 In this lecture, we are going to introduce classification of second order partial differential equations. It is also known as classification by characteristics. Outline of today's lecture is we start with a review of lectures 3.1 and 3.2 then we introduce define what is called characteristic curves using which you will define what is meant by classification of second order partial differential equations. In fact, we are going to classify second order quasi linear partial differential equations and then a few remarks on classification. Recall the second order quasi linear equation we denote by 2ql it stands for this equation AUXX plus 2BUXY plus CUYY plus D equal to 0 coefficients ABC of the second order derivatives and D are functions of X, Y, U, X, U, Y that is why this is quasi linear. When the coefficients depend only on X, Y particularly ABC depend only on X, Y and D may depend on all the 5 variables here X, Y, U, X, U, Y in that case 2ql is actually called a semi-linear equation. So, this is a linear equation which we will also be looking at. So, in this A, B, C note they depend only on X, Y even D its dependence on U, U, X, U, Y is linear. U here, UX here, U, Y here and a function of X and Y equal to 0. So, this is second order linear PDE. So, let us start with the review of the lectures 3.1, 3.2 what we did there. In lecture 3.1 we came across this ODE, A of zeta Y minus 2B zeta Y d phi by dy plus C of zeta Y d phi by dy whole square equal to 0 where zeta Y is 5 tuple phi Y, Y U of phi Y comma Y, UX at the point phi Y comma Y, UY at the point phi Y comma Y. So, the quasi data is prescribed on any curve gamma 2 given by X equal to phi Y. Yes, in lecture 3.1 to start with we considered a parameterized curve, regular parameterized curve gamma 2 given by X equal to FS and Y equal to GS and later if the curve is given by an equation of this form in other words it is a graph of function of Y, X equal to phi of Y then the equation that we got there namely delta S equal to 0 reduces to this equation this was already observed. So, if the quasi data is prescribed on any curve gamma 2 in the plane given by X equal to phi Y the equation where phi does not satisfy the above ODE. Derivatives of all orders for a solution to 2ql can be determined at all points of gamma 2. Note here the zeta Y of course involves the given solution U of the 2ql the quasi linear equation second order quasi linear equation. So, given such a curve gamma 2 such that the phi does not satisfy the solution where U is we have to input the given function U if it does not satisfy this equation then we can determine all derivatives of all orders for the solution U can be determined at all points of gamma 2. In lecture 3.2 also we came across the same ODE. I will not repeat exactly same ODE. How did we get that? We assume that a curve gamma given by X equal to phi Y is a curve of discontinuity for second order partial derivatives for a piecewise smooth solution to the second order quasi linear equation then phi solves the above ODE. So, the regular curves for which this equation is satisfied they can be determined for a second order quasi linear equations only if we are given a solution U. In other words an integral surface z equal to U x y is given this is another way of saying that a solution U is given because only then the coefficients are determined and once coefficients are determined we can try to solve for phi satisfying this ODE. For similar equations as we observed A does not depend on all the phi quantities namely X, Y, U, X, U, Y it just depends on X, Y same thing is with B and C. So, all the A, B, C functions they depend only on X and Y which means what this U knowledge of U is not required. Therefore, this ODE is meaningful without the knowledge of any integral surface since A, B, C are functions of X, Y only. Now, let us introduce characteristic curves for second order quasi linear PDEs. So, defining the notion of characteristic curves for 2ql they will be defined as curves satisfying the ODE that was obtained in lectures 3.1, 3.2 the equation that we saw just now. Defining characteristics curve for 2ql requires that a solution of 2ql is given. In other words characteristic curves for 2ql are defined with respect to a given solution or an integral surface. So, when the equation 2ql is actually semi linear that happens in that case A, B, C will be functions of X, Y only it turns out that the notion of characteristic curves becomes independent of any prescribed solution we do not require any solution to be prescribed. The ODE itself determines the characteristic curves and the differences between the quasi linear and semi linear cases will be brought out once we finish defining the characteristic curves. So, definition of a characteristic curve. We consider a regular planar curve described parametrically by X equal to FS, Y equal to GS and S belongs to some interval I and F and G are C1 functions and I want to capture the word regular. What does regular means? Both F dash and G dash cannot vanish simultaneously. So, we consider regular parametrized planar curve. Define delta of S by this expression where zeta S equal to this. Note you need to know U, Ux and Uy. So, given a U we can define zeta S. In the definition of characteristic curve we will see that U has to be defined as a solution of the 2ql equation then this is well defined. So, delta of S is denoted by this. Now the definition gamma 2 is said to be a characteristic curve for the PDE second order quasi linear equation 2ql with respect to a given solution of 2ql if delta S is identically equal to 0 for all S in I at every point on the curve delta S is 0. It is said to be a non-characteristic curve. It is exactly opposite of what is a characteristic curve. It is a non-characteristic curve means at no point delta S equal to 0 is satisfied. Of course, because we are dealing with 2ql we have to write non-characteristic curve for the PDE with respect to a given solution. If delta S is not equal to 0 for every S in I. So, when the PDE is actually semi-linear then delta S equal to 0 reduces to this. The zeta S has gone zeta S is actually FSGS in this case because ABCR functions of XY only. Therefore, delta S depends only on F and G. What is F and G? They are the ones who are describing the curve gamma 2. The curve gamma 2 is parameterized using F and G. So, if gamma 2 is given by y equal to psi x what will happen to this equation? It will be this. Psi will be a solution to this ordinary differential equation. It is a quadratic in dy by dx. So, it is a first order ordinary differential equation but degree 2. So, we may solve for dy by dx from this equation and get this expression dy by dx equal to B plus or minus root B square minus AC by A. So, the moment we see square root we have to become alert. Therefore, if this is negative then there are no real solutions for this equation for dy by dx. If it is positive B square minus AC is positive then we have 2 different ODEs given by this 1 corresponding to plus 1 corresponding to minus and this is equal to 0, we only have 1 OD. So, the equation 1 which is given above is called the characteristic equation for the semi-linear equation and its solutions are characteristic curves. So, characteristic curves are solutions to this OD. In fact, 2 ODs if B square minus AC is positive, 1 OD if B square minus AC is 0 and there is no OD real valued if B square minus AC is negative. So, in which case there will be no characteristic curves. So, if B square minus AC is positive 2 families of real characteristics exist which are transversal to each other because the slope at any point will be at any point of intersection will be 1 will have slope B plus root B square minus AC by A, other one will have B minus root B square minus AC by A both are different. They are one and the same if and only B square minus AC is 0. If B square minus AC is 0 there is only 1 family of real characteristic curves because there is only 1 OD here. If B square minus AC is negative there are no real characteristic curves. So, this is what leads to classification of second order equations. So, let U be a solution of the quasi linear PDE here define a function delta actually it is B square minus AC at this point x, y, u x, y, u x of x, y, u y of x, y that is delta at the point x, y. So, with respect to the given integral surface z equal to u x, y this delta makes sense now it is a function of x, y because u is already given to us therefore all the members of this phi 2 pull are known at any point x, y we say that the given equation is of hyperbolic type at the point x, y if delta x, y is positive it is a parabolic type if delta x, y equal to 0 it is a elliptic type if the delta of x, y is negative at this point remember hyperbolic type at the point x, y parabolic type at the point x, y elliptic type at the point x, y it is very important equation can change times from point to point we will see such examples also. So, if you want to define classification only for semilinear equations we need not start with a solution of the quasi linear PDE we say consider the semilinear PDE a of x, y, u x, x to be of x, y, u x, y plus c of x, y to be u, y, y plus d of the same thing x, y, u, x, u, y equal to 0 defined delta of x, y equal to b square minus ac at the point x, y and exactly the same definition. So, normally people want to avoid this kind of unpleasant thing that you have to say all the time that you have to fix a solution to the quasi linear PDE then only classification is defined people simply start with semilinear equations where you need not mention this but it is of relevance even for quasi linear equations therefore we are presenting it here. Let us look at some examples look at this equation u t, t minus u x, x equal to 0 is called wave equation it is a hyperbolic type at every point x, t see here notice the coefficients here are constants therefore if you go back to the definition if a, b, c are constants b square minus ac will be constant therefore given any equation it will be of exactly one type at all the points. So, here what is delta of x, t b square minus ac b is 0 here a is minus 1 c is 1 therefore delta is 1 this equation u, t minus u, x, x equal to 0 is called heat equation it is a parabolic type at every point since delta x, t here b and c are 0 only a is there a is minus 1 therefore delta is 0 b square minus ac is 0 the Laplace equation u, x, x plus u, y, y equal to 0 is called Laplace equation it is of elliptic type at every point because delta of x, y here b is 0 a and c are 1 so it is minus 1 delta is minus 1 in fact these three equations are the ideal equations they are the simplest examples of each of these types of equations wave equation is a canonical candidate for a hyperbolic equation whenever you think of hyperbolic equation second order you think of wave equation similarly for a parabolic equation if you want to think of it is a heat equation elliptic equation means Laplace equation therefore given any other second order partial differential equation where a, b, c are not constants we will ask the following question is it possible to do a change of coordinates so that after changing the coordinates at least the part which involves second order derivatives will have constant coefficients and looks like this for a hyperbolic equation looks like this for a parabolic equation looks like this for a elliptic equation this is a question that we will ask later on this is a very important example trichomy equation this is also very simple equation u, y, y minus y, u, x, x equal to 0 but these are not with constant coefficients here a of x, y is minus y b of x, y is 0 because u, x, y is absent c of x, y is 1 so b square minus a c is y and b square minus a c can become positive can become negative can become 0 so that is why this equation will have all the three properties of course at different points so trichomy equation is a hyperbolic type in the upper half plane because that is where y is positive parabolic type on the x axis because y equal to 0 is the equation of the x axis it is a elliptic type in the lower half plane because in the lower half plane the y coordinate is less than 0 negative so trichomy equation is one of the simplest linear partial differential equations which is of mixed type on R2 in fact not only in R2 take any domain any domain which contains x axis and therefore it contains some part of upper half plane as well as some part of the lower half plane of course some part of the x axis so it will have all the three types the equation will be of all the three types therefore there will be natural equations what kind of problems I should be able to solve for these kind of equations we will not be discussing trichomy equation as such in this course in fact no mixed type equation will be discussed a wealth of literature is available on trichomy equation now definition of classification is symmetric in A and C why the definition is based on B square minus A C okay A is a coefficient of U x x C is a coefficient of U y y but B square minus C A C will remain the same even if A and C are interchanged so it just says that there is no preferred variable between x and y the type of a quasi linear PDE of second order depends on the terms involving second order derivatives only because it is involving B square minus A C namely the function A B C what are A B C these are the coefficients of the second order derivatives in the partial differential equation of course in any given real number has to be positive negative or 0 that is what is called law of trichotomy therefore a quasi linear PDE always we have to say with a given integral surface it will be one of the three types the type of a quasi linear PDE might vary from point to point we already saw that in the trichomy equation if A B C are constant function the type does not change because B square minus C A C is a constant function if the quasi linear equation is similiar then the type of the equation depends only on the point x y in the plane for such equations the characteristic curves are called characteristic curves for the PDE without any reference to a given integral surface now we ask the following question whether the classification type of an equation changes if you change coordinates what we are asserting here is its invariant under change of coordinates we will consider second order linear equations only for this purpose just for the simplicity in the notations same assertion also holds for 2quiel with same computations so change of variables suppose that we have a change of coordinates from x y to xi eta and vice versa given by these equations xi equal to phi x y eta equal to psi x y and x and y are given as functions of xi and eta a function u of x y gets transformed to a function of xi eta and vice versa by this relation this is a picture we have already come across this in the first order equations discussion where we employed change of variables to solve some problems for linear and semilinear equations to find general solutions of linear and semilinear equations in the first order case now how PDE changes under a change of variables that needs to be understood so let us transform the given equation which is the second order linear equation into xi eta coordinates that requires for us to we know what is x and y in terms of xi and eta so we need to compute u x x u x y u y y u x and u y so differentiating u x y equal to w of phi x y comma psi x y with respect to x and y gives you an expression for u x and u y let us see what is u x so differentiate w with respect to this variable which is xi and then differentiate phi with respect to x differentiate w with respect to eta and then differentiate psi with respect to x this is the chain rule similarly we get u y please do all these computations pause the video do the computations by yourself because the computations are spread over many slides so it is very difficult to show you all the time all the equations so differentiate once again because we need to compute u x x u x y and u y y so that we can go back and substitute in the given linear equation so compute u x x u x y and u y y substitute them in the equation then you get an equation of the following type capital A w xi xi plus 2 times capital B w xi eta capital C w eta eta plus dw xi plus e w eta plus f w plus g equal to 0 now we have to tell what the coefficients a b c d e f g are but we want to say what is the assertion that we wanted to make the type of the equation does not change and the type of the equation is determined by b square minus a c therefore it is enough to know what is this capital A capital B and capital C we do not have to know what d e f g are therefore we do not present so we just write what are the expressions for a b c which are this please do the computations on your own as I pointed out there is no need to compute other coefficients d e f g because the classification type of a p d e depends only on the coefficients of the second order derivatives so b square minus a c you get this expression this is the new b square minus a c right capital letters these are small letters b square minus a c multiplied by with somebody is square okay so as long as this is non-zero the sign of b square minus a c here you will be the same as the science sign of the b square minus a c here if this is negative this will also be negative if this is positive this will also be positive if this is 0 this will also be 0 so we need to know that phi x psi y minus phi y psi x is non-zero but it is non-zero because it is a determinant of the Jacobian matrix corresponding to the change of coordinates therefore it is non-zero therefore the type of the equation does not change under a change of coordinates. Now we give some remarks on classification it is a different basically different perspective the following perspective is presented in the book of Helwig on partial differential equations let us ask some questions is the classification necessitated by the way the mathematical problem is posed is it inherent to the problem itself or is it because we have different methods for different types of equations is it because you know how to solve a certain type of equation therefore you give it and classify it and then solve another type of equation you know how to solve separately so you give it another name in other words you put it in a different classification and do it is that the reason or is it something inherent to the way mathematical problem is posed. So second one could be due to our inability to find a common method to deal with all equations because we are unable to deal with all the equations by a common method we compartmentalize the equations and then solve them separately is that the case that is a question posed here. These questions are purely mathematical therefore it will be ideal to have answers based on the first two criteria of Hadamard Hadamard gave a criterion for a problem to be well posed there are three aspects to that the first criterion is existence of a solution second aspect is uniqueness of a solution third aspect is continuous dependence on the data in the problem. Often the third one is desirable because the equation that we are going to deal with or the problems that we are going to deal with are coming from some physical models and measurements in the physical situations are only approximate therefore by making some errors in the measurements that namely the data and the equation are in the problem we do not get solutions which are very far from the actual solutions. Therefore the third property of Hadamard is desirable but Helmick says can we justify using just existent and uniqueness because these are mathematical questions how many solutions are there whether there is a solution using only these two can you justify the classification. Often people give an explanation for classification by saying that different kinds of problems are well posed for different kinds of equations that is a justification people give. So Helmick does not seem to agree with such a justification therefore he asks himself this question can I justify using only existent and uniqueness parts because these are mathematical. So we present this perspective in the context of systems of first order PDEs. PDEs of any order scalar or a system of equations can be converted into a system of first order PDEs. In fact the same statement was made even for ODEs this is due to Dahlambar we use PDEs in two independent variables denoted by X and Y because arguments are dimension independent. In fact we do not give arguments I request you to consult the book of Helmick for more details my idea was to bring this kind of thinking to your notice. So theorem one existence theorem of Kosi-Kowalowski this we have hinted at in lecture 3.1. Now we are going to pose this theorem for a very special quasi-linear system. Consider this quasi-linear system in this unknowns are u1, u2, un and there are n equations ui, x means derivative of u with respect to x ui with respect to x i equal to 1 to n. n equations there are n unknowns these are quasi-linear because the uk y coefficient depend on the unknowns also in fact x is not considered here okay x and y could also be included but in this theorem it is not included. So consider this system what is if you look at what is that is different from the kind of equation that we are considering here the derivative of x is explicitly given in terms of other derivative okay that is the difference there is no coefficient sitting in the front of this okay let phi i that is phi i of y be analytic functions in a neighborhood of the line y equal to y0 denote ui0 equal to phi i of y0. Now let f i k be analytic in a neighborhood of this vector u0 which is given by u10, u20 and u10 which is nothing but this phi 1 y0, phi 2 y0 up to phi n y0 these are the hypothesis then the quasi-linear system has exactly one solution with the property that is analytic in a neighborhood of this point 0, y0 and this vector solves the quasi-linear system and initial conditions are satisfied. A proof is available in Koran Hilbert's book on methods of mathematical physics the main ideas in the proof were already presented in lecture 3.1 ideas can be implemented without any difficulty we need to ask the following question we had a difficulty in the form of delta s equal to 0 in lecture 3.1 we try to determine all the derivatives first derivatives came up very easily using the Cauchy data second derivatives there was a problem delta s equal to 0 we could not determine so there was a difficulty that is not there here okay so you have to remember the Cauchy-Kowalski theorem is the theorem that we stated so compare the differences between the PDEs considered in both these contexts I already gave a hint what is the difference Cauchy problem handled by theorem 1 looks similar to one of the illustrative examples that we have presented in lecture 3.1 go and find out what that is and what the similarities are in theorem 1 the PDE has a special property X derivative of the unknowns UIs is solved for explicitly that is why delta s does not appear and therefore we can solve the problem without classifying into types it means that even in the analytic setup which is the best of things right because we have nice series for everything that that you have we cannot do away with the classification that is what we observed in lecture 3.1 now we are going to look at one example of Perron this is a reference if you wish you can consult this journal or it was where it is very much explained in the book of Helwig okay for a real number a consider this problem it is a system of two equations u x minus u y minus v y equal to 0 a u y minus v x plus v y plus f of x plus y equal to 0 here f is a given function and you are given this condition u of 0 y and v of 0 y are 0 so the unknowns are u and v okay there is a way to classify first order systems of PDEs which we have not introduced but you can ignore for the moment these names what matters is these conditions so this a is a number depending on whether it is positive 0 or negative the properties of this term system changes we are going to give this assertions so system is hyperbolic if a is positive parabolic if it is 0 a 0 elliptic if a is negative system of first order PDEs have a classification see the book on PDE is by garabidian that book has so system is hyperbolic if a is positive parabolic if a equals 0 elliptic if a less than 0 do not worry if you do not know the definition of hyperbolic parabolic elliptic for a system of first order PDE because assertion will not need that so this is about necessary and sufficient condition for the existence of solution u v in c 1 it is a first order PDE you expert solutions to be c 1 what is the condition in the hyperbolic case that is when a is positive you require that f to be c 0 continuous c 0 means continuous function for parabolic case that is when a is 0 you require f to be c 2 for the same assertion what is the assertion necessary and sufficient condition for the existence of a solution which is c 1 u and v are c 1 functions that changes from case to case if a is positive you require f to be c 0 if a is 0 you require f to be c 2 if a is less than 0 you require more smoothness on f that is analytic. So it also turns out that these are the only solutions that is Hadamard first two criteria are set are met. So this example once again illustrates that the idea of classification is for mathematical reasons not because of the third requirement of Hadamard. Let us summarize what we did in this lecture we introduce the notions of characteristic curves and characteristic equations for second order quasi linear equations. We observed how these notions get simplified for similar equations we classified the second order quasi linear PDEs as hyperbolic parabolic and elliptic this classification is also called classification by characteristics. We presented a different perspective of Helwig on classification of course we have also shown the classification type does not change with change of coordinates and in the next lectures we are going to take wave heat and Laplace equations as inspiration and ask the question whether given any PDE can I do a change of coordinates so that at least the part which involves second order partial derivatives looks like that of one of these three equations those are called canonical forms we will discuss them in the forthcoming lectures. Thank you.