Wave equations, examples and qualitative properties eduard feireisl abstract this is a short introduction to the theory of nonlinear wave equations. Pdf fourthorder and optimised finite difference schemes. In order to avoid the singularity, we split the nonlinear estimates into the low and high frequency sections and use biharmonic estimates instead of strichartz estimates. Systematic study of rogue wave probability distributions. The mathematics of pdes and the wave equation michael p. This paper is concerned with the cauchy problem for a class of fourth order wave equations in an ndimensional space.
Numerical study of the variableorder fractional version of. In the framework of the energy method, we further exploit the properties of the nehari functional. Solution of the wave equation by separation of variables. Finally, the global existence and finite time blowup of solutions are obtained. The convergence of the method for the problem under.
Bondarenko matrosov institute for system dynamics and control theory sb ras p. Fourth order wave equation in bhabhamadhavarao spin theory. The order of a partial differential equation is the order of the highest derivative entering the. Conference on digital audio effects dafx, maynooth, ireland, september 26, 20 fourth order and optimised finite difference schemes for the 2d wave equation brian hamilton, acoustics and audio group university of edinburgh. Jul 25, 2006 a modification of the exponential timedifferencing fourth order rungekutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as.
In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. The convergence of the method for the problem under consideration had been investigated. Global existence and asymptotic behavior of solutions to a. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. In this article, the 2d chebyshev wavelets cws are used for designing a proper procedure to solve the variable order vo fractional version of the nonlinear fourth order diffusion wave dw equation. Fourthorder and optimised finite difference schemes for. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Wave equation, compact finite difference method, consistence. This paper introduces the fourth order compact finite difference method for solving the numerical solution of onedimensional wave equations.
Lasiecka, uniqueness of weak solutions for the semilinear wave equations with supercritical boundaryinterior sources and damping, discrete contin. Some nonlinear wave equations are more difficult to solve analytically. Fourth order schemes for timeharmonic wave equations. For musical instrument applications, we are specifically interested in standing wave solutions of the wave equation and not so much interested in investigating the traveling wave solutions. Decay estimates for fourth order wave equations citeseerx.
By constructing a variational problem and the socalled invariant of some sets, we get global existence and nonexistence of the solutions. Fourthorder numerical method for the spacetime tempered. This paper is devoted to the finite time blowup phenomena for a class of fourth order wave equation with viscous damping term. Pdf fourth order compact finite difference method for. The main aim of the current paper is to develop a highorder numerical scheme to solve the spacetime tempered fractional diffusionwave equation.
K rigidity constant the larger k, the more rigid the beam and the faster the vibrations. Partial differential equations generally have many different solutions a x u 2 2 2. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. It looks like a one dimensional wave equation, but im unfortunately very bad at pdes. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t. A class of fourth order wave equations with dissipative. This discretization satisfies a summation by parts. Solving the heat, laplace and wave equations using nite. Numerical study of the variableorder fractional version. By solving cnls4 numerically, we are able to obtain quantitative predictions for the wave height distribution as a function of key environmental conditions such as average. Introduction and study of fourth order theta schemes for.
Random data cauchy problem for the fourth order schrodinger. Fourth order accurate scheme for the space fractional. The resistance is responsible for changing the wave equation to the fourth order beam equation utt. Thus, in order to nd the general solution of the inhomogeneous equation 1. Pdf fourthorder and optimised finite difference schemes for the. Solutions for a class of the higher diophantine equation p.
A class of fourth order damped wave equations with. Mechanics and information engineering numerical solution of the nonlinear wave equation. Intersections between spheres, cylinders, or other quadrics can be found using quartic equations. Numerical solution of the nonlinear wave equation via. In this paper we consider the initial boundary value problem for more general fourth order. We develop the new fourth order accurate method for the elastic wave equation in second order formulation. Introduction and study of fourth order theta schemes for linear wave equations. Fourth order wave equations with nonlinear strain and source terms. Closely related to the 1d wave equation is the fourth order2 pde for a vibrating. We study groundstate traveling wave solutions of a fourthorder wave equation. The unconditional stability of the scheme is established and the performance of the. Another classical example of a hyperbolic pde is a wave equation. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract.
The key ingredient of the method is a boundary modi. Pdf this paper investigates some fourthorder accurate explicit finite difference schemes for the 2d wave equation obtained using, 17, 21, and. The string has length its left and right hand ends are held. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. In particular, on the fourth order wave equations with nonlinear strain terms there are only works mentioned above 35. Wave equation like 4th order pde mathematics stack exchange. The resistance is responsible for changing the wave equation to the fourthorder beam equation utt 2uxxxx. Compared with the second order schrodinger equation, the singularity occurs in strichartz estimates for. First we develop a general scheme for fdm and focm for the damped. We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. This extends the f we use cookies to enhance your experience on our website.
Apr 15, 2018 its like asking why newtons law fma is second order. Laplaces equation recall the function we used in our reminder. This paper investigates some fourth order accurate explicit finite difference schemes for the 2d wave equation obtained using, 17, 21, and 25point discrete laplacians. Global solutions and finite time blowup for fourth order. Recent advances in mathematical and statistical methods, 279289. Also, we prove the unconditional stability and convergence of the developed method. Arbitrarily positive initial energy blowup and blowup time. The numerical scheme is fourthorder accurate in both time and space. Wave equations, examples and qualitative properties. In the first lecture, we saw several examples of partial differential equations that arise in. A class of fourth order damped wave equations with arbitrary. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. Wave equation like 4th order pde mathematics stack. The vibrating beam fourth order pde the vibrating beam fourth order pde the major difference between the transverse vibrations of a violin string and the transverse vibrations of a thin beam is that the beam offersresistance to bending.
The vibrating beam fourthorder pde the vibrating beam fourthorder pde the major difference between the transverse vibrations of a violin string and the transverse vibrations of a thin beam is that the beam offersresistance to bending. This handbook is intended to assist graduate students with qualifying examination preparation. Mar 10, 2020 in this article, the 2d chebyshev wavelets cws are used for designing a proper procedure to solve the variableorder vo fractional version of the nonlinear fourthorder diffusionwave dw equation. Some examples are given to demonstrate the validity of the present study. In this paper, we study the initial boundary value problem and global wellposedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term, which was introduced to describe the dynamics of a suspension bridge. Introduction and study of fourth order theta schemes for linear wave equations juliette chabassier, sebastien imperiale to cite this version. Dec 17, 2007 in this article, the adomian decomposition method has been used to obtain solutions of fourth. Evidently, the sum of these two is zero, and so the function ux,y is a solution of the partial differential equation. Exponential time differencing etd technique requires minimum stages to obtain the required accurateness, which suggests an efficient technique relating to computational duration that ensures remarkable stability characteristics upon resolving the nonlinear wave equations.
Numerical solution of the nonlinear wave equation via fourth. Fourth order wave equations with nonlinear strain and. Rungekutta 4th order method for ordinary differential equations. Construction analysis of fourthorder finite difference. Fourth order symmetric finite difference schemes for the. Several numerical methods have previously been developed for solving the elastic wave equation. The characteristic equation of a fourth order linear difference equation or differential equation is a quartic equation. In the presented model, fractional derivatives are defined in the caputo type. Consider the following initial boundary value problem for the wave equation. Strong instability of ground states to a fourth order. Like heat equation and laplace equation, the solution of second order wave equation can also be obtained using the standard method of separation of. Highamplitude nonlinear wave equation the wave equation for the highamplitude nonlinear acoustic waves in a viscous, heat. Analytic solutions of partial differential equations university of leeds. Chapter 4 derivation and analysis of some wave equations wavephenomenaareubiquitousinnature.
In particular, we examine questions about existence and. Traveling wave solutions of a fourthorder generalized. Chisholm university of toronto institute for aerospace studies the research institute for advanced computer science is operated by universities space research association, the american city building, suite 2. In this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. Based on the decay estimate of solutions to the corresponding linear equation, a solution space is defined. We take the model problem as the twodimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. We prove the global existence and optimal decay estimate of the solution in the corresponding sobolev spaces by the contraction mapping principle provided that. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient. A compact fourth order time discretization method for the. Like heat equation and laplace equation, the solution of secondorder wave equation can also be obtained using the standard method of separation of variables or fourier transform.
Rungekutta 4th order method for ordinary differential. Lawrence livermore national laboratory a fourth order. Existence and uniform decay estimates for the fourth order. The wave equa tion is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves.
Fourthorder timestepping for stiff pdes siam journal on. Then we develop an existence theory for a class of semilinear wave equations under suitable restrictions on the structural properties of the nonlinearities. Pde and boundaryvalue problems winter term 20142015. By using the concavity method, we establish a blowup result for certain solutions with arbitrary positive initial energy. Box 1233, 664033 irkutsk, russia abstract within the framework of the bhabhamadhavarao formalism, a consistent approach to the. The numerical scheme is fourth order accurate in both time and space. Acceleration is needed to define the equation of motion. A fourth order compact numerical scheme for threedimensional acoustic wave equation with variable velocity. A compact fourth order time discretization method for the wave equation wendy kress abstract a fourth order accurate discretization in time and space for the wave equation in rst order system formulation is investigated. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Fourth order wave equation in bhabhamadhavarao spin3 2 theory yu. The formula described in this chapter was developed by runge.
An example arises in the timoshenkorayleigh theory of beam bending. Compact finite difference scheme for the fourthorder. In this note, we prove the instability by blowup of the ground state solutions for a class of fourth order schrodinger equations. Finite difference method fdm and fourth order compact method focm are presented in this paper for the solutions of the well known one dimensional homogeneous damped wave equation. Rungekutta methods for linear ordinary differential equations. It is worthwhile pointing out that while these equations. We have discussed the mathematical physics associated with traveling and. Fourth order finite difference methods for the wave equation. Fourth order schemes for timeharmonic wave equations with discontinuous coef. In figure 3, we are comparing the exact results with eulers method rungekutta 1st order method, heuns method rungekutta 2nd order method, and rungekutta 4th order method. Pdf fourth order compact finite difference method for solving. By continuing to use our website, you are agreeing to our use of cookies.
In this paper, we investigate the initial boundary value problem of the nonlinear fourth order dispersivedissipative wave equation. It is shown that the high initial energy finite time blowup associated with the upper bound of the blowup time is established by introducing a new auxiliary function and employing an adapted concavity method. Numerical solution of the nonlinear wave equation via fourth order time stepping p. An exact solution for variable coefficients fourthorder wave equation. Lyon and bruno 18, 19, 4 proposed a framework based on fouriercontinuation and alternating direction methods, and devised high order accurate discretizations for the scalar wave equation and the heat equation.
In general, we allow for discontinuous solutions for hyperbolic problems. The mathematics of pdes and the wave equation mathtube. Messaoudi, exponential decay of solutions of a nonlinearly damped wave equation, nodea nonlinear differential equations appl. Solving the heat, laplace and wave equations using. Optimisation is conducted in order to minimise numerical dispersion and. Rungekutta methods for linear ordinary differential equations david w. An example of a discontinuous solution is a shock wave, which is a feature of solutions of nonlinear hyperbolic equations. A fourth order accurate finite difference scheme for the. Decreasing of thermal losses of the lightweight building envelope.
Examplesincludewaterwaves,soundwaves,electromagneticwavesradiowaves. Traveling wave solutions of a fourth order generalized dispersive and dissipative equation xiaofeng li 1. This research builds on our previous work in seismic wave simulations, which has been based on a second order accurate, also energy conserving, method. The fractional derivative is described in the caputo sense. Fourth order compact finite difference method for solving one dimensional wave equation.
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