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Laplace Equation

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13:17   |   May 06, 2016

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  • Today I'm speaking about the first of the three great
  • partial differential equations.
  • So this one is called Laplace's equation, named after Laplace.
  • And you see partial derivatives.
  • So we have-- I don't have time.
  • This equation is in steady state.
  • I have x and y, I'm in the xy plane.
  • And I have second derivatives in x and then y.
  • So I'm looking for solutions to that equation.
  • And of course I'm given some boundary values.
  • So time is not here.
  • The boundary values, the boundary
  • is in the xy plane, maybe a circle.
  • Think about a circle in the xy plane.
  • And on the circle, I know the solution u.
  • So the boundary values around the circle are given.
  • And I have to find the temperature
  • u inside the circle.
  • So I know the temperature on the boundary.
  • I let it settle down and I want to know the temperature inside.
  • And the beauty is, it solves that basic partial differential
  • equation.
  • So let's find some solutions.
  • They might not match the boundary values,
  • but we can use them.
  • So u equal constant certainly solves the equation.
  • U equal x, the second derivatives will be 0.
  • U equal y.
  • Here is a better one, x squared minus y squared.
  • So the second derivative in the x direction is 2.
  • The second derivative in the y direction is minus 2.
  • So I have 2, minus 2, it solves the equation.
  • Or this one, the second derivative in x is 0.
  • Second derivative in y is 0, those are simple solutions.
  • But those are only a few solutions
  • and we need an infinite sequence because we're going
  • to match boundary conditions.
  • So is there a pattern here?
  • So this is degree 0, constant.
  • These are degree 1, linear.
  • These are degree 2, quadratic.
  • So I hope for two cubic ones.
  • And then I hope for two fourth degree ones.
  • And that's the pattern, that's the pattern.
  • Let me find-- let me spot the cubic ones.
  • X cubed, if I start with x cubed, of course the second x
  • derivative is probably 6x.
  • So I need the second y derivative to be minus 6x.
  • And I think minus 3xy squared does it.
  • The second derivative in y is 2 times
  • the minus 3x is minus 6x, cancels
  • the 6x from the second derivative there, and it works.
  • So that fits the pattern, but what is the pattern?
  • Here it is.
  • It's fantastic.
  • I get these crazy polynomials from taking x plus iy
  • to the different powers.
  • Here to the first power, if n is 1, and I just have x plus iy
  • and I take the real part, that's x.
  • So I'll take the real part of this.
  • The real part of this when n is 1, the real part is x.
  • What about when n is 2?
  • Can you square that in your head?
  • So we have x squared and we have i squared y squared,
  • i squared being minus 1.
  • So I have x squared and I have minus y squared.
  • Look, the real part of this when n
  • is 2, the real part of x plus iy squared,
  • the real part is x squared minus y squared.
  • And the imaginary part was the 2ixy.
  • So the imaginary part that multiplies i is the 2xy.
  • This is our pattern when n is 2.
  • And when n is 3, I take x plus iy cubed, and that
  • begins with x cubed like that.
  • And then I think that the other real part
  • would be a minus 3xy squared.
  • I think you should check that.
  • And then there will be an imaginary part.
  • Well, I think I could figure out the imaginary part as I think.
  • Maybe something like minus-- maybe it's
  • 3yx squared minus y cubed, something like that.
  • That would be the real part and that would be
  • the imaginary part when n is 3.
  • And wonderfully, wonderfully, it works
  • for all powers, exponents n.
  • So I have now sort of a pretty big family of solutions.
  • A list, a double list, really, the real parts
  • and the imaginary parts for every n.
  • So I can use those to find the solution
  • u, which I'm looking for, the temperature inside the circle.
  • Now of course, I have a linear equation.
  • So if I have several solutions, I can combine them
  • and I still have a solution.
  • X plus 7y will be a solution.
  • Plus 11x squared minus y squared, no problem.
  • Plus 56 times 2xy.
  • Those are all solutions.
  • So I'm going to find a solution, my final solution
  • u will be a combination of this, this, this, this, this, this,
  • this, and all the others for higher n.
  • That's going to be my solution.
  • And I will need that infinite family.
  • See, partial differential equations,
  • we move up to infinite family of solutions instead of just
  • a couple of null solutions.
  • So let me take an example.
  • Let me take an example.
  • We're taking the region to be a circle.
  • So in that circle, I'm looking for the solution u of x and y.
  • And actually in a circle, it's pretty natural
  • to use polar coordinates.
  • Instead of x and y inside a circle that's
  • inconvenient in the xy plane, its equation
  • involves x equals square root of 1 minus y squared or something,
  • I'll switch to polar coordinates r and theta.
  • Well, you might say you remember we had
  • these nice family of solutions.
  • Is it still good in polar coordinates?
  • Well the fact is, it's even better.
  • So the solution of u will be the real part
  • and the imaginary part.
  • Now what is x plus iy in r and theta?
  • Well, we all know x is r Cos theta plus ir sine theta.
  • And that's r times Cos theta plus i sine theta,
  • the one unforgettable complex Euler's formula, e
  • to the I theta.
  • Now, I need its nth power.
  • The nth power of this is wonderful.
  • The real part and imaginary part of the nth power
  • is r to the nth e to the in theta.
  • That's my x plus iy to the nth.
  • Much nicer in polar coordinates, because I
  • can take the real part and the imaginary part right away.
  • It's r to the nth Cos n theta and r to the nth sine n theta.
  • These are my solutions, my long list of solutions,
  • to Laplace's equation.
  • And it's some combination of those,
  • my final thing is going to be some combination of those,
  • some combination.
  • Maybe coefficients a sub n.
  • I can use these and I can use these.
  • So maybe b sub n r to the nth sine n theta.
  • You may wonder what I'm doing, but what I'm achieved,
  • it's done now, is to find the general solution
  • of Laplace's equation.
  • Instead of two constants that we had
  • for an ordinary differential equation, a C1 and a C2,
  • here I have these guys go from up to infinity.
  • N goes up to infinity.
  • So I have many solutions.
  • And any combination working, so that's the general solution.
  • That's the general solution.
  • And I would have to match that-- now here's
  • the final step and not simple, not always simple--
  • I have to match this to the boundary conditions.
  • That's what will tell me the constants, of course.
  • As usual, c1 and c2 came from the matching the conditions.
  • Now I don't have just c1 and c2, I have this infinite family
  • of a's, infinite family of b's.
  • And I have a lot more to match because on the boundary,
  • here I have to match u0, which is given.
  • So I might be given, suppose I was
  • given u0 equal to the temperature was equal 1
  • on the top half.
  • And on the bottom half, say the temperature is minus 1.
  • That's a typical problem.
  • I have a circular region.
  • The top half is held at one temperature,
  • the lower half is held at a different temperature.
  • I reach equilibrium.
  • Everybody knows that along that line,
  • probably the temperature would be 0 by symmetry.
  • But once the temperature there halfway up, not so easy,
  • or anywhere in there.
  • Well, the answer is u in the middle, u of r and theta
  • inside is given by that formula.
  • And again, the ANs and the BNs come
  • by matching the-- getting the right answer on the boundary.
  • Well, there's a big theory there how do I match these?
  • That's called a Fourier series.
  • That's called a Fourier series.
  • So I'm finding the coefficients for a Fourier series, the A's
  • and B's, that match a function around the boundary.
  • And I could match any function, and Fourier series
  • is another entirely separate video.
  • We've done the job with Laplace's equation in a circle.
  • We've reduced the problem to a Fourier series problem.
  • We have found the general solution.
  • And then to match it to a specific given boundary
  • value, that's a Fourier series problem.
  • So I'll have to put that off to the Fourier series video.
  • Thank you.

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Description

MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: http://ocw.mit.edu/RES-18-009F15
Instructor: Gilbert Strang

Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region.

License: Creative Commons BY-NC-SA
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