TokenMonkey

I found the answer: a procedure is described in Kubicek's Algorithm 502 in ACM Trans. Math. Software. dl.acm.org/citation.cfm?id=355675

Hi there, I have a quick question regarding pseudo-arclength continuation. As some background, I am a chemical engineer, not a mathematician, applied or otherwise, so my knowledge of numerical methods is limited to the "standard" stuff. I've been reading up as extensively as I can on pseudo-arclength continuation, but unfortunately, it's all from second-hand sources; I don't have access to Keller's original paper. Here's what I understand at this point: We want to solve a problem [math]F(x,\lambda)=0[/math]. We assume that the solution is known at [math]x^0[/math] and [math]\lambda^0[/math]. To avoid the singularity of the Jacobian, and therefore the breakdown of Newton's method, at turning points, [math]x[/math] and [math]\lambda[/math] both become parameterised by arclength ([math]s[/math]), and we end up with an augmented system of equations to solve: [math]F(x,\lambda)=0[/math] [math]\left(u-u^{0}\right)\mathrm{d}u^{0}/\mathrm{d}s+\left(\lambda-\lambda^{0}\right)\mathrm{d}\lambda^{0}/\mathrm{d}s-\Delta S=0[/math] While this seems simple enough, how does one obtain the derivatives w.r.t [math]s[/math]? Not a single text seems to mention this. Ideas that spring to mind are forward differences using, say, cubic splines; however, that seems horrendously inefficient to me. There must be a better way! Thanks, TM

Thanks for the reply. The system I'm considering can definitely be linearized at the fixed points, so that will work fine. I haven't got access to the book you mentioned, but I did find a nice summary (again, for 2-D systems) in James Robinson's "An Introduction to Ordinary Differential Equations" (Cambridge University Press). To clarify my question further, what if, for example, you look at the eigenvalues in a 4-D system, two of which are real and > 0, and two of which are complex conjugates of each other, with real parts < 0. The real parts would then dictate that it is a saddle, but that combination (mixture of real and complex) isn't covered by any 2-D case. Is it then a special type of saddle? I hope that makes sense! If not, I'll try to clarify even more.

I'm familiar with the classification of fixed points of linear dynamical systems in two dimensions; it's readily available in many a book, as well as good ol' Wiki (http://en.wikipedia.org/wiki/Linear_dynamical_system#Classification_in_two_dimensions). However, what happens with higher-order systems, say, three-dimensional? In that case, you'll end up having three eigenvalues -- presumably, different combinations of their signs give rise to different fixed point types. Has this been investigated? I've looked at numerous books, and all I ever seem to find is classification for two dimensions. Any help with finding a book/paper/URL dealing with this would be much appreciated!

Haha, I did notice, but I wasn't about to go nitpicking with someone who was trying help me.

Thanks, Timo. I guess that that's what it must mean. Still, it's a weird, weird way of expressing a simple concept.

Hi all, I hope this is the right forum to post this question in, rather than one of the subforums, as it deals with notation, not actual mathematics. I've attached a pic of the problem I'm looking at: In Eq. (31), an implicit ODE is given; note how the last two derivatives are superscripted with a summation/capital sigma. What does this mean? I've Googled as best I can, but no luck. The only clue I have is that one of the terms superscripted with the sigma is a matrix of coefficients. There are more examples of this notation later on in the attachment. Any thoughts on this? Even a pointer in the right direction would be helpful. Thanks, TM

Thanks for the reply, D H. I've been using optimisation thus far (both the toolbox (patternsearch, specifically) and fminsearch), and it works OK, except that (a) that approach is slow, because it's a sort of "brute force" method where it calculates a trajectory at every iteration of the optimisation, but more importantly (b) it doesn't seem to find the global minimum, but a local one, so that the solution to which it converges is remarkably sensitive to the initial guess. I don't think I explained that very well, but suffice it to say that it's rather clumsy and I'm hoping for something more elegant than that, which will hopefully either be quicker so that I can try many initial guesses, or that it'll find the global optimum the first time round. I'll keep going with the optimisation route so long and see what I can do with it, but it just strikes me as odd that I can't find much (any!) info on this type of problem.

Hi, all! Google's turned up nothing relevant (except my post of the same thing on another non-SFN forum) to help me with this problem. I'm at a bit of a loss as to how to do the following elegantly—I've had some success with a slow, "nested" optimisation for what I'm trying to do, but I'm not happy with the results. I have a trajectory described by: [math]\frac{{dx}}{{d\tau}} = f(x,y,a,b,c)[/math] [math]\frac{{dy}}{{d\tau}} = g(x,y,a,b,c)[/math] where [math]a[/math], [math]b[/math], and [math]c[/math] are parameters. The functions [math]f[/math] and [math]g[/math] cannot be calculated analytically (iteration is required), but what happens internally is unimportant anyway, I think. If I select values for these parameters, I can plot a trajectory in the [math]x[/math]-[math]y[/math] space. What I'm trying to do is find out which combinations of parameters [math]a[/math], [math]b[/math], and [math]c[/math] will make the trajectory pass through two fixed points, say [math](x_1,y_1)[/math] and [math](x_2,y_2)[/math]. Of course, this will more than likely lead to many solutions to the problem, and I will choose a set of parameters that minimises another function, say [math]h(x_1,y_1,x_2,y_2,a,b,c)[/math]. Does anyone have any idea how I'd go about finding appropriate combinations of parameters? My weapon of choice is MATLAB, in case that happens to have built-in functions for what I need. Any help, or even just a pointer in the right direction, would be greatly appreciated. This must have been done before by someone somewhere; I just have no idea what the principle is called. Thanks! TokenMonkey

An old, but still very good, book on the topic of heat exchanger design is "Process Heat Transfer" by D. Q. Kern. If you can find a copy, have a look. That should have everything you need.

My plan of attack was this: An energy balance on A will reveal dU + PdV = dQ. Integrating that whole lot and using dU = CvdT, I should get Q, not so? HOWEVER, the issue arises in trying to calculate the PdV term in B (or A for that matter). W = ∫PdV, where P = nRT/V. Great, except for the fact that T isn't constant... How do I account for this in the integration?

Seeing as [math]\gamma[/math] cannot be determined with the given information, I'm sure there is a way to do it without, I'm just not quite sure how...

Damn... So without knowing what gas is in question (or at least whether it's monoatomic or diatomic), I have no real way of determining what [math]\gamma[/math] is?

A-ha, so that (and Wikipedia) would lead me to believe that for an ideal gas, I can use [math]\gamma = \frac{5}{3}[/math]... Correct?

A-ha, so my belief is a lie! Umm... I could have sworn I read it in some notes somewhere, but chances are I remembered that statement and not the conditions which make the statement true! I thought it was a bit sketchy, but it was a convenient simplifier, not so? Hehe... Anyway, that aside, what "If the cylinder and piston are heat insulators with negligible heat capacity ..." tells me is that compartment B is adiabatic, so the energy added by the heater will not directly affect it, only indirectly. The only work done on it is P-V work of its compression. However, before I can even start thinking about that, I need to solve the first part, which is the conditions in compartment B... I still need to know one more variable... I need to work out either VA, VB or TB somehow. Is there any way I can relate the pressure in A before the heating to the pressure in A after the heating? That would be of immense help.