 So, last class we started looking at the linear one-dimensional first order wave equation, right, that is where we left off. We were looking at a problem, let me just write the equation dou u dou t plus a dou u dou x equals 0, a is greater than 0, okay. And we saw, we called this, we also called it the advection equation, we also called it the advection equation, advection equation. So, it was basically it captured the idea of just pure propagation. The propagation speed is A, right, the propagation speed is A. And we were looking at a problem, I sort of started talking about the boundary conditions, but I did not quite state what the problem was, so let me just state the problem, okay. So, we could have say a pipe of length L at the, this of course is the x direction, at the left hand side, there is a valve and maybe there is hot water to the left of that, okay. Or there is hot air to the left of that, whatever, fine. One-dimensional problem, this pipe may go on forever, but we are only interested in a length, we are only interested in a length of pipe of length L, okay. The pipe could go on for quite some distance. So, the idea is that t equals 0, we are going to open the valve, okay, right. So, at t equals 0, so what are the boundary conditions that we have and the initial conditions, so at t equals 0 we have an initial condition, at t equals 0, the u throughout this pipe has a certain value, right. You can move the datum to that, you understand. So, we can move our reference temperature or whatever it is property that you are talking about, so at t equals 0, u of x comma 0, that is x comma t is 0, okay. So, we are interested only in this length, wear that in mind. And for all t at x equals 0, right, for all time, for all time t greater than 0, in fact we do not know how long the water has been getting hot, so we do not go there, right. So, for t greater than 0, greater than or equal to 0 if you want, at x equals 0, u of 0 comma t equals, you always scale the problem, equals 1, is that right? So, this is the simple problem that we are talking about. If you were to pose it in the xt plane, in the xt plane, so that is the length L, we have prescribed conditions, we have prescribed conditions from 0 to L on the x axis and we have prescribed conditions for all positive t, that is basically what we have done, okay, is that fine? And we know that our characteristics go out in this direction, so the characteristic that goes out here, I do not care for, right, it is going out. So, we know these characteristics go out in this direction. So, this is the orientation of the characteristic, okay, is that fine? Everyone? So, whatever is the value here on the axis is the value that is going to be propagated along the character at the speed a. So, yesterday I called this point x0, I will call it psi today for my own reason. So, what happens to this, what happens to this function now, so the initial function, if I plot just the function, I plot the function u versus x for the same length L, right, it is a rather simple function just to the left of it, which we are not interested in, apparently the function value is 1, right and then it drops to 0 and there is indeed a discontinuity at that point, okay. And I do not actually care the fact that there is a discontinuity, if it bothers you, you can choose some other, we can choose something else, right, but let us start with this, let us just start with this. So, this valve is open because this is the problem we have been looking at so far, the valve is open and what you would expect is that the water propagates left to right. So, whatever value you have here is going to propagate out, whatever value you have here, I have not drawn the characteristic going from that step, you have a step there, that step is going to move along that characteristic, it is going to propagate along that characteristic, is that fine? So, in time as you go along, this step is going to come out, so at some time t, the step is going to propagate out. So, what you would expect is if this were an animation, you would expect that this step is going to travel, right, left to right, is that fine? Right, and if you think about it, you have a tap, you open the valve and what water is going to travel and you would expect something like this to happen, okay. So, pure propagation, that is all it picks up, no diffusion, nothing else, no other physical phenomenon, pure propagation, there is no decay in the size of the step, there is no smoothening out of the step, nothing, okay, it is just going to propagate left to right, is that fine? Okay, so this property is important, so if you had instead of your initial condition, instead of being a step, if it had been some other function, you can imagine now that you can choose other functions. So, you can choose a function, the initial condition, for example, to go from some value, right, go down, so some value, it can be anything, I mean, you can just pick, right, you can pick any initial condition that you want and that condition will be propagated, it will just basically flow out of the pipe. What I am trying to say is that if in your pipe, whatever it is that you have, right, whatever, there is a temperature distribution, that water is going to flow out, right, so if it turns out that the pipe, because it has been sitting there is warmer, there is a region where it got warm due to conduction through the valve or whatever, then when you open the valve, it is all going to flow out, okay, so this function is just going to propagate left to right, fine? So, you can try out a few different functions. The other thing that you can do is, at, you can try this out, for t greater than or equal to 0, you can replace the function at x equals 0, you can replace the function 0, t, instead of it being constant, which is what I have done so far, pick a function and see what happens. So, you can take that to be say cos t or you can take this along with cos t or sin t, right, or try another one. So, this is one function, try a different function, right, 0, t equals cosine 2t, try various functions and see what happens, okay, try various functions, these are varying in time, see what happens, is that fine, okay, let us get back to this picture, so this characteristic, a typical characteristic intersects the x coordinate, right, at a point psi, yesterday's class I called it x0, today I will call it psi, at a point psi. So, what is the equation of this line, right, remember that this line, you are going to propagate a distance a in unit time, right, the property is being propagated at a speed, a units per second. So, it is going to be, x is going to be psi plus a t, in fact in reality the differential equation, the differential equation that we are actually solving is dx dt equals a at t equals 0, at t equals 0 that tells you x equals psi, that is what we are doing, you understand. And I am integrating that to get essentially x equals psi plus a t, because a is a constant, this is integrated out, but I want to make that statement a little more precise, right, I want to be a little more careful with that, okay, because I will use this fact later, a in this case is a constant everywhere, but what is more interesting to me is, a is a constant along that characteristic, a is a constant, right, but I want to say, I want to put it in a peculiar fashion, a is a constant along that characteristic, that it happens to be constant everywhere else I do not care, right, okay. So, a is a constant along that characteristic and in fact it just turns out that x equals psi plus a t, is that fine, okay, so that is the equation of that line. Now, so what does this equation do, what have we discussed so far, what this equation does is, if on this length L you prescribe some function, this equation will propagate that function, because a is greater than 0, it propagates it left to right, okay. So, I repeat, at t equals 0, if you prescribe u equals f of psi, u of x, 0 equals f of psi, this is going to be propagated how, what is going to happen to this f, so something of the form f of psi, but what is psi, psi in fact from our equation is x minus a t, so given the initial condition, so now we make the jump, given the initial condition f of psi, in fact given any function f, right, which has the appropriate derivatives as far as we are concerned, then f of x minus a t is a solution of this equation, is that fine, okay, right and you can verify that, what is dou f dou x, dou f dou psi dou psi dou x, chain rule, right, what is dou f dou t, dou f dou psi dou psi dou t, substituted into that equation, which is how we would verify whether it is a solution or not, what is dou psi dou x, dou psi times 1, dou psi dou x is 1 and what is dou psi dou t times minus a, is that okay, everyone, so this is indeed a solution, so you can just substitute there and you see that dou f dou t plus a dou f dou x equals 0, so it is satisfied, the equation is satisfied, okay, so any function of this form, any function of this form, equation is satisfied, so you remember where we started this, earlier when we started off I said can we guess the solution, right, so now we are probing, we are looking at, there is a way by which we can construct a solution, there is a way by which we can construct a, geometrical way by which we can construct the solution using characteristics, right, those lines were called characteristics and using those lines we can actually construct the solution, geometrically, okay and from there maybe we can get some, but now what we have done is, we have seen from that just using a little analytic geometry that we are able to say that any function of this form is a solution as long as these derivatives make sense, okay, you can complain that I took a step function what is going on, right, okay, we will see, we will encounter a lot of those situations but as long as these derivatives make sense, right, something of this sort is going to be a solution, everyone, right, so how do we do, where do we go now, I want to still be able to guess the solution, we have a general form, I want something a little more specific, okay, so we will repeat what we did, we will repeat what we did with Laplace's equation, you can of course if you give me a function f of psi, it is possible it is on a finite interval of length l, if you give me a function f of psi then I can use possibly Fourier series, I can use periodic extension, okay and I can use Fourier series to represent this function, okay, why do I use Fourier series, maybe I am getting a little ahead of myself, if I look at this function do I have to do t, so see please remember I am now, I am trying to explain the process of how we go about guessing, see this equation is simple, you can easily, I am pretty sure you can sit down, but the process that we go through is very important, it is more important, okay, so I see first derivatives here, so from my differential equations I am thinking exponential, right, I see first derivatives I am thinking exponential, so okay I am going to get an exponential, but an exponential can go two ways, an exponential can go two ways, in this case the function value did not decrease, in this case the function value did not decrease, right, so I do not want an exponential in the form of e power minus x kind of a thing because the function value will decrease, I want only pure propagation because that is what this equation represents, fine, which means that, so if I have an oscillation it is going to continue oscillating, that is what, that oscillation is going to be propagated, that is all that is going to happen, okay, so the minute you say oscillation not decay, we have something that looks like a Fourier series, is that fine, so what we will do is, we will represent, if you give f of psi, I can write this f of psi in terms of Fourier series, okay, and how does that go, that goes over a summation over n, not going to bother with the limits right now, okay, go from minus infinity to plus infinity because I am going to write it in a, or because I will take it from 0 to l, I meant from minus l to plus l, it depends on the lengths of the interval, you have to be careful, so I will just leave it vague, I will just leave it as n, right, depending on what exactly we pick, the n range will have to be picked appropriately, whether it goes from 0 to infinity or minus infinity to plus infinity, okay, so then we have an exponent i n wave number 2 pi psi by l, is that fine, everyone, is that fine, okay, and then you can do, you can take l is 2 pi, you know the 2 pi's go away, all that kind of stuff, that is fine, so you can have a function of the sort as a solution, so let me swing over here to the other side, so what we have is we will get back to the, so what it basically says is that my u in general, I can write it as summation over n, a n exponent i n 2 pi x minus a t by l, fine, okay, so we started off looking at this equation, right, so and by sequence of investigations trying to find out what it is, we were able, we were lucky, not always possible that we were able to do this, but we were lucky, we were actually able to guess the form of the solution, okay, a general form of the solution, fine, okay, so we can construct it geometrically using characteristics or given the initial conditions, you can try to see whether you can use Fourier series to actually construct the solution, okay, right, are there any questions? This course of course is not about analytic, it is not about analytic solution, we are interested in analytic solutions only because we want to know how the solution to the equation behaves, so that when we do the numerics, we are able to compare the, compare it to the solution, right and where the numerics does not work that I mean, I am sorry, where the, where you do not have an analytic solution, right, generally then we will appeal to some theory of differential equations, so that we can figure out, right, theory of differential equations, for example, when we talked about Laplace's equation having a maxima or a minima on the boundaries, that is the kind of a result that would come out of theory of differential equation, right, then we would appeal to theory of differential equation saying this is the problem that I have, does the mathematical theory tell me anything about the solution before I even solve it, okay, that is what the theory does for you, it will tell you something about the solution before you even solve it, okay, it is very important, right, so the theory part is very important, so then we do the numerics and as a first cut then you can compare it to see whether what the theory has told you is satisfied by the numerics, okay, in this case we have constructed a solution because this is actually a simple model equation for the kinds of equations that we are going to solve at a later point and we will run into some of the difficulties that you would run into with say the full Navier-Stokes equations or whatever, you will run into them with this equation, right, there are certain elements of this equation that so we start with a simpler equation for that reason, okay, so we have a solution but now to the basic point of this course I do not want to solve this analytically, if I did not know the analytic solution how would I solve it numerically, what would be the method that we would use to solve it numerically, fine, okay, the usual progression would be well we use central differences for Laplace's equation that seemed to work, why not use central differences here, that is you use for the equation dou u dou t plus a dou u dou x equals 0, you use a central difference representation for this and a central difference representation for that but I look at the boundary condition and you know at t equals 0 I have a condition but I do not have anything beyond that, I have a concern, okay and for the central difference here on the right hand side there seems to be some issue but I will ignore that, right, but it is clear that I do not have if I am going to, if I am going to march in time I am given conditions at t equals 0 I have nothing beyond behind that, right, so if I were to draw grid lines like I did for Laplace's equation, if I were to draw grid lines I really do not know how to take, I really do not know you know whether I do not know this value, I do not want to guess that value, the future value that would be like guessing the future value, you understand the issue here and I do not want to start here because then I do not know anything behind it, if I represent the equation here I do not know anything behind it, right, I am only giving you a complaint, I am not saying that it is impossible to use central difference, right, I am sure we can think of ways of getting around it but my objective is that is not my objective, right, so I have central differences here, I want to keep life easy, so I will use a forward difference here, is that okay, is that fine, that seems reasonable, that I use a forward difference, I use these two points, I use a forward difference at that point, I use a forward difference at this point and I use a central difference for the spatial derivative, okay, so for any arbitrary grid point, for any arbitrary point okay, so p, q, we can use central differences, so I will zoom in on this, so for any arbitrary point p, q and as we did in Laplace's equation maybe at least in the x direction we will take equivalent rules, right, in the t direction also we can take equivalent rules but we will see what happens here, so p, q, this is p plus 1, q, this is p minus 1, q, this is p, q plus 1, these are the points, right, I have gone with p, q because I have already used i for square root of minus 1, right, up here I have already used i for square root of minus 1, so I do not want any confusion, that is why I have gone to p, q, okay, okay, so this very often in books, textbooks and so on you will see this referred to as a stencil, okay, this is just for you to get the jargon, you will see this referred to as a stencil, right, so this stencil occurs everywhere, so you can take these four points and presumably solve wave equation and what we are doing here, very important, what we propose to do is we propose to represent the wave equation at the point p, q using p, q and these three other points, okay, so dou u, t, dou t, dou u, t, dou t for difference u, p, q plus 1 minus u, p, q divided by delta t plus the truncation error which I am not, so I should actually write an approximate there as a truncation error, dou u, x, dou u, dou x is approximately up plus 1, q minus up minus 1, q divided by 2 delta x, is that fine, so what do we have substituting into our governing equation, so you have up, q plus 1 minus up, q divided by delta t plus a times up plus 1, q minus up minus 1, q divided by 2 delta x should be approximately 0, we set it equal to 0 and this is the objective, this is what I am trying to get, this is the objective, the objective is up, q plus 1 can then be written as I take everything else, you notice this is only one that is a time level q plus 1, all of these are a time level q, so all the ones that are a time level q I am going to shift over to the right hand side, so I have up, q minus a times up plus 1, q up minus 1, q by 2 delta x, I miss something a delta t, a delta t, is that fine, is that fine, everyone, so this is what we have, so in fact this expression a delta t by delta x is going to appear so often, we are going to give it a symbol, we are going to call it sigma, this appears so often that we are going to call it sigma, so that this is going to be up, q plus 1 is up, q minus sigma by 2 up plus 1, q minus up minus 1, q, okay, we have what I would call an automaton, right, so given something a time level q, you can then increment it to the next time level, you understand what I am saying, you can find out what happens at the next time, given at q plus 1 you can go to the next time level, right, so at each time level you can move forward, you have an automaton, so if you are given an initial condition, you are given an initial condition, you can march that initial condition forward in time, right, q plus 1 is given explicitly in terms of, explicitly in terms of expressions that are occurring at level, at time level, time q, time level q, okay, so this scheme is called an explicit scheme, the scheme is called an explicit scheme, simply because you have q plus 1 occurring and this is, these are all at time level q, we are able to solve for it, it does not occur in an implicit fashion, is that fine, it does not occur in an implicit fashion, so this is called an explicit scheme, explicit scheme, sometimes you will hear it being called an Euler explicit scheme, we will call it forward time central space, FTCS, forward time central space, is that fine, okay, everyone, right, so you can actually code this, you can go ahead and code this, right, but this is different, this is a little different from Laplace's equation, so we are going to do a little analysis, right, there are two things that we have got going for us now, one is when we did the stability analysis for Laplace's equation, we substituted exponentials for the error term, we substituted exponentials for the error term and the error check whether the error term decays or not, what happens to the error term, okay, in this case, we have a similar set, they are there, in that case that was, they were if you think about, if you remember back, they were Eigen functions of the, the exponentials were Eigen functions of the Laplace operator, in this case, they actually represent the solution, the equation is a linear equation, the equation is a linear equation, you understand, remember linear equation basically means, equation is linear basically means that, equation is linear basically means that if you have, if l is linear, right, what does it say, l of u plus v is l of u plus l of v, right, l of u plus v is l of u plus l of v, fine, okay, that is basically what, so l again and usual stuff, I mean the l of alpha u plus beta v is l of alpha l of u plus beta l of v, I mean this is, this is the usual description of something being linear, right, which is why we know that A x equals A x, y equals y is linear, right, or the classical, when we talk about a straight line, this is the, this is the usual trap that people fall into, so if you take the equation y equals, and this is an aside, y equals A x plus b, this is not linear, right, linear it necessarily passes through the origin, this is not linear, it is a straight line, it is a straight line, it is not linear, so if you say linear, wait a minute, it is a straight line, does not that mean linear, if that confuses you then think of curvy linear, right, linear, all it means is a line, curvy linear means curved line, do you understand what I am saying, so but in this case linear, this is, this is a, this is a technical term, right, the function is linear, the operator is linear, basically means that l of alpha u plus beta v is alpha l of u plus beta l of v, okay, it is a very precise definition, as a consequence of which y equals A x plus b is not linear, though it is a straight line, okay, unless b equals 0, fine, okay. So that equation is linear, we ask the question again, so if you have dou u dou t or just for, just to keep this, just to keep this clean, let me write a very general equation, so if you have dou v dou t plus A dou v dou x equals some h of x, it could be h of x, t, h of x, keep it as h of x, right, the solution is v, the solution is v, okay, the solution is v, but v is disturbed, a perturbed v, it has an error, right, in Laplace's equation in case I called it e, in this case I choose to call the error u, for obvious reasons that I already have an equation for u. So if the solution, if our candidate solution is v plus u, if our candidate solution is v plus u, okay, and you substitute into that equation, because this is a linear equation, what is going to happen, what is the equation governing u, our original equation, you understand what I am saying, so if I substitute this back in, I get dou v dou t plus dou u dou t plus A dou v dou x plus dou u dou x equals h of x, I am sorry A dou u, oh it is very important A dou u dou x, thank you, right, this combination, this, this and this, the combination of these three satisfy each other, they knock each other out, the combination of the three knock each other out, leaving us dou u dou t plus A dou u dou x equals 0 as the equation that governs the error term, looks the same, suspiciously the same as the original equation, right, and you would expect that because it is a linear equation, even in Laplace's equation case, the equation governing the error was the same as the equation governing our original function, our solution, is that fine, okay, right, so what do we have, we go from here, so I want to substitute now in dou u dou t plus A dou u dou x equals 0, which is now the equation for our error and I want to ask the question, when does this decay for this particular scheme that we have chosen does it decay, okay, for the particular scheme that we have chosen does it decay, I want, what was our u, u is like summation over n A n exponent i n 2 pi by l x minus E t, what do you say, okay, and because the equation is linear, right, also equals summation of u n over n, right, where the individual u n's, where the individual u n's are A n exponent i n 2 pi by l x minus E t, okay, we will play the same game we have played for Laplace's equation, we will say for which wave number, is there a wave number, is there a bad wave number, right, if we can say that for the worst, for the wave number which has the largest gain, as we march in time, see this is what we are doing, we have a perturbation or an error u in our solution, okay, we want to know that if I march from time t equals 0 to t equals delta t, 2 delta 3, 3 delta 3, so on, as I march, is that error going to grow or is that error going to die out, does that make sense, is the error going to grow or is the error going to die out, that is the question that we are asking, so basically what we will do is we will pick one wave number, then ask the question for the wave number which has the largest gain, what happens, right or is there a wave number that has a problem, that is essentially what we are looking for, is there some particular wave number for which we are going to have a problem, okay, so because you know I have already got subscripts p and q, right, so you please allow me to drop this n, we understand now that I am going to take a look at the nth wave number, right, so you just otherwise we will just be carrying along a lot of these subscripts, so let us just assume that we understand that I am dealing with the nth wave number, so what do I have now, my automaton was upq plus 1 is upq minus sigma by 2 up plus 1q minus up minus 1q and as I did earlier what is the relationship between up plus 1q and upq, so if my x is p delta x, right, if my x is p delta x and q could be you know we do not really need it but q could be t could be q delta t, you could keep delta t constant if you want but it does not matter, if my x is p delta x and t is q delta t, okay, if my x is p delta x t is q delta t, what do we have, what do we have up plus 1q equals, where do I get this now, an exponent i n x is p plus 1 delta x minus a t at the time level, I will leave this as t, okay and this is nothing but an, so you can just check, so it is e power exponent i n, I forgot to 2 pi by l, i n 2 pi by l delta x into upq and in a similar fashion because there is a p delta x here which is x, this is basically x plus delta x, right, this is basically x plus delta x and up minus 1q in a similar fashion is exponent minus i n 2 pi by l delta x upq, fine, so I can now substitute into my forward time central space upq plus 1 is upq minus sigma by 2 up plus 1 is this one, this is a mess but these two expressions are the same, I will just redefine something, I will redefine that as theta equals n delta x 2 pi by l, right, I mean I could take l equals 2 pi and that will go away, it does not matter, so this is e power i theta minus e power minus i theta into upq, everybody with me, okay, I can divide through by upq, right, I divide through by upq and I get the gain G as upq plus 1 divided by upq 1 minus sigma by 2, what is e power i theta minus e power minus i theta 2i sine of theta, remember I am using Euler's formula, e power i theta is cos theta plus i sine theta, so this gives me the gain as being 1 minus i sigma sine theta, is that good news or bad news, what do you say, so you want the gain, the magnitude of the gain to be less than 1, right, you want mod G to be less than 1, okay, but what is the mod G, what is the magnitude of G, mod G is or mod G squared to be less than 1, what is mod G squared, 1 plus sine squared theta, sigma squared, 1 plus sigma squared sine squared theta, see the only parameters that we have to choose, right, now this brings this whole thing start, you know, we now face the fact, you cannot change A, the only parameters that you have to choose are delta x and delta t, okay, so sure, I mean theta depends on delta x and so on, but there will be that, we know that there is a highest wave number that we can represent, right, I mean so that all of that is very clear, we have no problem issue with that, so the only parameter that you can change is sigma, right, it is not complex, I mean it is real, so sigma squared we are stuck with it, am I making sense, we are stuck with it, so sigma squared, there is no sigma value for which this is going to be less than 1, so this does not work for us, theta equals 0, yeah it equals 1, right, but theta equals 0 is that dc component essentially, so theta equals, it does not help us, we are not getting anything, right, is that fine, so forward time central space, FTCS is unconditionally unstable for, be very careful now, be very careful, when applied to one dimensional linear first order wave equation, right, yeah, you have a question, why should mod g be less than 1, see when you go from time, when you go from time q to q plus 1, if you go from time q to q plus 1, mod g is greater than 1, right, so you are basically, look at this as a sequence, so you have a, you are generating a sequence, I will leave out the p part, you are generating a sequence which is uq, you are generating a sequence of u's indexed on q and we are asking the question, does this converge and we in fact wanted to go to 0, u in this case is the error, right, v was our original solution, u in this case is the error and we are asking the question, does u go to 0, u is the error in v, you can have a u which is non-zero, this can still satisfy this equation, you can have a u which is non-zero in the Laplace equation case because we have a unique solution for Laplace equation given a boundary condition, u equal to 0 was the e equal to 0 was the only solution, but here you can have a conditional boundary solution which is non-zero for this same equation. u can be a solution which is not, but it does not satisfy our original equation, does it satisfy our original equation including the boundary condition, well if it has homogeneous boundary conditions and homogeneous then you can add it, then you can add it, right and then the question is what happens at the initial condition, so it has to be 0 at the initial condition and somehow it magically came up, you understand, okay, right. What we are basically saying is even from our numerics, typically see where these things come from is even from our, the source of this error, it is not that there is an error that you have injected into the solution and does it decay, there is a source of the error which comes from our, say from our round off, right, from various reasons, there is a source of error that you have, what happens to that source of error, right, it is no different from the whistling that you hear in an amplifier, the whistling that you hear from an amplifier, it is not that there has to be a input, just thermal noise, there is thermal noise in the resistors in the circuitry, that thermal noise is enough, so the issue is the question that you have is it is not that you have, the disturbance is there, you have the thermal noise which is the equivalent here would be I have round off error, right, every step I have these errors, I am making these errors, what I want to know is do those errors, errors grow or those errors do not grow, so in a sense I am not actually answering the global question, what you are talking about is answering the global question, right, so the stability analysis that we are doing, both what we have done here, what we are doing now and what we did in Laplace's equation is a local one, we are only asking at a given grid point what is happening to the, what is happening to the solution if I were to integrate it out, but not for all x, am I making sense, okay, say I see where you are coming from, you are saying it is a homogeneous the equation, this equation is a homogeneous equation, it basically does not disturb, you can add any amount of this homogeneous equation solution that you want, some constant time this homogeneous equation, but normally in your differential equations if you look at it, the way it works is the homogeneous part actually takes care of the boundary conditions, in this case even the boundary conditions are 0, so in a sense this does not, this is truly sort of in the null space of that operator, it does nothing, okay, but potentially if it grows it can blow up, the problem that you have is if I, the difficulty that I have is if this uq, if the gain is greater than 1, the sequence that I get is diverging, so I am going further and further away from my v that satisfies the original equation. No, no, there is no issue of boundary condition here because I am only working at a grid point. No, no, I am not looking at a solution, that is the whole point, no, no, no, no, the solution is v, the only question that we are asking is there a wave number that is unstable in a sense, is there a wave number that is going to grow, that is going to become unbounded, if I disturb that wave number or there is an error in that wave number, is there a wave number that is going to grow, that is all, that is the only thing that is happening, is that fine, right, so that is basically it, so we are generating at this point p, so though I have removed it, I will stick it back there, at this point p, we are generating a sequence indexed on q and we are asking the question what happens now at the point at that x location, what happens as q goes, but it is the analysis that we are doing as a local analysis, it is not a global analysis, maybe at one of the other classes I will do a global analysis where I take the boundary conditions also into account, right, that is really what we should do, okay, here is sort of I beg of saying well this we are engineers, this is, you know, we get this, this gives, and this is already unstable, forget doing the full thing, this is already unstable, right, we are already out of luck, this is already unstable, fine, and you can try to implement this and you will see that it is unstable, try to implement this, so this analysis actually works, okay, this analysis actually works, which is not justification enough, I cannot just say that it works and therefore it is right, but what I am saying is this analysis actually, so this gives you this q's, the sequence of q's that you are going to generate using that automaton which is FTCS, if there is any disturbance it is just going to blow up, is that fine, it is going to diverge, if there is the smallest disturbance that disturbance will grow, that is the key, fine, is that okay, is that okay, so this is where we are, FTCS did not work, okay, so the next thing is, next obvious possibility is we did forward time, why do not we try forward space, now we are groping in the dark, if it did not work, now we are sort of groping around, trying to figure out what to do, so forward time, we use forward time, let us try forward space, is that fine, so Monday's class we will do forward time forward space FTFS, right and let us hope that it works, fine, okay, I will see you then.