Introduction to Partial Differential Equations

hello everybody and welcome to another video today this is actually the first video in our series discussing partial differential equations this is going to be a multi video endeavor where we're going to be talking about some of the subjects such as these namely we'll be talking about modeling engineering problems using partial differential equations we'll talk about a few famous partial differential equations such as the 1 & 2 D wave equations and heat equation will also look at solutions of these equations using techniques such as a separation of variables we'll also talk about eigenvalues and eigenfunctions of partial differential equations we'll also talk about Fourier analysis Bessel functions and numerical solutions to partial differential equations so if all of this sounds good why don't we jump right in and talk about what is a partial differential equation alright so the easiest way to start talking about partial differential equations is to actually back up a little bit and remember our discussion on ordinary differential equations so if you remember here our I deal with ordinary differential equations or let's just refer to these as OD E so we don't have to keep writing that out here is the idea with an OD e is you might have some function let's call it u and this might be a function of one independent variable T here and this differential equation that describes how this function evolves here might involve derivatives of this function so in other words you might have something like you might have the second derivative of this function with respect to the independent variable T in this case and then maybe that's added to I don't know the first derivative of that function with respect to time plus maybe the function itself here right U of T here right and this might even equal some external forcing function let's call it maybe F of T here and you know what all of these terms the function itself its derivative its second derivative they might have constants in front of it like an a a B and a C these coefficients and finally with ordinary differential equation you might have some initial conditions associated with this if you remember our lecture earlier we discussed that with the second-order differential equation you actually need two two initial conditions to fully satisfy this so you might say that this problem also comes along with this function at a certain time like usually time zero is the easiest spot is equal to some value and maybe its first derivative here or UDOT at that same time is equal to some other value here so this is our typical format of an ordinary differential equation this might be used to describe something like the deflection of a block all right you might have some mass here attached to a spring and a damper and I think you've all seen the situation where if I want to measure the deflection let's call it U as this block moves up and down this differential equation might describe that motion here right so the idea with an ordinary differential equation here in fact this one is actually linear in this case here but the concept with this differential equation is that we are trying to find this function U of T here right this is equal to something some function but the key here is that there's one independent variable right time is the only independent variable if you know what this function is as a function of time that will describe how does this block move up and down at any given point in time all right so the goal with solving this ordinary differential equation again was just find this function u now what if instead of just this single point here like a single mass that's moving up and down what if we had like something more complicated so for example let's say you had a continuous string or rope maybe it looks I don't know what it actually looks like I'm gonna draw it kind of in some exaggerated fashion maybe this thing has length L here so let's draw X maybe as the this x axis here and then the vertical thing is still u then maybe the deflection of how much has this string moved away from this x-axis here right so now in this case we see that if I also want to describe the deflection of this string so if I want you to describe the deflection of string we see now in this case as this string moves up and down you see it's actually a function of two variables it's a function of when you look at the string and it also is a function of where along the x axis or the spatial variable do you look at it so now U is actually a function of time and space so there are two independent variables right so in this case you have two independent variables you've got T here which is time here and X here which is some sort of I guess we can call it space right because it's some spatial location here at an x distance here so in this case the deflection of this string is not going to be governed by an ordinary differential equation I need to find this function U which is a function of two independent variables and that's what makes this a partial differential equation so let's just go ahead and write that down here and in order to again avoid writing the words partial differential equation over and over and over let's just call these things PDE s from here on out okay so a partial differential equation or a PDE is basically an equation involving one or more derivatives or partial derivatives of an unknown function of two or more independent variables so let's just write that out down here right so this is an equation involving one or more partial derivatives an unknown function of two or more independent variables okay and the other thing that's kind of key to look at here is now we have this word partial derivative right earlier in the ordinary differential equation we were able to just use a whoops sorry gosh back here I missed a D here whoops sorry I should have had a D over there in my differential equation equal expression but anyway these were DS right because there was only one independent variable so you can only take a derivative with respect to one variable here but now we have a partial here so instead of using the D terminology we're going to start using the partial terminology here right to determine which variable are you taking a derivative with respect to okay so one other thing that we may want to talk about right now here's let's talk about the order of a given partial differential equation so the order of a PDE is the order of the highest derivative of whichever partial shows up in the equation and we'll look at an example here in a second but maybe let's just go ahead and write that down here so the order of the highest derivative okay is called the order of the equation okay so let's take a gander at what a general form of a PDE might look like so let's go ahead and maybe let's leave our OD e up here and we'll go ahead and erase some of this down below and just write out a general form of a partial differential equation here so in general we saw that this partial differential equation let's write those PDE like a general format so we saw that this function let's call it maybe big f here right this is going to involve potentially more than one independent variable so you could have an X 1 and X 2 all the way up to an X n so these are independent variables that could be time-space whatever the independent variable is here for your expression here right but now what makes this interesting here is that we're again looking for this function U but the you could now be dependent on all of these as well as their partial derivatives so you might have the partial derivative of U with respect to the first independent variable x1 you might have the partial derivative of U with respect to the second independent variable etc all the way up to the partial derivative with respect to the nth here right and you know what you don't have to stop at the first derivatives write this equation might have second derivatives it might have mixed partials so to keep this general right you might have something like partial squared u with respect to x1 and then with respect to x2 alright and you see you can start mixing all these partials up as much as you'd like to get all of these different terms in here and then maybe we'll just leave it as that like this is a general expression and this might equal I don't know 0 or something like that right it might equal some forcing function but you could always subsume that forcing function over here so in general this is sort of a very open-ended way that you might go ahead and define a partial differential equation so partial differential equations we're gonna see can be used to actually describe a wide variety of engineering and physical phenomena such as sound you know heat electronics things like that why don't we go over to PowerPoint real quick and I'll show you a couple movies of some example PDEs particularly ones that we might take a look at here and during the course of this lecture so I wanted to give a couple of examples of some systems that are modeled by partial differential equations so one example here is a vibration of a string so we're actually going to take a look at this in a little it and this is what's known as the 1d wave equation we'll also see that you can extend this idea to multiple dimensions so in other words you could have a vibration of instead of just a simple string you could have a vibration of a two-dimensional plate another place where we see partial differentials in engineering is things like the diffusion of heat so we'll take a look at a system where you've got some heat distributed in some fashion and we'd like to see how that heat spreads out over time so again the point of this is just to show a couple of movies to give you an idea of what solutions to partial differential equations look like so again if you remember we are just trying to find the functions which describe this motion or this diffusion of heat and we have to find these functions that satisfy our particularly a particular governing partial differential equation so let's jump back to the blackboard here and dive into a little bit more of the introduction of PDE s all right so now might be a good time to discuss a little bit of notation so notation with PDE s it's very common to use partials or derivatives right and instead of writing partials all the time here it's very common to use subscripts to denote these partials so in other words what you might see is something like again there's a function let's call it u that we're looking for this function you had multiple independent variables like T X Y Z and anything you'd like right so one quick way to denote the partial of this function with respect to one of those dependent variables is just to denote this as a subscript so for example use of X is shorthand for partial of U with respect to X right and you might things like whoops you might see things like u XY here right so again this is shorthand notation for partial squared u with respect to x + y right which again I guess is even shorter hand notation for the very verbose way of saying take the partial of U with respect to X right do that first and then take the partial of the result here with respect to Y so again all of these are equivalent in a notation here all right um so in physics and engineering it's very common to also see instead of just these subscripts you might see what's called dot and prime notation here so what that means here is usually we're gonna see later on that the two most popular independent variables are you have T for time and then something else for space here so in other words what you might see here is you might see something trying to write down the second derivative of U with respect to the time variable again usually that's T here right and what you're gonna tip eclis see people right here these engineers and scientists will write something like this you might see u dot dot here right so the dots on top here are basically saying this is the same thing as saying UT T here right which is partial square u with respect to T squared so again the dots on top are going to be used to denote a derivative with respect to time here right and we can contrast this with say the second derivative of U with respect to the spatial variable maybe again in this case let's let's use X as our example here right so the alternative notation you might see for that here is u prime prime here which again you could use our subscript notation we talked about earlier and say this is just UX X here right which is partial u squared with respect to x squared here so again the dots usually denote a derivative with respect to time and the Prime's denote derivative with respect to space ok with some of that notation out of the way why don't we to also talk a little bit about how about classification of P DS we saw earlier when we were discussing our OD e's there are all kinds of ways you could describe an OD even to call it linear you could call it nonlinear there are all these other ways a homogeneous non homogeneous what order all that kind of stuff here right well there's a very sim lexicon here for four PDEs here so let's just go ahead and say that it looks very similar to Odie's so if you remember earlier here let's go ahead and write down just like a general PDE so again it's probably impossible to enumerate all the different possibilities here so let's just use an example PDE with one or two independent variables and partials to kind of illustrate the fact here so again you said the idea with the PDE is I'm searching for some function U let's make it a function of two independent variables you know for giggles let's make it how about X and wire right okay and we said the idea with the PDE is I'm searching for this function which satisfies some relationship of its derivatives I don't know let's make something a partial of this with respect to X here right plus maybe you leave a little bit of a gap here how about this same function with respect to say y plus I don't know maybe maybe the function itself X Y right is equal to some maybe maybe some forcing function let's call DX Y something like that here right now when we were looking at OD YZ we saw that you know you could also tack on coefficients here in front of all of these and if all of these coefficients were constant and if all of the functions the use here didn't have anything crazy like squares or square roots or sine or cosines this was what's referred to as a linear OD e well it's very similar in the PDE case here except now these coefficients are actually they don't have to be a constant they can be a function of x and y here just not the function U that you're looking for itself here so you could have an A which is a you know a function of x and y you could have a B here which is a function of x and y and you could have a see coefficient here which is a function of x and y here okay and as long as you can write it in this format in the sense that the there are no squares or anything like that or second or square roots or cosines here in other words if the if the degree is is one here and I mean we should make that make a note of that here so if the degree function you and derivatives whoops derivatives I spelled that a little bit wrong rivet is or one and the coefficients are as described above right then this PDE is called linear and we're going to see there's some nice consequences of having linear PDE s now the thing I want to mention here is this is the degree here let's just make a note here that that the degree is not the order of the PDE remember we talked about the order earlier the order is the how many derivatives are you taking the order of the derivatives here so you could have second third fourth plenty of mixed partial derivatives here of order greater than one as long as the degree and as long as there is no squares or cubes or or or you know ease or anything like that to any of them the degree will be one right and then we can call this thing linear alright okay great there's also another classification here within PDE s here and then M again it involves these coefficients here so instead of rewriting this whole thing here maybe let's just make a little line here and we'll note here that the other way that you could write this here is if this coefficient you know earlier we had a of XY right this was like let's just look at the a coefficient if you could instead write this here of a as a function of X Y and the function you're looking for U of XY and similarly for the B of XY and the C of XY these coefficients if they could all be rewritten very slightly to be a little bit more complicated right like this well in this case this is no longer referred to as linear here linear was over here right this is how you write the coefficients for a linear system in this case this is referred to as quasi linear okay the so very small change here right if a system can if a PDE cannot be written as linear or non-linear or a quasi linear it's basically referred to as nonlinear so the last bucket here is nonlinear right which is basically not linear and not quasi linear then it's basically nonlinear the last may be adjectives that we could can use to describe this here is coming back to sort of the the forcing function right this was the external input that you might have seen in an ordinary differential equation here right so again you could maybe very roughly think of this as a forcing function right and the idea here is that if this is zero here right you have what's referred to as a homogenous one otherwise it's non homogeneous here so maybe we're kind of running out of space or maybe we can just kind of keep that in the back of our head here right if D of X Y if this is zero gosh I really don't wanna rewrite this whole thing well let's just state it right if this is zero you have the homogeneous PDE if it's nonzero you have a non homogeneous PD just like you had with ordinary differential equations so tell you what let's look at a couple examples I think we have enough of this notation and classification down let's look at a couple examples here all right so let's look at some famous ones here or popular examples of PDE s all right so and actually you know what we're gonna I should probably make a note here that all of these examples here are actually linear PDEs and actually i think they're all second order right of second order so the highest partial you're gonna see here is a two here okay so just some examples um here example one here how about something like this partial u squared with respect to t squared is equal to some coefficient let's call it C squared partial squared u with respect to x squared ok so again let's just be compose this we see here that again we're looking for this function u it's a function of two independent variables time and space here this is actually the famous one-dimensional wave equation and we are going to be taking an in-depth look of this here in a second okay so again we see that it's linear here it's a second-order right you have a second partial here and it yeah it's basically linear okay let's look at another example here how about if all you had difference here was instead of a second derivative here on the x side you had a single duritz derivative here so this could again equals c-squared partial u squared partial x squared so again this is still a linear second order PDE here right because there's a second derivative here but we're gonna see that this one little difference in partials on the x side makes a makes a big change in the solution space here this is what's referred to as the 1d heat equation okay how about number three here another example here this is the two dimensional laplace equation you might have something like partial square u with respect to x squared it plus partial u squared with respect to y squared 0 okay so this is the to deal applause okay another example how about number four here how about the two dimensional Poisson which is basically the two dimensional Laplace with a nonzero forcing function so it's partial squared u with respect to x squared plus partial u squared with respect to Y squared is equal to some function f of X Y here so now instead of having the to deal applause here you the to deal the Plus on equation how about another one here let's go look at two more examples just so we all get the flavor here how about the two-dimensional wave equation so it's going to look very similar to the 1d wave equation but there's gonna actually be two spatial variables here so we still have partial U with respect to T squared is equal to some coefficient let's call it C squared but now we've got partial u with respect to x squared plus partial U with respect to Y squared okay so this is now the 2d wave equation and also this is one that we're going to take a greater in-depth look here later on in the discussion here okay and then finally it's part of our sixth example how about let's look at the 3d Laplace equation so it's partial u with respect to x squared plus partial U with respect to Y squared plus partial U with respect to Z squared so what if you had three spatial variables here and that's equal to zero here so this is the 3d Laplace so again this is what partial differential equation sort of look like and as you can see what they all involved here is trying to find this function U and U has a very specific property that when you take its mixed partials with respect to whichever variable whichever independent variable you combine them in all the fashions that are described here you get the left-hand side is equal to the right-hand side all right yeah and maybe again maybe the only other thing to note like we said all of these are linear PDE s of second-order and actually example four is the only non-homogeneous PDE that we've we've written up here everything else is is homogeneous okay now with this in mind what we should also think about his now that we're able to define the PDE we can start thinking about how can we go ahead and solve the PDE and again I think it's so helpful to recall our discussion on ordinary differential equations you remember here with Oh des if someone just gave you an Eau de and said solve it that's actually yields an infinite amount of solutions and that's similar in many of these PDE case here in the sense that many dissimilar functions will solve these these equations of motion or these partial differential equations so for example let's go ahead and how about consider the the to deal applause okay so the 2d Laplace equation what do we say that was that is partial U squared with respect to x squared plus partial u squared with respect to Y squared is equal to zero so this actually yields a family of solutions so for example I can just pull out a thin air a whole bunch of these actual use that would solve this so for example let's go ahead and try you know here's solution one U of XY is equal to x squared minus y squared how about a second example here u2 of XY is equal to e to the X cosine of Y that looks nothing like the first set of solutions alright how about you three here u 3 of XY how about this how about natural log of x squared plus y squared why don't we run over to Mathematica and verify that actually all three of these very dissimilar functions here will solve the 2-d Laplace equation alright so here we are with Mathematica let's go ahead and just quickly how about define these functions that we talked about just now on the board here so I think we had a u1 of x and y here this was x squared minus y squared was our candidate here and then I think we had two other ones right at you two when au 3 and u 2 we claimed was e to the X cosine of whoops that should be an X all right X cosine of Y and then finally you three was natural log whoops the log of x squared plus y squared right okay that's our functions and what we claimed is all of these would solve the to deal applause so let's go ahead and how about define the 2d laplacian operator right that was what we needed here so that was our del 2d let's say and let's say you pass this thing a function u here and what this thing does and I'll actually use the set delay so : equals what it did was it took the function U here and it took its derivative with respect to X twice and then what it did here was added that to the functions derivative with respect to Y twice here right so what this is here is this is effectively u xx plus u YY here all right so I'll shift enter that I don't need a semicolon there because I use the set delay operator and now let's go ahead and just go ahead and do a a check here so let's verify that this solves the PDE here so what that meant here is I just need to pass in del 2d let's give it these three functions so you want of x and y here right so what this should happen what should this should do is once I shift enter this let's cross our fingers and hope this equals zero yep that worked here let's do the same for the other two functions and see if this works here so u 2 & u whoops u2 and u3 shift inch up okay I worked for two but three here I think three just needs a little bit of persuasion here I think those actually are the same thing here but if I go ahead and let's say slash slash simplify and give Mathematica a little bit of a chance to clean that up and yep there we go so all of these satisfy the to deal applause so as we see here this family of dissimilar functions for this PDE are all able to solve the governing PDE so I guess the natural question is how do we narrow down the solution space here so we can get a single or a unique solution alright so we saw that yell these are dissimilar families of functions but they all solve the PDE here so in order to actually reduce the solution space here so in order to reduce solution space we're gonna see that we're going to need to apply just like we did it in ordinary differential equations we'll have to apply some initial conditions or let's call them ICS as well as in many cases we saw earlier that an initial condition just by the name of it it seems to imply initial time here right so that we need to fix what happens in these temporal variable but we're also going to have to apply probably what are called boundary conditions or we'll call B C's so hopefully if we now have the original PDE as well as some initial conditions and some boundary conditions that will allow us to choke down the solution space and get something more like a unique situation here okay so with that that actually brings us to an interesting discussion here that we saw that okay a PDE might have multiple solutions here one question you might ask yourself is well if I find myself two solutions to a PDE could I generate a third solution based on that those those those two solutions that I already found all right so let's write down actually one of the key theorems here of PDE s that deals exactly with this case so this is in fact it's so important that sometimes this is referred to as the fundamental theorem which is basically the idea of superposition ok so what this states here is that again like we said if you want in you two are solutions of a linear homogeneous where that's actually a little bit key so this doesn't work for all PDE s here but if you have a linear homogeneous PDE so you want it you two are solutions to this PDE and let's say in some region R of the solution space then you know what you could do here is you could take u 1 and you could add it to you two in fact you can even jam on coefficients in front of these C 1 and C 2 and this let's call it maybe you hear right it's a linear combination of u1 and u2 if this is the case then then you know what u is actually also a solution so this again should be very similar and bringing me some bells - when we were discussed ordinary differential equations where we did a very similar thing where we we added up multiple solutions and created a third solution and you know what why stop at three right maybe you add three four five six seven eight different you know infinite number of solutions to create yet another solution so in other words you could go ahead and say you know what if I have let's say u n and I'll good tack on a coefficient cn and I'm gonna sum this thing up from N equals one down oh no big end so I have big end solutions to my PDE I can just add them together to obtain yet another PDE okay I want to remember this here because we're gonna come back and discuss this within the context of Fourier analysis when we start trying to build up general solutions to arbitrary initial conditions and boundary conditions we are going to be using this fact extensively down the road here so again hopefully this has nothing to earn a Turing right now we've seen things like this before here in fact you know we could even jump back to Mathematica and verify that this works here so why don't we do that right now let's use it exact same example as we did earlier and just start adding up linear combinations to them and verifying that they still solve the the PDE all right so we're back at that same notebook so let's go ahead and apply the fundamental theorem in other words we're gonna apply superposition to start creating additional solutions so for example I could make au 4 of X&Y right which is going to be I don't know a C 1 times u 1 of x + y plus a C 2 times the U 2 of X&Y and again you know what I don't even need to pick C 1 and C 2 because theoretically work for any c1 and c2 as long as they're constant here so let's go ahead and define this as my fourth solution and let's run it through this operator again so del 2d which is our 2d laplacian of u four of X&Y I'll pass this in and look at that it's still 0 so it sure looks like superposition works and you know what we can even come back here and get a little bit more complicated you all right see three times u 3 of x + y let's add those all together so now I've got something really complicated let's try this again and again yeah we'll probably need to simplify this thing a little bit but you know what it's still 0 so here we go superposition held for this case and actually it should hold for any linear homogenous partial differential equation all right so this is all very exciting we saw that if I have some solutions of a PDE I can easily generate other solutions based on those and I can do this as many times as I want however here's the rubber I hit is how did I get those solutions to the PDE in the first place and that's where the whole art and the elegance comes in here right in general PDE s are there they're difficult beasts here right if you have a general arbitrary partial differential equation it is very difficult to solve and a lot of times there won't be a general solution however if the system or the PDE is simple enough you might be able to solve for a general solution so I want to take a look at a very simple example of a reasonable easy to solve PDE and how you might be able to do this analytically here so let's talk about how about an example here of a simple PDE and how can I go ahead and solve this analytically so let's go ahead and consider a PDE so the PDE I'd like to look at here is u XY is equal to minus Hugh X here so again recall this is shorthand notation for I'm looking for a function U that satisfies partial X or partial U with respect to X Y here is equal to the negative partial of the function U with respect to X here right that's all the same thing so I'm looking for the goal here is find this function U of X and Y here which satisfies this PDE here right I've got to be able to take its partial derivative with respect to X and then with respect to Y and that better equal the same thing if I took the partial of that thing with respect to X and then stuck a minus sign in front of this here right okay how are we gonna go ahead and solve this here this PDE is is simple enough that we might be able to do this using analytical techniques okay so the first thing I want to do here is let's go ahead and assume that we found you so assume we have this function U then we're done here well let's go make sure it satisfies this PDE here so the first thing I want to do here is let's take the derivative with respect to X so let's go ahead and do UX and let's say that what this kicks out is this generates some function let's call it P here so this is a function obtained by taking partial U with respect to X here right you with some complicated function of both x and y you take its partial you get something a little bit less complicated but it's still a function of x and y okay all right let's go ahead and now take the derivative of both sides here with respect to Y here so let's write that down so take derivative with respect to Y next year so in other words I'm gonna go ahead and calculate partial of U X with respect to Y is equal to partial with respect to Y of P something like that here right so again left hand side shorthand notation I could write this thing as u XY right that's the same thing here and I can rewrite shorthand notation and this is just P sub y right okay so if you recall here what is the whole point of what does u XY have to satisfy it has to satisfy this here right so let's make a note here so recall here that we said U of XY has got to equal negative UX right so let's substitute negative UX over here on the left hand side right because it theoretically needs to be the exact same thing it right so we end up with the left hand side becomes minus UX is equal to peace of Y okay so if you recall earlier here what we said here is that actually U of X is equal to a to P here right so that's what we said up here so again let's make a little note here and recall from previous discussion here that we said UX is equal to P right so again let's go ahead and substitute this in on the left hand side so I can write here how about minus P is equal to negative P sub y right ok great so I can also rewrite this how about as let's move everything to one side I could rewrite this as piece of Y over P is equal to negative 1 right same thing okay great now let's go ahead and integrate both sides with respect to Y here so let's write that down so int with respect to Y of both sides here right so what I'm doing here is I'm got py over P I'm going to integrate this side with respect to Y this has got to equal negative 1 integrated with respect to Y right okay if you remember here this is basically a natural log all right so we end up with a was a left hand side this is natural log of P here is equal to what do we end up with over here that's just oh actually we better be a little careful so this is minus y here right plus some constant of integration actually you know what both of these these things would have would have kicked out some constant of integration let's move both constant of integrations over here right plus maybe let's call it a C here tilde here right so it's both of those constant integration kicked over here and the other thing that's interesting about this constant of integration right it's not just a constant because we're integrating with respect to Y this thing could actually be any function as long as it's only a function of X here right so it's not a constant it can be a function of X here so I'm gonna denote it like such so so we keep that in the back of our head right ok let's raise both of these things to the e right so to simplify this let's go e to the natural log of P is equal to the e to the negative y plus C till the X right okay so simplifying this thing a little bit what do we end up with we get P on the left side here and then over here we end up with you could write this whole thing as e to the minus y times e to the C tilde X here right and okay let's look at this little C eetu the C tilde X this is just again some arbitrary function of F our start of X here right so let's go ahead and just rename this we don't have to keep rewriting this whole thing let's just rename this whole thing how about C of X right it's just some other arbitrary function of X all right all right so what do we got here now we've got we've got P here is equal to the e to the negative Y times C of X right now if you remember again right we said that P is equal to the use of X right so let's put that in on the left hand side right this is what we said way back in the beginning let's put this in on the left hand side here so what do we end up with we end up with use of X is equal to I guess I'll just switch the order this to make the look a little bit nicer let's go see the X e to the negative Y right okay great now let's let's go on to the other side of the board here let's integrate both sides with respect to X so now int with respect to X where I both of these sides so I'm gonna go integrate use of X DX here this has got to equal integral of C of x times e to the negative Y DX like such right um all right so left hand side that's pretty obvious here what we end up with here this is just you great that's what I'm looking for now the right hand side let's take a look at this a little bit e to the Y luckily that is not a function of X here so I can pull that out of the integral right so let's get e to the negative Y outside here and now we end up with integral of C to the xdx right okay perfect um this thing here right this integral we can do the exact same trick we did earlier we could say you know this is just some integral some arbitrary function of X so it's gonna be some other function of X here plus a constant of integration and that constant of integration can be anything as long as it's just a function of Y here right so in other words let's rewrite this like such eetu the negative Y and then inside here this is gonna be f of X here all right so f of X is just the whatever happens when you integrate this function of X with respect to X you get some other function let's just call this thing f for now right plus something else some constant of integration let's call it R but this concept of integration R can be a function of Y here right because because of the way we integrate it okay perfect okay um you know what let's just distribute this thing through here so we end up with E to the minus y f of X plus e to the minus y r of Y okay and again you look at this thing over here this is just some crazy function of Y here right so you know what and so we'll just call this thing how about what did I end up using a G sub y it's just some other function right okay great so at the end of the day what we end up with here is this function U of X and Y here right that we're looking for it's just e to the negative Y times some function of X plus some function of Y right so box this up so this is really fascinating if you look at this here it's basically saying any function that is going to solve that PDE that we started out with it really can be anything you want as long as you have some function that you know you're at your function f is just a function of X you then have to multiply it by e to the Y in the front then you have to add it to whatever the heck function you want as long as it's only a function of Y here right so for example let's just choose a couple of random functions just to see if this thing works right so I don't know let's pick some arbitrary functions let's make a table here so f of X and then G sub y so let's do a couple of cases how about f of X what if you what have you pick something like I don't know cosine of X here right and then G sub y this could be something like Y squared again the concept here is everything in this column the the s it can only be a function of X these things they can only be a function of Y here right and then I don't know let's look at another test case I think I'm gonna look at how about a natural log of X here and then how about tangent of Y squared that'll apparently work how about something even crazier how about I don't know X cubed plus 3x plus 1 over cotangent natural log of X something like that right again something crazy but it's just a function of x over here on the y side I know something that's also Y squared e to the Y cubed or to the fourth Y to the fourth you know why okay this all of these should work here so what we want to do now is go ahead and compose this U of X Y using the expression we had earlier right you take whatever's in this first column you multiply by e to the to the Y and you add it to the second column here and let's go ahead and show that this will basically solve our original differential equation here which we said was u XY is equal to negative u X okay and that's a lot of boring busy work so I don't think I want to do this here on the board so let's run over to Mathematica and see if this works all right so here we are over in Mathematica again so let's go ahead and first how about choose some of these f of X and G sub y functions that we just talked about here and use that to compose this solution maybe let's call u one is X of Y let's do this all in one go here right we said that all right the general solution of this was e to the minus y times f of X plus Gy right and now let's go ahead and just use a one of the F and the G's that we discussed so let's slash dot and say replace any where you see an f of X let's use the first example of how about cosine of X and then G Y we said was gonna be Y squared here alright so here's you one that's my first candidate function let's go ahead and make sure that this satisfies so we're gonna verify that this satisfies what was our PD I think it was you XY is equal to minus u X did I get that right I think I think that's that's what our PDE was so what I need to do here is alright let's do the left hand side let's take the derivative of the function here my candidate solution X Y I'm gonna take the derivative of this here first with respect to X and then with respect to Y here right so here's the left hand side here right this is UX Y here okay and I really better make sure that this equals negative the partial derivative of U 1 of X Y my solution with respect to X here right great so here we go i this should check if the left hand side equals the right hand side and hey look at that it's true that ended up working let's go back up here and choose a couple of more complicated examples that we had earlier so let's make a u 2 and a u 3 so I think our replacements here for F and X we tried something a little bit more complicated for the second one I think we said this was u natural log of X and then I think we'd say tangent of Y squared something like that and then for our third case this was our most complicated thing I think we said f was going to be x cubed plus 3x plus 1 over cotangent natural log of X something like that something really complicated and then Y here was going to be Y squared e to the Y for all right something like that so here's our a bunch of complicated candidate solutions and again let's just verify that these work so what that meant is just plugging it into the original governing partial differential equations so I'll plug all three of those in here and shift-enter this and look at that true true true all of these work here right so in this case we were able to find a analytical solution to this general partial differential equation here and that's really satisfying here although we're not going to be able to do this in general here so we're gonna require more formal method methods to solve some of these additional PDE s so one popular technique is actually to treat an OD e like a PDE and we're gonna investigate this in our next lecture alright so we saw that this actually works and while this solution method while elegant I'm sure you can kind of think to yourself that you know what this isn't really practical nor tractable especially for some more complicated partial differential equations here it it happened to work nicely in this case but in general this might be a difficult technique so with that being said I think this is a good spot to stop this first video lecture here and this sets the stage nicely for our next topic here namely how can we come up with a more formal technique to solve some of these linear partial differential equations so I hope you'll join us at our next video where we are going to start discussing the one-dimensional wave equation and we'll look at techniques such as separation of variables to go ahead and solve more complicated PDE s so with that being said I hope you enjoyed the video if so please subscribe to the channel here because we're gonna be having a ton more of these discussions here related to partial differential equations and other math topics in the future so I hope you catch you at one of those future videos thanks so much I'll talk to you later bye

Channel: Christopher Lum

Views: 34,245

Keywords: Introduction to Partial Differential Equations, Partial Differential Equations, PDE

Id: THjaxvPBGOU

Length: 52min 37sec (3157 seconds)

Published: Sun Nov 11 2018