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Showing papers on "Partial differential equation published in 2004"


Journal ArticleDOI
TL;DR: Barrault et al. as discussed by the authors presented an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence, replacing non-affine coefficient functions with a collateral reducedbasis expansion, which then permits an affine offline-online computational decomposition.

1,265 citations


Book
12 Feb 2004
TL;DR: A priori estimates 4. Variational solutions 5. Pressure and temperature estimates 6. Fundamental ideas 7. Global existence 8. Mathematical preliminaries and abstractions as discussed by the authors 11].
Abstract: 1. Physical background 2. Mathematical preliminaries 3. A priori estimates 4. Variational solutions 5. Pressure and temperature estimates 6. Fundamental ideas 7. Global existence

956 citations


Book
01 Jan 2004
TL;DR: The Navier-Stokes equations under initial and boundary conditions were studied in this paper, where they were shown to be incompressible in the spatially periodic case and in the constant-coefficient case.
Abstract: Preface to the Classics Edition Introduction 1. The Navier-Stokes equations 2. Constant-coefficient Cauchy problems 3. Linear variable-coefficient Cauchy problems in 1D 4. A nonlinear example: Burgers' equations 5. Nonlinear systems in one space dimension 6. The Cauchy problem for systems in several dimensions 7. Initial-boundary value problems in one space dimension 8. Initial-boundary value problems in several space dimensions 9. The incompressible Navier-Stokes equations: the spatially periodic case 10. The incompressible Navier-Stokes equations under initial and boundary conditions Appendices References Author index Subject index.

764 citations


Journal ArticleDOI
Ji-Huan He1
TL;DR: In this article, the semi-inverse method was used to obtain variational principles for generalized Korteweg-de Vries equation and nonlinear Schrodinger's equation.
Abstract: Variational principles for generalized Korteweg–de Vries equation and nonlinear Schrodinger’s equation are obtained by the semi-inverse method. The most interesting features of the proposed method are its extreme simplicity and concise forms of variational functionals for a wide range of nonlinear problems. Comparison with the results obtained by the Noether’s theorem is made, revealing the present theorem is a straightforward and attracting mathematical tool.

601 citations


Journal ArticleDOI
TL;DR: A problem of boundary stabilization of a class of linear parabolic partial integro-differential equations in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts.
Abstract: In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

523 citations



Journal ArticleDOI
TL;DR: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed in this article.
Abstract: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.

482 citations


Journal ArticleDOI
TL;DR: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form as mentioned in this paper.
Abstract: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

474 citations


Journal ArticleDOI
TL;DR: It is proved that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis, and developed two examples of nonlinear complex processes, useful in image processing.
Abstract: The linear and nonlinear scale spaces, generated by the inherently real-valued diffusion equation, are generalized to complex diffusion processes, by incorporating the free Schrodinger equation. A fundamental solution for the linear case of the complex diffusion equation is developed. Analysis of its behavior shows that the generalized diffusion process combines properties of both forward and inverse diffusion. We prove that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, we develop two examples of nonlinear complex processes, useful in image processing: a regularized shock filter for image enhancement and a ramp preserving denoising process.

459 citations


Journal ArticleDOI
01 Jan 2004
TL;DR: The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace and the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.
Abstract: This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.

445 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived a partial differential equation that approximates solutions of Maxwell's equations describing the propagation of ultra-short optical pulses in nonlinear media and extended the prior analysis of Alterman and Rauch.

Journal ArticleDOI
TL;DR: By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth.


Journal ArticleDOI
TL;DR: In this article, the authors considered quasilinear stationary Schrodinger equations with the form (1) − Δ u− Δ (u 2 )u=g(x,u), x∈ R N, where f is suitably chosen.
Abstract: We consider quasilinear stationary Schrodinger equations of the form (1) − Δ u− Δ (u 2 )u=g(x,u), x∈ R N . Introducing a change of unknown, we transform the search of solutions u ( x ) of (1) into the search of solutions v ( x ) of the semilinear equation (2) − Δ v= 1 1+2f 2 (v) g(x,f(v)), x∈ R N , where f is suitably chosen. If v is a classical solution of (2) then u = f ( v ) is a classical solution of (1). Variational methods are then used to obtain various existence results.

Book
01 Jan 2004
TL;DR: In this paper, the authors present some techniques for Difference Equations and Partial Differential Equations, as well as some results and techniques for the Diffusion Equation with Linear Kinetics.
Abstract: 1. Single Species Population Dynamics.- 2. Population Dynamics of Interacting Species.- 3. Infectious Diseases.- 4. Population Genetics and Evolution.- 5. Biological Motion.- 6. Molecular and Cellular Biology.- 7. Pattern Formation.- 8. Tumour Modelling.- Further Reading.- A. Some Techniques for Difference Equations.- A.1 First-order Equations.- A.1.1 Graphical Analysis.- A.1.2 Linearisation.- A.2 Bifurcations and Chaos for First-order Equations.- A.2.1 Saddle-node Bifurcations.- A.2.2 Transcritical Bifurcations.- A.2.3 Pitchfork Bifurcations.- A.2.4 Period-doubling or Flip Bifurcations.- A.3 Systems of Linear Equations: Jury Conditions.- A.4 Systems of Nonlinear Difference Equations.- A.4.1 Linearisation of Systems.- A.4.2 Bifurcation for Systems.- B. Some Techniques for Ordinary Differential Equations.- B.1 First-order Ordinary Differential Equations.- B.1.1 Geometric Analysis.- B.1.2 Integration.- B.1.3 Linearisation.- B.2 Second-order Ordinary Differential Equations.- B.2.1 Geometric Analysis (Phase Plane).- B.2.2 Linearisation.- B.2.3 Poincare-Bendixson Theory.- B.3 Some Results and Techniques for rath Order Systems.- B.3.1 Linearisation.- B.3.2 Lyapunov Functions.- B.3.3 Some Miscellaneous Facts.- B.4 Bifurcation Theory for Ordinary Differential Equations.- B.4.1 Bifurcations with Eigenvalue Zero.- B.4.2 Hopf Bifurcations.- C. Some Techniques for Partial Differential Equations.- C.1 First-order Partial Differential Equations and Characteristics.- C.2 Some Results and Techniques for the Diffusion Equation.- C.2.1 The Fundamental Solution.- C.2.2 Connection with Probabilities.- C.2.3 Other Coordinate Systems.- C.3 Some Spectral Theory for Laplace's Equation.- C.4 Separation of Variables in Partial Differential Equations.- C.5 Systems of Diffusion Equations with Linear Kinetics.- C.6 Separating the Spatial Variables from Each Other.- D. Non-negative Matrices.- D.1 Perron-Frobenius Theory.- E. Hints for Exercises.

Journal ArticleDOI
TL;DR: A linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation.
Abstract: Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation. In this study, a linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation. A new definition of the fractional Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional Laplacian. Under the Szabo's smallness approximation, where attenuation is assumed to be much smaller than the wave number, the linear model is found consistent with arbitrary frequency power-law dependency.

Book
23 Jul 2004
TL;DR: In this article, the authors demonstrate how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations and provide a detailed comparison with earlier results simultaneously.
Abstract: The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation.

Journal ArticleDOI
TL;DR: This paper establishes a link between reachability, viability and invariance problems and viscosity solutions of a special form of the Hamilton-Jacobi equation to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon.


Journal ArticleDOI
TL;DR: This paper is devoted to proving that the mixed finite elements of this P k -P k-1 type when k > 3 satisfy the stability condition-the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedral meshes.
Abstract: A natural mixed-element approach for the Stokes equations in the velocity-pressure formulation would approximate the velocity by continuous piecewise-polynomials and would approximate the pressure by discontinuous piecewise-polynomials of one degree lower. However; many such elements are unstable in 2D and 3D. This paper is devoted to proving that the mixed finite elements of this P k -P k-1 type when k > 3 satisfy the stability condition-the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra. This type of mesh simplifies the implementation since it has no restrictions on the initial mesh. The new element also suits the multigrid method.

Journal ArticleDOI
TL;DR: The generalized diffusion tensor formalism is capable of accurately resolving the underlying spin displacement for complex geometrical structures, of which neither conventional DTI nor diffusion‐weighted imaging at high angular resolution (HARD) is capable.
Abstract: Diffusion tensor imaging (DTI) is known to have a limited capability of resolving multiple fiber orientations within one voxel. This is mainly because the probability density function (PDF) for random spin displacement is non-Gaussian in the confining environment of biological tissues and, thus, the modeling of self-diffusion by a second-order tensor breaks down. The statistical property of a non-Gaussian diffusion process is characterized via the higher-order tensor (HOT) coefficients by reconstructing the PDF of the random spin displacement. Those HOT coefficients can be determined by combining a series of complex diffusion-weighted measurements. The signal equation for an MR diffusion experiment was investigated theoretically by generalizing Fick's law to a higher-order partial differential equation (PDE) obtained via Kramers-Moyal expansion. A relationship has been derived between the HOT coefficients of the PDE and the higher-order cumulants of the random spin displacement. Monte-Carlo simulations of diffusion in a restricted environment with different geometrical shapes were performed, and the strengths and weaknesses of both HOT and established diffusion analysis techniques were investigated. The generalized diffusion tensor formalism is capable of accurately resolving the underlying spin displacement for complex geometrical structures, of which neither conventional DTI nor diffusion-weighted imaging at high angular resolution (HARD) is capable. The HOT method helps illuminate some of the restrictions that are characteristic of these other methods. Furthermore, a direct relationship between HOT and q-space is also established.

Book
26 Jun 2004
TL;DR: In this article, Fourier Transforms Laplace Transforms Linear Ordinary Differential Equations Complex Variables Multivalued Functions, Branch Points, Branch Cuts, and Riemann Surfaces Some Examples of Integration which involve multivalued functions Bessel Functions What are Transform Methods?
Abstract: THE FUNDAMENTALS Fourier Transforms Laplace Transforms Linear Ordinary Differential Equations Complex Variables Multivalued Functions, Branch Points, Branch Cuts, and Riemann Surfaces Some Examples of Integration which Involve Multivalued Functions Bessel Functions What are Transform Methods? METHODS INVOLVING SINGLE-VALUED LAPLACE TRANSFORMS Inversion of Laplace Transforms by Contour Integration The Heat Equation The Wave Equation Laplace's and Poisson's Equations Papers Using Laplace Transforms to Solve Partial Differential Equations METHODS INVOLVING SINGLE-VALUED FOURIER AND HANKEL TRANSFORMS Inversion of Fourier Transforms by Contour Integration The Wave Equation The Heat Equation Laplace's Equation The Solution of Partial Differential Equations by Hankel Transforms Numerical Inversion of Hankel Transforms Papers Using Fourier Transforms to Solve Partial Differential Equations Papers Using Hankel Transforms to Solve Partial Differential Equations METHODS INVOLVING MULTIVALUED LAPLACE TRANSFORMS Inversion of Laplace Transforms by Contour Integration Numerical Inversion of Laplace Transforms The Wave Equation The Heat Equation Papers Using Laplace Transforms to Solve Partial Differential Equations METHODS INVOLVING MULTIVALUED FOURIER TRANSFORMS Inversion of Fourier Transforms by Contour Integration Numerical Inversion of Fourier Transforms The Solution of Partial Differential Equations by Fourier Transforms Papers Using Fourier Transforms to Solve Partial Differential Equations THE JOINT TRANSFORM METHOD The Wave Equation The Heat and Other Partial Differential Equations Inversion of the Joint Transform by Cagniard's Method The Modification of Cagniard's Method by De Hoop Papers Using the Joint Transform Technique Papers Using the Cagniard Technique Papers Using the Cagniard-De Hoop Technique THE WIENER-HOPF TECHNIQUE The Wiener-Hopf Technique When the Factorization Contains No Branch Points The Wiener-Hopf Technique when the Factorization Contains Branch Points Papers Using the Wiener-Hopf Technique WORKED SOLUTIONS TO SOME OF THE PROBLEMS INDEX

Journal ArticleDOI
TL;DR: This work discusses the appropriate extension of cubature to Wiener space and develops high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.
Abstract: It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.

Journal ArticleDOI
Jie Li1, Jianbing Chen1
TL;DR: In this paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability, and the PDEE is further reduced to a one-dimensional partial differential equation.
Abstract: Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.

Journal ArticleDOI
TL;DR: A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems and it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.
Abstract: A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L 2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

Posted Content
TL;DR: In this paper, a unified theory of invariant manifolds for infinite dimensional dynamical systems generated by stochastic partial differential equations is presented, where a random graph transform and a fixed point theorem for non-autonomous systems are introduced.
Abstract: Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional {\em random} dynamical systems generated by {\em stochastic} partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

Journal ArticleDOI
TL;DR: A new Carleman inequality is presented for the linearized Navier–Stokes system, which leads to null controllability at any time T >0.

Journal ArticleDOI
TL;DR: This work uses partial differential equation techniques to remove noise from digital images using a total-variation filter to smooth the normal vectors of the level curves of a noise image and finite difference schemes are used to solve these equations.
Abstract: In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.

Journal ArticleDOI
TL;DR: In this paper, a level set approach for elliptic inverse problems with piecewise constant coefficients is proposed, where the geometry of the discontinuity of the coefficient is represented implicitly by level set functions.

Journal ArticleDOI
TL;DR: For large classes of vorticities, this article showed that a steady periodic gravity water wave with a monotonic profile between crests and troughs must be symmetric.
Abstract: For large classes of vorticities we prove that a steady periodic gravity water wave with a monotonic profile between crests and troughs must be symmetric. The analysis uses sharp maximum principles for elliptic partial differential equations.