noc21 - ma33 - lec06

[Music] [Music] today let us talk about laplace equation situation laplace equation now ah in the first few lectures we have talked about first order equation right so ut plus uh ux equals to that sort of equation yes transport equation vertex equation that sort of thing but now we are going transitioning in to the second order equation why laplace equation let us understand that this thing first why laplace equation see the point is this is the most basic the second order equation of elliptic type if you remember from ah i mean earlier in pde courses what happens is most of the if given given a general linear pde linear okay pde okay of the form of the form so second order second order how does it look like ah if you remember it will look like this a u x x plus b i am not writing all of this but a depends on x y so i am talking about two dimensions and b u of x y plus c u of 1 y plus d u of x plus e u of y plus f u uh plus let's say g equals to zero yeah that is a ah expression so all of this here all of these coefficients a b c these are all asymptotes i'm not writing all that so okay so let's just do that all of these coefficients g these are all continuous let's say these are smooth coefficients in r2 here so that depends on x and y where where a b and all of this is in c infinity rp right now the point is this if you remember your canonical form so the canonical form of this particular ah general equation canonical form okay i hope you understand so what i meant by this is see any second order linear pde can be written in this form here canonical form let's say that is your one let's say that is your star let's just call it star so canonical form what it does is actually if you in a different if you in a more suitable um coordinate system okay so what it does is under us suitable coordinate coordinate system right under the suitable coordinate system start gets okay so under the suitable coordinate system star gets reduced reduced to either of the following either of the following so what is it it is either w x x plus w y y plus some lower order terms are lower terms containing w x w y and w x and y that is equals to 0 or w x minus w y y plus lower terms containing w x w y w x y equals to zero yeah or you can have this form number three w x x minus w y y plus some lower term containing w x w y w x and y okay what this canonical form does is it does not change the nature of the uh equation if you remember you if you look at the discriminant of this the equation yeah and depending on the i mean sign of the discriminant if it is negative it is positive is that the hyperbolic problem and if it is zero it's a parabolic problem okay depending on that nature if you do a suitable coordinate change okay causing coordinate change then what happens is you can reduce this to something like this okay this this particular equation is an example of an elliptic equation right okay this is this equation is an example of a ah heat equation or a parabolic equation parabolic equation and this equation is an example of a hyperbolic equation let us understand what i mean by this so let me again recall what i am doing c essentially we have a linear second order pd which looks like this yes all the coefficients are smooth this is what we are assuming what canonical form does is see this is in some coordinate system x and y right you can change the quadratic system to u v so here you can change x y to u v but i mean it does not matter these are all variables right you can write x y you can write q so essentially under the new coordinate system you can actually change a f star okay to one of the following this expression so let's say if the if this star is a elliptic equation to begin with you there is a change of coordinate which will actually transform this to this yeah this is much easier to follow yeah this expression so essentially we are only interested in the higher order terms lower terms i mean those we can handle higher order terms so it will look like w x x plus w y okay that's the elliptic problem again if the original problem is a parabolic problem under the change of variable uh it will look much simpler which will look like w x x minus w y y y plus log data so essentially this is also a lower term but i am just writing it in this way just to make it look special okay so familiar it's like a heat equation so that's parabolic form because otherwise i mean you can just take this w x over here also it's not a problem and w x x minus w y y plus l of this is equal to zero that's a hyperbolic form so essentially this is like a wave equation from two second order terms plus lower terms like this okay so where let me put it like this where of are the lower order terms okay now so let me make a small remark i am assuming here that since this is an advanced pd course i am assuming that you guys know all of this okay if you do not know please choose a suitable suitable uh textbook okay whatever you want you see most textbook contains this thing yes please choose a suitable textbook for example meantime they've not meet and they've not to review it okay see this is the thing which i said if you guys are familiar with this thing there is absolutely no issues there if you are not then obviously you can just choose anything you you want i mean this is just an example okay means and then partial differential equation this book you can just look at that book and just give you this whatever i say that is not very difficult thing to do okay now what's the point of all of this the point of all of this is see if we want to study most linear equation you do not need to know you understand you do not need to know how to solve a general equation like this ok all you need to know is how to solve this three prototype equation if you know how to work with this three prototype you are done right because ultimately any linear equation can be reduced to something like the second order okay any linear cellular equation it can be reduced to one of these three forms right so if we can just study these three forms these are the fundamental forms if you can just study these three forms so basically we call this an operator l of w is w x plus w y okay how these operators behave yeah then we are done so then we can actually work out our problems okay so let us start by first of all with the example so we define defined l of u so this is what i mean by an operator okay define l of u to be u x x plus u y y ok before i move on doing anything i just want to want you to take 15 seconds 10 15 seconds and think about if what is l so l is supposed to as you can understand is kind of a function right it is taking something and putting it into something like it is taking a function and giving back another function which looks like this so can you tell me where is l defined from just take 10 15 minutes in seconds and just think about it right so let me just tell you what where if you see l is an operator what do i mean by operator here so a is an is a linear operator okay what do i mean by linear operator what i mean by this is essentially c it is taking values from some vector space u see if i am taking a u u x s and u y has to be satisfied right so u has to be at least twice differentiable so it is c2 of whatever denominator i don't care some domain let us say omega i am starting out with an omega which is subset of r2 so l is from c2 r2 i mean you are choosing an element of twice differentiable element okay and it is giving back u x x plus c y if u is twice differentiable u x x is continuous and u yy is continuous the sum of two continuous function is continuous so it is giving you back something like this okay so so l from is a is a linear of second order second order operator ok operator this is what i mean by operator so you see this is also a function but in a special way what is so special about it that is not taking elements from any ordinary set but a vector space ok of functions ok it is taking elements from a vector space of functions right now let us look at where all of this comes from so let us look at some physical motivation yeah so before i do this let us put some name to this particular operator so laplace equation ok laplace equation if you are looking at equation which looks like this laplacian of u that is given by divergence of gradient tube okay you if you remember gradient u is let's let's just start with two dimensions okay don't worry about any n is exactly the same okay let's just start with two dimensions so in two dimension ua the gradient of u gradient of u is ux and ui right so divergence of gradient of u that is u x x plus u yy ok ok so this is u x x plus c y y ok that's your laplacian now there is something this equation this is not an equation anymore i mean this is just an operator now this if it is 0 equals to 0 then it's called a laplace equation now we also talk about a similar equation which is called the poisson equation okay so the name is poison equation this is uh i mean most probably i am not hundred percent sure but poazo is just is like a fish in french okay so in the equation looks like this minus laplacian of u ok minus there is nothing special about minus you can write it you may not write it this is just a convention minus laplacian of u equals to f okay that is called the possible equation right okay here here we will assume x is in from omega which is subset of rn right so what i meant by x is an integral x 1 x 2 x n ok um ok i have used a something ok so let me let me change this thing so maybe let me do it for um from now on i will do it for rm itself okay okay um actually this is the problem you see i should write it like this let let me change this pattern let me change this part so in 2 dimension it will look like this x 1 x 1 ok x 2 x 2 so essentially your i mean x 1 x 2 ok so i am just writing a tuple in r 2 as x 1 x 2 you can write it as x y also but in that case you can't i mean i just want to i mean reserve x for a element in r m okay so x is x 1 x 2. so essentially when i am saying it is a element of r n what i meant is ah so let me put it here to be rn so this is u x and x see n equals to 2 is just this one i mean here i just want to write x is in omega subset of r n means x so essentially x is x 1 x 2 x n okay i mean you do not have to worry about x n on the n components just two component whatever it does encompass same sort of thing okay now poison possible equation is minus laplacian of u equals to f this f will be given to your u from omega bar to r is the unknown function which you need to find unknown function okay and if is any given function function on the open subset of rdt so just think about it what i am saying what i said is you are looking at a u which is from omega bar omega bar is omega is the open set right as i told you if i am not specifying what exactly omega is most of the all the time in this course just assume omega to be an open set in rna so omega bar is a closed set the closure of omega to r that is an unknown function and f is any given function on an open subset of r here okay now the question is you just have to find what u is that's that's the question okay so i will start with a small definition here definition so let us say a c2 function or c2 function u is called a harmonic function if it satisfies the laplace equation right so if you are looking at a u such that laplacian of u equals to zero then that is that sort of function is called a harmonic function okay now example take five seconds just think about an example of a harmonic function okay let me tell you what is a very easy example let's say if you are let's say omega to be r n okay and u of x one x n to be identically equals to a constant okay constant ok now if that is the case what do you think laplacian of you should be any derivative of this is zero so laplacian equals to zero so that's a trivial ah harmonic function right and of course one special thing about harmonic functions are okay 0 is always included so 0 so u x equals to since the constants always hold 0 therefore u x equals to 0 is always harmonic always and this is harmonic in any domain right i mean it does not matter whenever you can take any omega you want omega is any open set and u of x is always going to be a harmonic function ok so that's one example can give think of another example ah so let us assume that omega is in open set and u of x one um may be i don't know maybe x 1 x 2 x n let's just call it i'm writing it for x n you can just think of two dimension also no issues so let's say that is x i whatever i is here i can be 1 2 whatever you now if that is the case let's look at what laplacian of u is laplacian of u in this case is u x 1 x 1 plus u of x n x n now i don't have to calculate this thing you guys can understand that this is going to be zero okay so any coordinate function this sort of function is called a coordinate function right any coordinate function is harmonic okay let's take another non-trivial function here so in two dimension let's just think two dimensional x one x two to be x one square minus x two square let's see if this is harmonic or not it may be it may not be i let's just as a look at it laplacian of u is u x x x 1 x 1 minus u y 1 y 1 plus u y 1 ok now u x 1 x 1 as you can see the first derivative is 2 x 1 second derivative is x 1 2 so that is 2 plus u y 1 sorry this is not y 1 y 1 it is x 2 x 2 x 2 x 2 ok plus 2 x 2 x 2 is minus 2 ok that will give you 0 so laplacian of u is 0 so hence this is an example therefore ah u x 1 x 2 given by x 1 square minus x 2 square is also harmonic so there are i mean there will be other examples also but these are more or less the basic examples of harmonic function okay before we move on let us look at what is the special about harmonic function so uh first of all properties properties ok we define h h to be the set of all c 2 omega bar functions such that laplacian of u equals to 0 okay so this is the set of harmonic functions set of harmonic function harmonic functions okay h is the set of all c2 omega bar functions as the laplacian equals to 0 so that is the set of harmonic function now i want to see what are some special properties of this set see let us say that let let f and g are in h okay let's see what happen define and define and define a plus g yeah acting at x to be f of x plus g of x okay if i define it like this clearly clearly see ah if this is the case then laplacian of a plus g yeah this is going to be laplacian of f plus laplacian of g ok how is this true because of linearity yeah if you are not convinced about this thing i am not proving all of this here it is going to be time consuming i mean it is not it is just two lines but please check this part yeah its not very difficult moreover moreover you see for a c in r for any c in r if you define c times f acting at x to be c times f of x and f is in h ok you start with a f in h and c in r and you define c f like this then laplacian of cf let's see what happens it is definitely c times laplacian of f1 this also you can check please check this pattern check this check check okay now think about this if you consider this property that you can given two functions f and g the sum is ah linear and the constant times f is also in h right so this this definitely belongs to h it means that cf belongs to h okay what does that say it says therefore h is a vector space okay h is a vector space so this is very special about it this is all of this is happening because the operator is a linear operator okay since since delta so this is called a delta this symbol is called delta is a linear operator linear operator okay okay now what we are going to do is we are going to look at some physical interpretations how where do we use this sort of ah operator okay see this is the most widely used operator in all of partial differential equation probably one of the most important objects in all of mathematics laplacian okay so it is very important that we know understand and i mean appreciate what how beautiful this operator is so let us look at some physical interpretation physical interpretation ok so first of all let us assume that u is the density of some chemical concentration in equilibrium okay let us assume u is sub function so this represents the density of some chemical composition okay i mean some substance is given you are looking at the chemical concentration composition of that thing okay in equilibrium equally brilliant here okay equilibrium is when it is stable ok now let us say that the chemical is contained in some i mean you know region which we will call as omega so the region we are where the chemical is contained clear so that's your omega omega is our domain okay so let us see what happen see uh let's say that's your omega that's your omega yeah now let us assume that v is some v yeah so let let v be a smooth okay v be a smooth region region content in omega right so that's your v let's say any smooth region which can which is contained in omega now if that is the case there seems to be some problem with the software okay let's just take another page okay so if that is the case then so basically what i am saying is if v is a smooth is a smooth so containing you smooth is smooth okay so what am i doing that's your u okay and uh this is where the chemical is so the chemical is and u is the density of the chemical and that chemical is in equilibrium and you are just looking at another small sub region okay which is v sorry this is not calling it omega v is continuing right okay now um you see since the i mean the chemical yeah the concentration that is in equilibrium the chemical is in equilibrium okay so therefore the net flux the net flux so net plus means whatever is going in or coming out ok of you of u through the boundary of v del v 0 right this is quite easy and if that is hap true what does that say del v if dot gamma d s is 0 right so basically there is no flow of um the flux density is not changing yeah in the unit direction so let us say gamma is this gamma is the unit outward direction okay so where where f is the flux density flux density and gamma is the unit okay outward direction so what i mean by this is c see the the whole liquid yeah or whatever the chemical is it's in equilibrium right so if you look at a small region wherever on the omega small smooth region the net flow okay of the chemical the density the change in the density basically uh the net flux of u okay through this boundary of the the v yeah that is going to be zero yeah so if if f denotes the flux density then ah i mean in that direction in any direction i mean whatever the gamma is here in any direction gamma the flux density if you take the integral of that that is going to be zero yeah now if that is the case in from green steeler or gaussian actually from from gauss divergence gauss divergence if you remember when he looked at i mean integration by parts ok i said that it is going to be one of the most uh important you know how do i put it uh most exa important thing you can learn integration by part so okay we are going to use that integration that pipeline integral del v f dot gamma d s is 0 yeah then you use gauss divergence and say that you see integral over v the divergence of f okay d of x that is equals to integral of del v f dot gamma d s right and that is going to be 0 yes how are you getting this is gauss divergence here so basically gt gauss divergence theorem okay gaussian theorem says this and from here we get this the flux is zero okay now see this v is arbitrary right this v can be anything so you are saying that ah object when you integrate that object over any sub region smooth sub region that is going to be zero what does that say that the divergence of f is going to be zero in v okay v is arbitrary of course rb sorry in omega arbitrary okay in omega i have to say it is in omega y t is one omega c v is containing omega right v is arbitrary so you are saying that you are taking the divergence you are taking some object and integrating it in any smooth sub region of omega and that integration is zero yeah so definitely the object in question that is going to be zero right if you are not convinced here please check this part okay so check this check check that let us say u is in c let's say 2 of omega b such that integral over let us say u ok and dx over v this is 0 for for any smooth v containing omega okay then u has to be 0 in omega okay see one thing is this why am i talking about smooth v in omega if you remember gauss divergence theorem says that you have to have this region where you are integrating that region has to be a smooth region right at least c1 so that is why i am just assuming to be smooth okay now you see there is something called so since we are talking about chemical concentration there is something called a fixed law okay fixed law of diffusion this is from your physics ok fixed log diffusion that says that the flux density f is proportional to um gradient of u right so basically what it is saying is minus some constant times gradient of u okay right so y minus because uh the flow the flux density the flow ok ah is from is from the region of higher to lower concentration see if there is a flow in the system here it is now if there is a flow in the system that is always going to be from a region of higher concentration to the lower concentration right so that is why this negative sign is there is just say that the flow is somehow you know in a negative way and this c of course is a constant or proportionality i mean we are just saying that f is proportional to gradient of u okay and gradient of u is just the change of uh i mean the flux so basically the changes are density so and this c is positive we are assuming this otherwise this minus c we look at it so if f is minus theta gradient let's just put it here therefore divergence divergence of minus c times gradient of u that will give you minus c times divergence of gradient of right that will be zero because divergence of f is zero f is f look like this so divergence of gradient u is 0 so that will give us that laplacian of u is 0 because c is positive right since c is positive that will give you or minus laplacian i don't care i mean okay now see the very important thing is this is from fick's law we got that laplacian of u is equal to zero now you just uh if you want to so this isn't laplace equation okay so that's a laplace equation the whole idea of this is c ah here if it is a chemical concentration you are talking about fixed law ok fick's law will give us the laplace equation actually if it is the fourier law of heat conduction that is also similar to this here if the fourier law of heat conduction then that also will give you a laplace equation okay in that case you will consider so ah let us say for heat conduction u will be your temperature the function u and if it's a ohm's law electricity conduction then you will be the electrostatic potential okay so this is a very very important thing to understand okay ah so with this what we are going to do is in the next lecture set of lectures we are going to talk about ah how to work with laplace equation yeah so with this we are going to end this particular video [Music] so [Music] [Music] you