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hello and welcome to the first video in.my video series on partial differential.equations so I'll start with a.definition and that is a PDE is an.equation relating the partial.derivatives of some unknown function so.hopefully you've studied some ordinary.differential equations already and you.know that an ordinary differential.equation relates the ordinary.derivatives so when there's only one.independent variable now we have perhaps.many independent variables and so we.have partial derivatives and the.equation will relate those partial.derivatives now oftentimes the goal when.we study PDE is to learn something about.this unknown function and this unknown.function will call a solution and the.solution is often something that we want.to find an explicit formula for or.sometimes it's something we just want to.know something about we want to know.some of its properties now we will have.three basic goals in our study of.partial differential equations let me.describe those now so the first goal is.one that's kind of easily overlooked and.that is to learn how to formulate.partial differential equations so what.do I mean by this well if you're.studying some field of science and.engineering you often encounter some.kind of dynamical process you may be.observing data from laboratory.experiments you may be driving something.from physical principles but you're.going to encounter a dynamical process.and you're going to want to convert that.into a mathematical description of the.process and that is the art of.formulating a PDE coming from a.dynamical process based on perhaps.laboratory data and writing down a.mathematical equation to describe that.process so this is a very useful skill.the second goal will be pretty much the.most obvious one and that is to solve.the partial differential equations.that is to find the unknown functions.that satisfy the relation that the.partial differential equation describes.so we're going to spend most of our time.doing this we won't be able to solve.every PDE that we encounter but we will.be able to develop techniques that will.solve large classes of important.equations and so that will be most of.our focus now the third goal is to study.the solutions to partial differential.equations now I don't necessarily mean.to study the formulas that we get from.part two oftentimes we will have trouble.obtaining formulas or we may get.formulas which don't give us a lot of.information and so we would like to have.techniques for learning things about the.solutions to partial differential.equations without doing a lot of hard.work to find formulas for the solutions.so things that we can learn about a.partial differential equation just from.the equation itself.I'll introduce now a little bit of.notation that I'll use in the rest of.the videos to be consistent so for.independent variables I'll use things.like X and y and z and T so these are.the independent variables.for dependent variables I'll use letters.like you and perhaps V if I have more.than one and I'll use subscripts for.partial derivatives so when I write.something like UX I mean the partial.derivative of U with respect to X or if.I write u X T I mean the partial.derivative second partial derivative.with respect to X and then respect to T.okay so these V subscripts for partial.derivatives and one thing I should point.out here is that I will use things like.XY and Z for spatial coordinates so.these will be spatial coordinates and.something like T will be time well often.have partial differential equations.which describe some relationship between.derivatives with respect to space and.with respect to time so right now we.should just kind of get our feet wet a.little bit by looking at a particular.partial differential equation and.examples of solutions to that PDE so the.equation we're going to look at is UT.equals u xx this equation is called the.heat equation and I'll explain in the.next video why we would want to call it.the heat equation but for now let's just.take it as it is.u T equals u xx now I claim first of all.that the function u equals one-half x.squared plus T is indeed a solution to.the heat equation.let's check it okay so check how do we.check well we take this particular.function and we compute u t + u xx and.compare them and see that they're equal.and then it will be we know that it will.be a solution at that point.so first let's compute UT UT is the.partial derivative of U with respect to.T now that is one because the partial.derivative with respect to T of T is 1.and the partial derivative of one half x.squared with respect to T is zero now.we'll compute u xx but first let's.compute u X and when then we'll take one.more partial derivative so u X is.partial derivative of U with respect to.X and that is well the T goes away.because the derivative of T with respect.to X is zero and I get a two times.one-half X and so that's just X so.taking one more derivative I get u xx.which is the second partial with respect.to X that's just 1 so checking that the.heat equation is true amounts to.checking that UT is equal to u xx and.this is simply the identity 1 equals 1.so we get a solution so this is we've.checked this is a solution to the heat.equation now just just for fun let me.point out that this guy right here this.this this particular solution to the.heat equation this is called a heat.polynomial.and the reason it's called the heat.polynomial is because it is a polynomial.in X and T and it is a solution to the.heat equation there are many heat.polynomials and I challenge you to find.another one to find a different heat.polynomial there's there's many and.they're not that hard to find ok now.let's do something just a little bit.different let's see if we can find a.solution to the heat equation simply by.making a guess so this guess is often.called a non sots.and it's a little smarter than a guess.what it is is it's an assumption that.the solution has a certain form so what.we're going to assume is that the.solution has the form e to the ax plus.BT this kind of an assumption is a lot.more powerful than you might first.assume and we'll see that later so.what's the question here question is.what do a and B need to be in order for.this expression to be a solution to the.heat equation so in other words what are.a and B or what relationship do a and B.satisfy well let's figure it out so we.plug it in to the heat equation we have.u T equals well it's e to the ax plus B.T times by the chain rule the derivative.of this inside function with respect to.T so we have partial of ax plus B T with.respect to T and that gives us simply e.to the ax plus B T times B now we'll.compute u xx but let's do UX first.that gives us e to the ax plus B T times.the partial derivative of ax plus B T.now with respect to X so that gives us e.to the ax plus BT.a and similarly we would compute uxx to.be e to the ax plus B T times a squared.so together this information tells us.that this expression this function will.be a solution to the heat equation if.the following relationship is satisfied.so u T equals u xx which is the same.saying that e to the ax plus B T times B.is equal to e to the ax plus B T times a.squared okay well I can cancel these.guys off let's cancel off this e to the.X plus B T now maybe I should say.something real quick about that.cancellation these cannot be 0 if i.exponentiate anything at all i get.something which is nonzero.so i am perfectly it's perfectly.legitimate to cancel these terms so what.I end up with is B equals a squared so.if you choose any a and B that satisfy.this relationship you get a solution to.the heat equation so for instance a.could be -1 and B could be 1 or they.could both be 0 or maybe a could be I.and B could be minus 1 if you think.about that alright I'd like to conclude.this introduction with a quick.comparison of solutions to ordinary.differential equations and solutions to.partial differential equations so this.will be just a comparison by example and.a very simple one but when you're just.starting out simple as good so let's.start with the OD e which will be the.population equation.now this OTE is DP DT equals P.frequently there's a constant in front.of this P and we're going to leave it.off just to keep things simple now a.general solution to this code ee is P of.T equals C times e to the T the see here.is an arbitrary constant and this is a.thing that you see very frequently when.you solve ordinary differential.equations you find arbitrary constants.in the solutions in fact you often will.find as many arbitrary constants in the.general solution as the order of the OD.e so is the number of derivatives that.you find in the OD e just one derivative.here so just one arbitrary constant now.let's look at the same equation but.let's just make it a PDE so we'll say a.PDE that looks like the population.equation.okay.and what I mean by that well I'll just.write you t equals u and here we're.going to assume U is a function which.depends on more than one independent.variable so u depends on say X X and T.okay so this is now a partial.differential equation and it looks very.similar to this ordinary differential.equation let's see what we can figure.out about the general solution so first.of all notice that if I take the.function so I'll say a question mark if.I take the function u equals C e to the.T okay so just like I took P and let me.be explicit this is a function of X and.T so this is U of X comma T if I take U.of XT equals C e to the T indeed this is.a solution to this equation because when.I take the partial derivative with.respect to T and I and I plug it in I.will find that I get a solution to the.equation for any C that I choose but you.may ask yourself why didn't I.incorporate X somehow I should be able.to incorporate X and if you think about.it for awhile you realize that the.partial derivative with respect to T.treats X as a constant so by plugging in.X anywhere in this equation it's.essentially acts like a constant so what.I could do is I could take this C and I.could replace it with any function of X.at all.okay so let's call it capital f of X.where this guy now is an arbitrary.function of X arbitrary function of X.so arbitrary function appearing in this.general solution check it and make sure.that you believe that that's the case.take the partial derivative with respect.to T and compare it to the original.function so here in an ODI at an.arbitrary constant here in a PDE I have.an arbitrary function appearing in the.general solution now this is not a a.general theorem but this is a phenomenon.that you will see very frequently you.will find arbitrary functions appearing.in the general solutions let's just give.you a basic idea as to what the.difference is between solutions to OD e.and PD.

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