical treatment compare favourably with experimental tests visualized using particle imaging techniques.

Partial Differential Equations

Abstract: This book offers an ideal introduction to the theory of partial differential equations. It focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. It also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. It also explores connections between elliptic, parabolic, and hyperbolic equations as well as the connection with Brownian motion and semigroups. This second edition features a new chapter on reaction-diffusion equations and systems.

ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D

Abstract: The ADER scheme for solving systems of linear, hyperbolic partial differential equations in two-dimensions is presented in this paper It is a finite-volume scheme of high order in space and time The scheme is explicit, fully discrete and advances the solution in one single step Several numerical tests have been performed In the first test case the dissipation and dispersion behaviour of the schemes are studied in one space dimension Dispersion as well as dissipation effects strongly influence the discrete wave propagation over long distances and are very important for, eg, aeroacoustical calculations The next test, the so-called co-rotating vortex pair, is a demonstration of the ideas of the two-dimensional ADER approach The linearised Euler equations are used for the simulation of the sound emitted by a co-rotating vortex pair

Metamorphic testing of programs on partial differential equations: a case study

Abstract: We study the effect of applying metamorphic testing to alleviate the oracle problem for numerical programs. We discuss a case study on the testing of a program that solves an elliptic partial differential equation with Dirichlet boundary conditions. We identify a metamorphic relation for the equation and demonstrate the effectiveness of metamorphic testing in identifying the error. The relation identified should also be applicable to other numerical methods that yield better approximations on the refinement of grid points or step sizes.

Pointwise Green's function bounds and stability of relaxation shocks

Abstract: We establish sharp pointwise Green's function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([32, 108, 110]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, and transversality of the connecting profile, with no additional assumptions on the structure or strength of the shock. Restricting to Lax type shocks, we establish the further result of nonlinear stability with respect to small L 1 n H 2 perturbations, with sharp rates of decay in L p , 2 ≤ p ≤ ∞, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in W 3,1 n W 3, ∞ , we obtain sharp rates of decay in L p , 1 ≤ p < ∞, for (Lax type) shocks of arbitrary strength. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin-Xin and Broad-well models, for which spectral stability has been established in [61, 43], and in [52], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [107] for the study of traveling waves of parabolic systems; however, significant extensions are required to deal with the nonsectorial generator and more singular short-time behavior of the associated (hyperbolic) linearized equations. Our main technical innovation is a systematic method for refining large-frequency (short-time) estimates on the resolvent kernel, suitable in the absence of parabolic smoothing.

On Carleman estimates for hyperbolic equations

Abstract: We prove a Carleman inequality for the second order hyperbolic equation in the cylinder Q with the norm of right- hand side taken in the space (W 1 2 (Q)) ∗ .

Traveling wave solutions for nonlinear equations using symbolic computation

Abstract: In this paper, a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) is presented and implemented in a computer algebraic system. The key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh-function in the tanh method. It is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. More importantly, for some equations, with no extra effect we also find other new and more general solutions at the same time. By introducing appropriate transformations, our method is further extended to the nonlinear PDEs whose balancing numbers may be any nonzero real numbers. The efficiency of the method can be demonstrated on a large variety of nonlinear PDEs such as those considered in this paper, Burgers-Huxley equation, coupled Korteweg-de Vries equation, Caudrey-Dodd-Gibbon-Kawada equation, active-dissipative dispersive media equation, generalized Fisher equation, and nonlinear heat conduction equation.

An Unconditionally Stable ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions

Abstract: An unconditionally stable alternating direction implicit (ADI) method of O(k 2 +h 2 ) of Lees type for solving the three space dimensional linear hyperbolic equation u tt +2 f u t + g 2 u = u xx + u yy + u zz + f ( x , y , z , t ), 0 0 subject to appropriate initial and Dirichlet boundary conditions is proposed, where f >0 and g S 0 are real numbers. For this method, we use a single computational cell. The resulting system of algebraic equations is solved by three step split method. The new method is demonstrated by a suitable numerical example.

Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems

Abstract: This work is concerned with the boundary stabilization of an abstract system of two coupled second order evolution equations wherein only one of the equations is stabilized (indirect damping; see, e.g., J. Math. Anal. Appl., 173 (1993), pp. 339--358). We show that, under a condition on the operators of each equation and on the boundary feedback operator, the energy of smooth solutions of this system decays polynomially at $\infty$. We then apply this abstract result to several systems of partial differential equations (wave-wave systems, Kirchhoff--Petrowsky systems, and wave-Petrowsky systems).

Nonhomogeneous dirichlet problems for degenerate parabolic-hyperbolic equations

Abstract: The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type $$$$ in the case of nonhomogeneous boundary data a0. Here u=u(x,t)∈ℝ, with (x,t)∈Q=Ω× (0,T), where Ω is a bounded domain in ℝN with smooth boundary and T>0. The function b is assumed to be nondecreasing (allowing degeneration zones where b is constant), Φ is locally Lipschitz continuous and g∈L∞(Ω× (0,b)). After introducing the definition of an entropy solution to the above problem (in the spirit of Otto [14]), we prove uniqueness of the solution in the proposed setting. Moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type (specifically the diffusion function b is approximated by functions bɛ that are strictly increasing).

Controllability of partial differential equations and its semi-discrete approximations

Abstract: In these notes we analyze some problems related to the control- lability and observability of partial dierential equations and its space semi- discretizations. First we present the problems under consideration in the clas- sical examples of the wave and heat equations and recall some well known results. Then we analyze the 1 d wave equation with rapidly oscillating coef- ficients, a classical problem in the theory of homogenization. Then we discuss in detail the null and approximate controllability of the constant coecient heat equation using Carleman inequalities. We also show how a fixed point technique may be employed to obtain approximate controllability results for heat equations with globally Lipschitz nonlinearities. Finally we analyze the controllability of the space semi-discretizations of some classical PDE models: the Navier-Stokes equations and the 1 d wave and heat equations. We also present some open problems.

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

Abstract: We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

The Cauchy problem for a nonlinear stochastic wave equation in any dimension

Abstract: A nonlinear wave equation on $ \mathbb{R}^d $ driven by a spatially homogeneous Wiener process is studied Conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise are given for an arbitrary dimension d

Numerical Solution of Fuzzy Differential Equation

Abstract: In this paper numerical algorithms for solving fuzzy ordinary differential equations are considered. A scheme based on the 2nd Taylor method in detail is discussed and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy cauchy problems.

Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws

Abstract: We consider the Cauchy problem for n×n strictly hyperbolic systems of nonresonant balance laws each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that and are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.

hp-Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity

Abstract: We develop the a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for linear and nonlinear hyperbolic problems. Sharp a posteriori error bounds are derived using a duality argument. The bounds exhibit an exponential rate of convergence under hp-refinement if either the primal or the dual solution is an analytic function over the computational domain. We implement the error bounds into an hp-adaptive finite element algorithm.

The generation, propagation, and extinction of multiphases in the KdV zero‐dispersion limit

Abstract: We study the multiphases in the KdV zero-dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi-linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space-time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler-Poisson-Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler-Poisson-Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler-Poisson-Darboux solution. © 2002 Wiley Periodicals, Inc.

Lattice boltzmann scheme for hyperbolic heat conduction equation

Abstract: An extended lattice Boltzmann (LB) equation, the lattice Boltzmann equation with a source term, is developed for the system of equations governing the hyperbolic heat conduction equation. Mathematical consistence between the proposed extended LB equation and the governing equations are accomplished by the Chapman-Enskog expansion. Four illustrative examples, with both finite and semi-infinite computational domains and subjected to linear and nonlinear boundary conditions, are simulated. All numerical predications agree very well with the existing solutions in the literature. It is also demonstrated that the present scheme is stable and free of numerical oscillations especially around the wave front, where sharp change in temperature occurs.