Showing papers on "Hyperbolic partial differential equation published in 2009"
TL;DR: Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
Abstract: In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
599 citations
TL;DR: Many existing partial differential equation solver packages focus on the important, but arcane, task of numerically solving the linearized set of algebraic equations that result from discretizing a set of PDEs.
Abstract: Many existing partial differential equation solver packages focus on the important, but arcane, task of numerically solving the linearized set of algebraic equations that result from discretizing a set of PDEs. Many researchers, however, need something higher level than that.
373 citations
TL;DR: In this paper, the traveling wave solutions involving parameters of the combined Korteweg-de Vries modified KORTeweg de Vries equation, reaction-diffusion equation, compound KdV-Burgers equation, and generalized shallow water wave equation were constructed using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation.
Abstract: I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.
270 citations
Book•
15 Jun 2009
TL;DR: In this paper, the Cauchy problem in general relativity is reduced to the problem of solving a system of hyperbolic partial differential equations, and the initial value problem is formulated for the so-called Einstein-Vlasov system, and a recent future global nonlinear stability result in this setting is described.
Abstract: After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system, and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.
229 citations
TL;DR: In this paper, the authors consider hyperbolic Timoshenko-type vibrating systems coupled to a heat equation modeling an expectedly dissipative effect through heat conduction and show that the coupling via the Cattaneo law does not yield an exponentially stable system.
Abstract: We consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While exponential stability under the Fourier law of heat conduction holds, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier’s law by changing to the Cattaneo law causes a loss of the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property.
212 citations
TL;DR: In this paper, the integrability of nonlinear hyperbolic equations on quad-graphs is defined as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent.
Abstract: We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ℤ2. The fields are associated with the vertices and an equation of the form Q(x
1, x
2, x
3, x
4) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ℤ
N
. We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.
190 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is implemented for finding approximate solutions of the Burger and regularized long wave (RLW) equations, and the results of the proposed method are compared with the results obtained by HPM.
170 citations
Book•
10 Aug 2009
TL;DR: In this article, the wave equation and conservation laws in one-space dimension were discussed. But the authors focused on the energy equivalence for the wave expression and the variable-coefficient wave equation.
Abstract: Vector Fields and Integral Curves.- Operators and Systems in the Plane.- Nonlinear First Order Equations.- Conservation Laws in One-Space Dimension.- The Wave Equation.- Energy Inequalities for the Wave Equation.- Variable Coefficient Wave Equations and Systems.
163 citations
TL;DR: A novel high-resolution numerical method is presented for one-dimensional hyperbolic problems based on the extension of the original Upwind Leapfrog scheme to quasi-linear conservation laws, which is second-order accurate on non-uniform grids in space and time.
161 citations
TL;DR: In this article, the existence of solutions for a class of fractional differential equations was studied and a new and proper concept of periodic boundary value conditions was introduced to deal with the singularity of the possible solutions.
Abstract: We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.
150 citations
TL;DR: Sufficient exponential stability conditions with a given decay rate are derived in the form of Linear Matrix Inequalities for both systems, and these conditions are utilized to synthesize H"~ static output feedback boundary controllers of the systems in question.
TL;DR: Numerical experiments for a variety of hyperbolic partial differential equations, such as Euler equations and ideal magneto-hydrodynamic equations, confirm a significant improvement of shock resolution, high accuracy for smooth functions and computational efficiency.
TL;DR: An accurate and efficient Haar wavelet solution of Fisher's equation, a prototypical reaction-diffusion equation, which is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.
TL;DR: An Eulerian diffuse interface model for elastic solid-compressible fluid interactions in situations involving extreme deformations is derived andabilities of the model and methods are illustrated on various tests of impacts of solids moving in an ambient compressible fluid.
TL;DR: In this paper, the G ′ G -expansion method was used to construct exact travelling wave solutions of nonlinear evolution equations, the balance numbers of which are not positive integers.
TL;DR: A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed and is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions.
Book•
01 Jan 2009
TL;DR: This paper presents a method for adaptive mesh refinement of hyperbolic systems based on the results of a large-scale simulation of the response of the Higgs boson in three dimensions.
Abstract: Preface 1. Introduction to partial differential equations 2. Scalar hyperbolic conservations laws 3. Nonlinear scalar laws 4, Nonlinear hyperbolic systems 5. Methods for scalar laws 6. Methods for hyperbolic systems 7. Methods in multiple dimensions 8. Adaptive mesh refinement Bibliography Index.
TL;DR: In this paper, the authors studied the microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations and established that the rank of Hessian ∇cffff 2 istg u is of constant rank for any convex solution u of equation F(∇¯¯¯¯ 2¯¯ u,∇ u, u,u,x)=0.
Abstract: We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇
2
u is of constant rank for any convex solution u of equation F(∇
2
u,∇
u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed.
TL;DR: A general linear controller design method for a class of hyperbolic linear partial differential equation (PDEs) systems is presented and an optimal controller is designed for linearized fixed-bed reactor model, and the resulting closed-loop stability is analyzed.
TL;DR: In this article, a non-linear approximation type approach was proposed to solve high dimensional partial differential equations, and the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N.
Abstract: We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.
TL;DR: In this paper, a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component is introduced, which is unconditionally stable and solves the two-dimensional linear hyper bolic equation with high accuracy.
Abstract: In this article, we introduce a high-order accurate method for solving the two dimensional linear hyperbolic equation We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component The resulted method is unconditionally stable and solves the two-dimensional linear hyperbolic equation with high accuracy In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation © 2008 Wiley Periodicals, Inc Numer Methods Partial Differential Eq, 2009
TL;DR: In this article, a single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator.
Abstract: A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.
TL;DR: In this paper, the authors considered the Schrodinger equation with no radial assumption on real hyperbolic spaces H n and obtained in all dimensions n ⩾ 2 sharp dispersive and Strichartz estimates for a large family of admissible pairs.
TL;DR: In this article, a meshless method is proposed for the numerical solution of the two space dimensional linear hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions, which uses collocation points and approximates the solution employing thin plate splines radial basis functions.
Abstract: A meshless method is proposed for the numerical solution of the two space dimensional linear hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The new developed scheme uses collocation points and approximates the solution employing thin plate splines radial basis functions. Numerical results are obtained for various cases involving variable, singular and constant coefficients, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009
TL;DR: Application of homotopy perturbation method to Cauchy reaction-diffusion problem shows the rapid convergence of the sequence constructed by this method to the exact solution.
Abstract: In this paper, the solution of Cauchy reaction-diffusion problem is presented by means of the homotopy perturbation method. Reaction-diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution.
01 Jan 2009
TL;DR: In this article, the authors focus on the nonlinear partial differential equations and apply the Adomina decomposition method and the variational iteration method to obtain the solutions of nonlinear wave equations.
Abstract: So far in this text we have been mainly concerned in applying classic methods, the Adomina decomposition method [3, 4, 5], and the variational iteration method [8, 9, 10] in studying first order and second order linear partial differential equations. In this chapter, we will focus our study on the nonlinear partial differential equations. The nonlinear partial differential equations arise in a wide variety of physical problems such as fluid dynamics, plasma physics, solid mechanics and quantum field theory. Systems of nonlinear partial differential equations have been also noticed to arise in chemical and biological applications. The nonlinear wave equations and the solitons concept have introduced remarkable achievements in the field of applied sciences. The solutions obtained from nonlinear wave equations are different from the solutions of the linear wave equations [1, 2].
TL;DR: The stabilization of the wave equation on general 1-d networks is studied, using an interpolation inequality similar to the one proved in Begout and Begout (2006) to obtain explicit decay estimates of the energy for smooth initial data.
Abstract: In this paper we study the stabilization of the wave equation on general 1-d networks For that, we transfer known observability results in the context of control problems of conservative systems (see [R Dager and E Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math Appl 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems Then we use an interpolation inequality similar to the one proved in [P Begout and F Soria, J Differential Equations, 240 (2007), pp 324-356] to obtain the explicit decay estimates of the energy for smooth initial data The obtained decay rate depends on the geometric and topological properties of the network We also give some examples of particular networks in which our results apply, yielding different decay rates
TL;DR: In this article, it was shown that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.
Abstract: We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.
TL;DR: The traveling wave solutions are constructed in terms of the hyperbolic functions, the trigonometric functions and the rational functions by using a new approach, namely the G^'G-expansion method, where G=G(@x) satisfies a second order linear ordinary differential equation.
TL;DR: This paper studies the numerical approximation of bedload sediment transport due to shallow layer flows by considering a Roe-type first order scheme as well as a variant based on the use of flux limiters.