 Welcome to this course on partial differential equations. In this lecture, we are going to discuss some basic concepts and the nomenclature that is involved in the study of partial differential equations. So, the outline of today's lecture is first we discuss certain preliminaries, then we go on to define what is a PDE and what we mean by solution of the PDE and then we discuss classification of partial differential equations. So, study of partial differential equations involves functions of several variables, theory of differentiation and integration for such functions. Therefore, let us discuss what are the domains of those functions are going to be. So, we consider functions defined on an open subset omega of rd. Since we are studying partial differential equations, functions are of several variables, therefore, d is greater than or equal to 2. We use the notation omega d to remind us that the set is contained in rd because later on we are going to see that in a certain problem we will see many sets one is lying in higher rd, higher dimension and one is lying in lower dimension. Therefore, this will keep track or it will remind us that we are in certain dimension. So, an element of rd, x is written in bold phase, it is not difficult, it is not easy for us to write on paper, but in print it is possible. So, the bold phase x actually stands for d tuple of real numbers x1, x2, xd. We may also use normal x, x in rd and there should be no confusion because the moment you know you are dealing with functions in rd and you have a function let us say u of x, it means x is a d tuple. So, writing x takes more time and effort than writing a very simple x. So, when working with functions defined on subset of r2, elements of r2 will be denoted by xy, that is the standard product it takes and we will not use x1, x2, we stick to the standard notation xy. Similarly, when we are working with the functions defined on r3, we use the notation xyz, the 3 tuple is denoted as xyz instead of x1, x2, x3. And in the context of PDEs, the variables x1, x2, xd these are called independent variables. Now, we have discussed the domains of functions, let us discuss the core domains of the function that we are going to deal with. So, core domains of functions means the function where the values it takes. So, we assume that the functions take values in rp, p greater than or equal to 1. That is we consider functions u defined on omega d, I need not tell you that omega d is a subset of rd again because the notation tells us and takes values in rp. Thus, the function u consists of p real valued functions defined on omega d and we use this notation u equal to u1, u2, up. So, the function u is given in terms of the sometimes what is called coordinate functions u1, u2 etc up to up. In partial differential equations u1, u2, up are called dependent variable or unknown functions. In our course, we are going to deal with PDEs which have only one unknown function. That is only one dependent variable in other words p equal to 1. So, the PDEs that we are going to study in this course are going to be a single equations. There is only one unknown function not systems. Some more notations let u be a function defined on omega d taking real values. Then first order derivatives the notation we use here is du. It denotes all the first order partial derivatives which are described here dou u by dou x1, dou u by dou x2 up to dou u by dou xd. Then the second order derivatives are denoted with d2u. This is a notation d2u stands for all second order partial derivatives of the unknown function u which is dou to u by dou xi dou xj i and j vary from 1 to d. Now in general dku for k greater than or equal to 1 denotes all kth order partial derivatives of u. So, recall that the notation dku stands for the collection of all kth order partial derivatives. However, we need a notation for any specific kth order partial derivative. Therefore, we are going to introduce that notation to do that what we do is let us fix a d tuple of numbers which are either natural numbers or 0. That is alpha is a vector which is equal to alpha 1 up to alpha d. It is a d tuple. Where does it belong? N union 0 power d that means each of them is a whole number either a natural number or 0. For such an alpha of d tuple of numbers whole numbers we denote mod alpha equal to sum of the alpha i's. So, mod alpha equal to alpha 1 plus alpha 2 plus alpha 3 up to alpha d. So, the notation d alpha u means I am going to differentiate u with respect to x 1 alpha 1 times with respect to x 2 alpha 2 times and so on up to with respect to x d alpha d times. So, that is the notation here d alpha u where alpha is not a natural number or a whole number alpha is a d tuple of whole numbers. d alpha u is equal to dou mod alpha u divided by dou x 1 alpha 1 dou x 2 alpha 2 up to dou x d alpha d. Let us see an example assume that d equal to 4 that means we are dealing with functions of 4 variables. So, the independent variables are denoted by x 1, x 2, x 3, x 4. Let us take alpha equal to 0, 2, 1, 3 and we want to differentiate you this alpha d alpha u. What is that? It tells us that do not differentiate with respect to x 1 with respect to x 2 differentiate 2 times with respect to x 3 differentiate 1 time and with respect to x 4 differentiate 3 times. So, sum of the entries of alpha is 6, 0 plus 2 plus 1 plus 3 that is 6. That means what it defines is a particular 6th order partial derivative of u. So, d alpha u is dou 6 u by dou x 2 square because the second entry here is 2 that is why second variable differentiate 2 times. Third entry here is 1 therefore differentiate once with respect to the third variable and similarly with respect to x 4 variable differentiate 3 times because there is a 3 here. So, there is a room for confusion because we are using d k u as an at the top of the slide to stand for collection of all kth order partial derivatives and here also d alpha u to denote this particular partial derivative. But there should be no reason to be confused because this d k u stands or makes sense only when k is a natural number. This d alpha u makes sense only if alpha is a d tuple of whole numbers. So, there is no room for confusion here. So, the order in which partial derivatives are taken does not matter. For example, if I want to differentiate with respect to x 3 first in the above example and with respect to x 2 later there is no way to take care of that in this notation. In other words this notation does not care the order in which you take the differentiation which is not okay. We know that partial derivatives do depend the order in which they are taken but for reasonable functions they coincide. So, we are going to consider such functions only where the order in which you differentiate does not matter okay. For example, if you are considering a c 2 function then it does not matter in which order you differentiate that is a theorem in multivariable calculus. Maybe that is the reason why this notation does not respect the order in which you will differentiate. Now let us go to the definition of a partial differential equation and what do we mean by its solution. One normally says PDE is an equation involving partial derivatives. This can be accepted as a definition provided we understand this statement. Not only that everybody understand this statement in the same way. Question what do we understand by the above sentence right. We are saying involving partial derivatives. So question is how many derivatives finitely many or infinitely many is also allowed. There is one question there. Second question is what does the equation involving stand for what does that mean? No idea. However we seem to be knowing examples of PDE is therefore we do not question this kind of a definition if found we do not question because we know what our PDE is. We know okay. So how do we know? We have examples and this English statement seems to be okay but the problem is this English statement might allow more types of equations which you do not think is a PDE. We will see one of them as an example okay. Therefore these are the example that we seems to be knowing dou u by dou x plus dou u by dou y equal to u, u dou u by dou x plus dou u by dou y equal to 0 and another equation dou 2 u by dou x square plus dou 2 by dou y square equal to sin x plus cos phi and so on many such examples we know. And the above sentence PDE is an equation involving partial derivatives seems to be acceptable okay. Therefore there seems to be an answer to the first question. That means how many derivatives should be there in equation because whenever we are writing an example of PDE we are not thinking of writing an equation which involves infinitely many derivatives. Therefore we know that a PDE should have only finitely many derivatives appearing in the equation. What about second questions answer? Still no idea. So we need to capture what we are writing here in terms of some definition which is very rigid okay. So in mathematics as we know definitions are set in stone and vaguely worded sentences are not allowed as definitions such as this PDE is an equation involving partial derivatives. So we reformulate the above sentence or rather we give a mathematical meaning of this sentence or a possible mathematical meaning of the above sentence. And once we write down as a definition we stick to that that is a definition okay. So we understood that PDE will have only finitely many derivatives in it. Therefore find out the order of the derivatives which appear in the equation look at the equation ask which is the derivative appears and let m be the largest having the above property. Being the largest it will be unique and this largest derivative which appears largest order derivative which appears in this that particular order is unique and that is called the order of the partial differential equation. So now we need to define a PDE may be defined as an equation satisfied by all partial derivatives up to order m including the function as well okay. So a relation of this type capital F of x u du d2u dmu equal to 0 recall what is du collection of all first order derivatives d2u collection of all second order derivatives and so on this is collection of all mth order derivatives. So for some m and some function f we should be able to cast our equation in this form now where x is in omega d questions what is m what is the domain of f so m is the order of the PDE okay. So this is how we plan to define we can if you have a PDE which involves only first order you can also think I can write this equation with m equal to 2 but such a thing should not be allowed our definition will not allow because this m we would require that m should be the order of the PDE as identified on the last slide. m is the order of the PDE means some mth order partial derivative appears and no higher order partial derivatives appear in the equation that is why that m is unique. And now once you know this it must be easy to write down the domain of f once you see the equation it is easy to write let us look at some examples examples it is always easy okay. So for this PDE dou u by dou x plus dou u by dou y equal to u m is 1 that means first order PDE. So first order PDE means it has to be expressed as x y u du equal to 0 du means all first order partial derivatives which are dou u by dou x and dou u by dou y. So it should be written in this form what is f? f should be some function of how many variables 1, 2, 3, 4, 5 f should be a function of 5 variables. So this is always a good practice to write any such function like this f is f of x y z p q okay x y is independent variables it means we are working with a PDE in 2 independent variables here z is the location where you are going to substitute u p and q are the places where you are going to substitute for dou u by dou x and dou u by dou y respectively okay it is important to stick to this kind of notation understand this notation. So what is f of x y z p q in this example it is p plus q minus z therefore if you take f equal to this and consider this LHS that will become dou u by dou x plus dou u by dou y minus u and hence you get the PDE. Now larger domain on which this f is meaningful is r 5 because it is defined for every 5 to pull x y z p q this function is defined. However we may also consider the domain to be any open subset of r 5 that is also okay. Observe something nice about this f it is a linear polynomial in this variable p q and z of course it does not x y does not appear therefore it is still you can say it is a linear polynomial in all the variables okay. Let us look at second example it is dou u by dou x plus dou 2 u by dou y square minus dou 2 u by dou x dou y equal to cos of y by x. What is the m here it is 2 because second order derivative appears and higher order derivatives other than 2 are not appearing only 2 okay 3 4 5 etc nothing appears so m equal to 2. Once you have decided m equal to 2 what should be the f? f should look like this x y u this is corresponding to d u that is dou u by dou x and dou u by dou y and rest corresponds to all second order partial derivatives of u. Therefore f must be written like this I told x y z is a place we are going to put u p q for the first order derivatives or s t for the second order derivatives. I have included s and s tilde just a placeholder for dou 2 u by dou x dou y and dou 2 u by dou y dou x. As I mentioned earlier we are not going to really distinguish between these 2 so after sometime we will not write s tilde we will simply write f of x y z p q or s t s tilde will be forgotten we will not write that because we will maybe we will consider functions for which both are same for reasonable class of functions this mixture partial derivative is coincide and what is the formula for f one can easily write down it is this. Now we will ask what is the domain of f then there is a problem here because x equal to 0 should not be allowed everything else is allowed and what is the domain of f it is which how many tuples tuple of what size 2 here 3 plus 2 first order derivatives and 4 second order derivatives so 2 plus 1 that is 3 plus 2 5 plus 4 9. So domain of f will be in r 9 but it should not contain x equal to 0. Okay here I have written we did not what I am saying is we will not distinguish I mean I told you what that means already okay what is the domain of f so domain of f let us call omega 9 because it is sitting in r 9 as we have written because we have included f is a function of x y z p q or s s tilde t so 9 variables are there x equal to 0 is not allowed so that is what we say. Note that the given p d is meaningful for those x y is in r 2 for which x is not 0. So if we denote the projection of omega 9 to the first two coordinates by omega 2 then omega 2 is an open subset of r 2 which does not intersect the y axis thus we may write the domain omega 9 to be like this f is from omega 2 cross r cross r 2 cross r 4 this omega 2 stands for the independent variables x y vary here this r stands for the unknown function u which we do not place any restriction on what values it must take so therefore it can take any real number r 2 stands for the derivatives the p q variables where dou e by dou x and dou e by dou y are going to come r 4 is all 4 second order partial derivatives. So we can further write omega 9 in this form now dou f by dou s is not 0 in fact dou by dou t is also non-zero it means f is not a constant function in s it means s variable is used by f it is not constant it depends on s it means it is a second order p d this guarantees that some second order partial derivative appears in the equation. So now we are going to write the definition let omega d be an open subset of r d and let f be like this omega d cross r p and cross r d p d square p and d m p taking values in r q be a function so it means f is taking values in r q that means there are q functions in this okay this omega d is where x is going to sit that means I am considering x in r d so d independent variables and this is r p that means the unknown function has p components u 1 u 2 u p it is a vector valued unknown function and this corresponds to derivatives because u is a mapping from omega d omega d to r p therefore the derivative will be of size d p d p many first order derivatives will be there this will be corresponding to second order derivatives and this corresponds to mth order partial derivatives of the unknown function. So a system of partial differential equations why system because we are considering q greater than or equal to 1 so there can be more than one equation that is why in general we call it is a system of partial differential equation order m because we have d power m here is defined by this equation okay so d m u and all mth order partial derivatives which are so many okay I am not discounting the symmetries like earlier I have done I am writing full okay so we are at least one of the mth order partial derivatives of u appears in the system of equations only then we are going to say it is a mth order p d e so as I said if q is bigger than 1 it is actually system of p d e is q can be equal to 1 then it is a single p d e or a scalar equation and if p equal to 1 it is a single unknown function if p is bigger than 1 unknown function is vector value. So now we have to define what is the solution of the p d e a solution of the system of p d e which is defined here is a function solution is a function when we say function with some domain so it is a function phi with domain u of course here we have written system for vector variable dependent variable is a vector value therefore this phi is also vector value r p with what property u is a subset of omega d what is omega d omega d is that set where this equation is meaningful for x belongs to omega d the domain of f was omega d cross some things but a solution can be a function which is defined even on a proper subset of omega d it is a function with some properties what should be the properties all partial derivatives of phi up to order m should exist including m because I am thinking of a mth order equation therefore all partial derivatives should exist up to order less than equal to m. Second once it exists look at this tuple I am going to substitute inside this x phi x d phi x d 2 phi x d m phi x therefore I would require that this tuple belongs to the domain of f and I have the right to apply f to that when I apply I get 0 that means this equation is satisfied in this way by this function phi not phi we are demanding that it should be defined on a subset of omega d and we are not demanding that it should be defined on omega d that is very important to be noted. So, we do not insist that a solution must be defined for every x in omega d you may be proper subset of omega d this should be omega d now answer the following questions decide which are the following are PDEs write down capital F as in the definition how do you say somebody is a PDE you have to go to the definition of the PDE that involves a capital F it comes with this domain right you should be able to give such a function f and then this becomes a PDE. So, Laplace in U the first one is dou 2 U by dou x square plus dou 2 U by dou y square equal to 0 next is e power Laplace in U equal to 1 Laplace in U is defined above and this is U x plus U y equal to sin of Laplace in U this is U x plus U y equal to U composed with U. So, this circle stands for the composition of functions now at this point you must stop the video write down you think about the answers write down the answers if you have and then proceed because answers going to come on the next slide. What are the answers? The F is r plus t is a second order PDE e power Laplace in U equal to 1 F is e power r plus t minus 1 second order U x plus U y sin of Laplace in U f equal to p plus q minus sin of r plus t second order and this one is not a PDE. Think in your leisure time if you do not get this answer do not worry keep thinking about this question motivated by the following example of Arnold in ODE is d y by dx equal to y circle y he says it is not a ODE. So, the vague definition would allow this as a PDE because it just says it should involve unknown function and its derivatives which this equation does involve U x plus U y equal to U circle U. So, there is no point in admitting such equations if you cannot have a general theory for some special class of such equations. So, maybe that is the reason why Arnold called this as not as a ODE. So, if you understand the answer to this the answer of this will be immediate. Now, we move on to classification of PDEs. So, what is classification classification of PDEs means creating banks with labels and associate each PDE to exactly one of the banks. Examples of a few methods of classification we are going to see each one of them based on the number of equations, based on the order of PDEs, based on algebra I will mention this details when we come there. So, question is it important to classify. Recall that a complete theory is available for solving systems of linear equations A x equal to b and the same is not the case with transcendental equations which are also known as non-linear equations. You have to hold each and every equation and analyze for solutions. Now, coming to linear ODE is there is an oscillatory behavior of solutions that is different from first order ODE and second order ODE. Look at simple equation like y dash equal to y and y double dash plus y equal to 0 or y double dash equal to y. Think of the solutions some of them are oscillatory. So, it changes. So, it is important to identify a reasonably interesting or useful class of PDEs for which it is possible to obtain a larger understanding possible is possible may not be complete, but at least a larger understanding good understanding. So, first order PDEs we are going to see that classification leads us to uncover certain aspects of solutions starting from the quasi-linear to the general non-linear equations. We will see that in the next chapter. So, we will come back to this discussion on classification later on also in chapter 3 where we are going to classify second order PDEs we ask the same question and we have some better answers there we will see that. So, now a classification based on the number of equations recall that a PDE is defined through a function of this type. If q is bigger than 1 it is called system, if q is equal to 1 it is called scalars. So, this is an example is a Maxwell system of equations in 3 independent variables x, y, z and a time variable that means there are 4 independent variables here and e is 3 vector even e to e 3 functions b is 3 functions b 1 b to b 3 and dou e by dou t equal curl b dou b by dou t equal to minus curl e divergence of b and divergence of e are 0. So, this is indeed a PDE you must write in this setup and that is left as an exercise to you. As I told you when q equal to 1 that is called a scalar PDE but when q equal to 1 we also take p equal to 1 that means the unknown function also is a single unknown function. For this otherwise what happens is that you have one equation because you said q equal to 1. If you have more than one unknown it means that information is I mean conditions are very less. For example, just recall what happens with one linear equation two variables not so much interesting right I mean x minus y equal to 1 infinitely many solutions that is one of the aspects okay. So, this is an example of a single equation it is a very important example we will be studying a depth later on. Now, second thing is the classification based on the order of the PDE. So, order of PDE as we know is uniquely defined and hence they may be classified based on their orders. This is a first order PDE this is a third order PDE. Now, this is the important classification which uses algebra. So, this classification is based on the manner in which the unknown function and its derivatives appear in the equation. What are the possibilities? First possibility dependent variable and all its derivatives whichever appear in the equation they all appear as a linear combination with coefficients which are functions of only independent variables that equation such equations which feature which has this feature or will be called linear equations. Second option is highest order partial derivatives appear as a linear combination with coefficient which are functions of only independent variables and no conditions on the appearance of low order derivatives and the function u. This may be called semi-linear equations. Now, highest order partial derivatives appear as a linear combination but now the coefficients depend on independent variables and also on the lower order derivatives including u. So, that will be called quasi-linear equations. The highest order partial derivatives do not appear as a linear combination at all. So, such equations will be called non-linear equations. Now, we may ask this question why only highest order derivatives are important for the last three classifications. They play a major role in determining properties of functions is a vague answer. You should be in a position to appreciate this at the end of this course. Now, let us look at these equations these two equations. One is the first order PDE and this is the second order PDE. What is common between them is that u and du in the first equation, u, du and d2u in the second equation they appear linearly. It is a linear combination of those derivatives and the function. X and Y we do not ask any questions. The classification is based on ZPQ variables. In this for the second order PDE it will be ZPQ or SS tilde t. So, we will see very specific examples for first order PDEs, the classifications and we will see most general possible equations that we can write of that type. So, linear first order PDEs means the unknown function and the first order derivative should appear linearly that is coefficients are allowed to depend only on independent variables. So, they appear like this AXY dou by dou X plus BXY dou by dou Y plus CXY u plus DXY equal to 0. So, this is an example, semi-linear. Here what we ask is dou by dou X, dou by dou Y should appear linearly. Coefficients are allowed to depend only on the independent variables and u may appear in any way but not in the coefficients of dou by dou X and dou by dou Y. So, therefore the most general possible semi-linear equation of first order is like this. This is an example. Now comes the quasi-linear first order PDEs. Here the first order derivatives dou by dou X and dou by dou Y should appear linearly, fine. Coefficients are now allowed to depend on the independent variables as well as the lowest lower order derivatives. If this is the first order derivative what is the lower order derivative that is function itself, they are allowed. So, therefore this is the general possible quasi-linear first order PDE. This is an example. Now a PDE which is not quasi-linear is called a fully non-linear PDE. So, it is basically the complement of quasi-linear equations. This is an example, very important example, it is called Ikanl equation. Now this is the picture or Venn diagram, first is linear. This is the formula how the general linear PDE looks like. Linear is also semi-linear, that is a bigger class. Bigger class is quasi-linear. Out of quasi-linear will be full PDEs, this color in this color, red color but not in green will be fully non-linear. Otherwise these are all general first order PDE will look like this. That was the definition. Now for a linear PDE, first order PDE, F will be of this type. F of X, Y, Z, P, Q will be like this. This is for semi-linear and this is for quasi-linear. Now we can ask the questions like which quasi-linear PDEs are semi-linear. When this A does not depend on Z, B does not depend on Z, a quasi-linear equation becomes semi-linear equation. Similarly, this is another question, please answer yourself. Now we can classify the m-th order PDEs. We have explicitly seen first order PDEs classification but same thing can be done for m-th order PDEs. What is the guiding principle? The 1, 2, 3, 4 that we have written down. How unknown function and its derivatives appear? If all of them appear as a linear combination, it is called linear PDE. If that is not the case, we straight away go for highest order derivative and ask how they appear. Then we had 2, 3, 4. So we will write down here once again. Linear, see here, mod alpha less than or equal to m. So when alpha equal to 0, 0, 0, 0 that is m equal to 0, the 2-pull alpha is 0, 0, 0 that means no derivatives which means it is a function. So here it says function as well as the derivatives up to order m appear as a linear combination. Coefficient depend only on the independent variables. Semi-linear, once again highest order partial derivative that is m-th order appear linear combination. Coefficient are functions of independent variables. Rest however it appears no restriction. Quasi-linear, once again it is on the highest order derivatives. Coefficient are allowed to depend on apart from independent variable x, Udu up to dm-1 that is m-th order derivatives. Then this is called quasi-linear. Outside that it can appear in any way they want. But as coefficients are the highest order partial derivatives, they can only be like this. Now PDE which is not quasi-linear is called fully nonlinear PDE. So Laplace in U, its second order and linear. E power Laplace in U equal to 1. It is second order fully nonlinear. And this equation third order semi-linear. Of course you may also say quasi-linear but when I say semi-linear it has more information because semi-linear much smaller set. This equation second order fully nonlinear. So this is an exercise for you through examples show that the following inclusions are very strict inclusions. That means find a semi-linear PDE which is not linear, find a quasi-linear which is not semi-linear and show that there are PDEs which are not quasi-linear. That means given example which is not quasi-linear PDE. These are the questions for you and thank you.