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 Subject: 1-d Nonlinera Burger's Equation Category: Science > Math Asked by: javier5440-ga List Price: \$10.00 Posted: 15 Feb 2005 06:46 PST Expires: 17 Mar 2005 06:46 PST Question ID: 474879
 ```I want to know the "analytical" solution for the 1-d nonlinear Burger's equation, given by : with a computational domain of (0,1) and periodic boundary conditions : u(x=0)=u(x=1), the initial condition is a sine wave given by: u= 2 + (1/(4*pi))*sin(2*pi*x); thanks, I am looking for the mathematics of it, is for a CFD study and I am looking at the wavenumber of 3 numerical schemes, best regards, Javier``` Request for Question Clarification by mathtalk-ga on 15 Feb 2005 10:03 PST ```Hi, javier5440-ga: I suspect you accidentally omitted the partial differential equation itself. Are you referring to this 1-D nonlinear 2nd order PDE: u_t + u*u_x - c*u_xx = 0 where subscripts x,t refer to derivatives of u(x,t)? regards, mathtalk-ga``` Clarification of Question by javier5440-ga on 15 Feb 2005 10:30 PST ```thanks for your quick reply: the equation is : ut+ (0.5*(u^2u))x=0 where I have partial derivative of U w.r.t time (ut) and partial derivative of (0.5*(u^2)) w.r.t. space ((0.5*(u^2u))x) thanks``` Clarification of Question by javier5440-ga on 15 Feb 2005 10:32 PST ```In the last e-mail I accidenatlly miswrote the equation it should be: ut + (0.5*(u^2))x=0 where I have the pde of u , w.r.t. time and pde of (0.5*(u^2)), w.r.t space(x) thanks``` Request for Question Clarification by mathtalk-ga on 16 Feb 2005 05:57 PST ```Hi, javier5440-ga: The equation you've asked about is called the inviscid Burgers equation: u_t + u*u_x = 0 and is often given as a model for evolution of singularities from smooth initial conditions in finite time. It is possible to give for the case of periodic initial and boundary conditions an analytical solution in terms of Fourier series in x whose coefficients are functions of t, valid up to the time at which the singularity appears. Would a reference to the literature on this subject be satisfactory for your purposes? regards, mathtalk-ga``` Clarification of Question by javier5440-ga on 16 Feb 2005 06:24 PST ```Thanks for your reply, I am actually looking for an analyticl condition because I have a olution using numerical schemes using Runge-Kutta method and different spatial differencing methods, If it is too much work is alright, i have to have a solution for tomorrow, I thank you for you time and compliment you on your diligent work, best regards, Javier5440```
 Subject: Re: 1-d Nonlinera Burger's Equation Answered By: mathtalk-ga on 16 Feb 2005 17:08 PST Rated:
 ```Hi, Javier: Your question certainly brought back memories of graduate classes in numerical PDE's and fluid dynamics. Here's a recent paper that details the approach for analytically expressing solutions to the inviscid Burgers equation in 1-D, for smooth periodic initial data defined on the entire x-axis: [A Simple Illustration Of A Weak Spectral Cascade - David J. Muraki] http://www.math.sfu.ca/~dmuraki/research/cascade.pdf Given the inviscid Burgers equation: u_t + u*u_x = 0 and smooth initial data: u(x,0) = f(x) that is periodic in x, one can give an implicit "solution" for Burgers equation by: u(x,t) = f(x - u*t) It is easily verified that for t = 0, such a function u must satisfy the initial condition. Furthermore implicit differentiation with respect to x,t will show that the PDE is satisfied as well. For fixed time t > 0, the periodicity of u follows from the periodicity of f. For consistency with the notation in that paper, we will assume that f(x) is of period 2pi, rather than of period 1 as assumed in your problem. It is easy to show that a common change of scale for both x,t preserves the solutions of Burgers equation while adjusting the period of initial values f. As the above linked paper shows, Platzman's earlier analysis of the Fourier series for u(x,t): Platzman, G.W. An exact integral of complete spectral equations for unsteady onedimensional flow, Tellus, XVI (1964), pp. 422?431 can be generalized to give (up to the time of singularity): a_0 +oo u(x,t) = --- + SUM [ a_n(t)*cos(nx) + b_n(t)*sin(nx) ] 2 n=1 where for n > 0: a_0 = (1/pi) INTEGRAL f(z) dz OVER [-pi,+pi] 1 a_n(t) = ----- INTEGRAL sin(nz + nt*f(z)) dz OVER [-pi,+pi] pi*nt -1 b_n(t) = ----- INTEGRAL cos(nz + nt*f(z)) dz OVER [-pi,+pi] pi*nt Note that the "constant term" a_0 is truly a constant, independent of time t, corresponding to the conservation of the mean exhibited by Burgers equation. This Fourier series expansion of u(x,t) determines a single-valued function for all time t, but it only satisfies Burgers equation up to the point in time when a singularity develops. The author of the paper above, David J. Muraki, explains the distinction between the single-valued function defined by the Fourier series, the multi-valued function defined by the method of characteristics, and the "entropy" solution to which one ascribes "physical reality". Please feel free to post a Request for Clarification if I can be of further assistance. regards, mathtalk-ga```
 javier5440-ga rated this answer: and gave an additional tip of: \$2.00 ```Thanks for your time, I greatly appreciate your answer and compliment you on the refernce you attached, good work, best regards, Javier5440```

 ```The PDE which Javier has asked about is a nonlinear evolution equation. Without loss of generality we may view the initial conditions as extended periodically to the entire x-axis. The particular initial data consists of a constant plus a superimposed "fundamental" sine wave: u(x,0) = 2 + (1/(4pi))*sin(2pi*x) so that the period in x is length 1. As the solution evolves all the higher harmonics are "excited". In 1964 G.W. Platzmann published a paper in an obscure meterology journal Tellus which gives the exact solution (up to singularity) for the case of initial data which is purely sinusoidal. The "time" coefficients of the various harmonics are then given explicitly in terms of Bessel functions. This approach has recently been generalized to all smooth periodic initial conditions, but it would require some effort to work out the details for the particular initial condition posed here. Other approaches to obtaining exact solutions consider the extension of the domain to complex or even hypercomplex numbers. regards, mathtalk-ga```
 ```this information is enough for now, i greatly appreciate it and you CAN go ahead and bill me the amount I had agreed when I posted the response (\$10) your service is great and I am quite impressed, I will try to find your reference, best regards, javier5440```