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HARVARD MATH 21A - PDE's

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PDE’s Math 21aOn this handout we deal with functions f(t, x) of two variables. These functions satisfy equa-tions containing partial derivatives called partial differential equations or shortly PDE’s.The topic of PDE’s would fill a course by itself. Finding and understanding solutions of suchequations can be very difficult. The topic is used here only as an exercise for partial differ-entiation. No knowledge on PDE’s will be required from you in this course. You should beable to verify however that a given function f(t, x) satisfies a specific PDE.LAPLACE EQUATION. fxx+ fyy= 0 . A stationarytemperature distribution on a plate satisfies this equa-tion.1) Verify that f(x, y) = x3− 3xy2satisfies the Laplaceequation.ADVECTION EQUATION. ft= fx. Models trans-port in a one-dimensional medium. It is also called atransport equation. In the homework, you look at aslightly more general case.2) Verify thatf(t, x) = e−(x+t)2satisfy the advectionequation ft(t, x) = fx(t, x).WAVE EQUATION. ftt= fxx. For fixed time t, thefunction x 7→ f(t, x) describes a string at that time.3) Verify thatf(t, x) = sin(x − t) + sin(x + t) satisfiesthe wave equation ftt(t, x) = fxx(t, x).HEAT EQUATION. ft= fxxFor fixed time t, thefunction x 7→ f(t, x) is the temperature at the pointx. The heat equation is also called diffusion equation.The functionf(t, x) =1√te−x2/(4t)satisfies the heatequation.BURGER EQUATION. ft+ ffx= fxxDescribes onedimensional waves (i.e. at beach). In higher dimen-sions, it leads to the Navier Stokes equation. One ofthe millennium (106\$) problems is to solve the existenceproblem in 3D.There are N-wave solutionsf(t, x) =xt√1te−x2/(4t)1+√1te−x2/(4t)Without fxxterm, solutions will break (form shocks).KDV-EQUATION. ft+ 6ffx+ fxxx= 0 Describeswater waves in a narrow channel. First discovered byJ. Scott Russel in 1838.The solutionf(t, x) =a22cosh−2(a2(x − a2t))describesa wave with speed a2and amplitude a2/2. It is calleda soliton. Unlike linear waves, these nonlinear wavescan travel with different speed: higher waves movefaster. Solitons form a fancy research topic.SCHR¨ODINGER EQUATION. ft=i¯h2mfxxDescribesa free quantum particle of mass m.A solution is f(t, x) = ei(kx−¯h2mk2t)models a particlewith momentum ¯hk. The constant i satisfies i2= −1(it is an imaginary number), ¯h is the Planck constant¯h ∼ 10−34Js.”A great deal of my work is just playing with equations and seeing whatthey give. I don’t suppose that applies so much to other physicists; I think it’sa peculiarity of myself that I like to play about with equations, just lookingfor beautiful mathematical relations which maybe don’t have any physicalmeaning at all. Sometimes they do.” - Paul A. M. Dirac.Dirac discovered a PDE describing the electron which is consistent both with quantum theory and special relativity.This won him the Nobel Prize in 1933. Dirac’s equation could have two solutions, one for an electron with positiveenergy, and one for an electron with negative energy. Dirac interpretated the later as an antiparticle: the existence ofantiparticles was later

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