Welcome to Math 257/316 - Partial Differential Equations
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1 Welcome to Math 257/316 - Partial Differential Equations Instructor: Mona Rahmani mrahmani@math.ubc.ca Office: Mathematics Building 110 Office hours: Mondays 2-3 pm, Wednesdays and Fridays 1-2 pm. Course webpage: teaching/math / (all assignments and lecture notes will be posted here.) A list of course topics is given in the course outline on the webpage.
2 List of Topics 1. Review of techniques to solve ODEs 2. Series solutions of variable coefficient ODEs 3. Introduction to partial differential equations a. The heat equation b. The wave equation c. Laplace s equation 4. Introduction to numerical methods for PDEs 5. Fourier series and separation of variables. a. The heat equation b. The wave equation c. Laplace s equation 6. Boundary value problems and Sturm-Liouville theory
3 Text book (recommended but not required): Elementary Differential Equations and Boundary Value Problems by Boyce & DiPrima, Etext. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (4nd Ed), R. Haberman, (Pearson), Partial Differential Equations for Scientists and Engineers (1st Ed), S. Farlow, (Dover), Online resources: Professor Anthony Peirce s course material: ~peirce/ you can find all last terms lecture notes, previous assignments, exams and their solutions on this webpage. Professor Richard Froese s lecture notes: ~rfroese/notes/lecs316.pdf
4 Formula sheet - also posted on the course webpage: You will get this formula sheet on the exams. Please make sure you become familiar with the notation as we progress through the term. Math PDE Formula sheet - final exam Trigonometric and Hyperbolic Function identities sin( ± )=sin cos ± sin cos sin 2 t + cos 2 t =1 cos( ± ) = cos cos sin sin. sin 2 t = 1 2 (1 cos(2t)) sinh( ± ) = sinh cosh ± sinh cosh cosh 2 t sinh 2 t =1 cosh( ± ) = cosh cosh ± sinh sinh. sinh 2 t = 1 2 (cosh(2t) 1) Basic linear ODE s with real coe cients constant coe cients Euler eq ODE ay 00 + by 0 + cy =0 ax 2 y 00 + bxy 0 + cy =0 indicial eq. ar 2 + br + c =0 ar(r 1) + br + c =0 r 1 6= r 2 real y = Ae r1x + Be r 2x y = Ax r 1 + Bx r 2 r 1 = r 2 = r y = Ae rx + Bxe rx y = Ax r + Bx r ln x r = ± iµ e x [A cos(µx)+bsin(µx)] x [A cos(µ ln x )+Bsin(µ ln x )] Series solutions for y 00 + p(x)y 0 + q(x)y =0(?) around x = x 0. Ordinary point x 0 : Two linearly independent solutions of the form: y(x) = P 1 n=0 a n(x x 0 ) n Regular singular point x 0 : Rearrange (?) as: (x x 0 ) 2 y 00 + [(x x 0 )p(x)](x x 0 )y 0 + [(x x 0 ) 2 q(x)]y =0 If r 1 >r 2 are roots of the indicial equation: r(r 1) + br + c = 0 where b = lim x!x 0 (x x 0 )p(x) and c = lim x!x 0 (x x 0 ) 2 q(x) then a solution of (?) is y 1 (x) = P 1 n=0 a n(x x 0 ) n+r 1 where a 0 =1. The second linerly independent solution y 2 is of the form: Case 1: If r 1 r 2 is neither 0 nor a positive integer: 1X y 2 (x) = b n (x x 0 ) n+r 2 where b 0 =1. Case 2: If r 1 r 2 = 0: n=0 y 2 (x) =y 1 (x) ln(x x 0 )+ 1X b n (x x 0 ) n+r 2 for some b 1,b 2... n=1 Case 3: If r 1 r 2 is a positive integer: 1X y 2 (x) =ay 1 (x) ln(x x 0 )+ b n (x x 0 ) n+r 2 where b 0 =1. n=0 Fourier, sine and cosine series Let f(x) be defined in [ L, L]then its Fourier series Ff(x) isa2l-periodic function on R: Ff(x) = a P 1 n=1 a n cos( n x L )+b n sin( n x R L ) where a n = 1 L n x L f(x) cos( L L ) dx and b R n = 1 L n x L f(x) sin( L L ) dx Theorem (Pointwise convergence) If f(x) and f 0 (x) are piecewise continuous, then Ff(x) converges for every x to 1 2 [f(x )+f(x+)]. Parseval s indentity Z 1 L f(x) 2 dx = a 0 2 1X + a n 2 + b n 2. L 2 L n=1 For f(x) defined in [0,L], its cosine and sine series are Cf(x) = a X Sf(x) = 1X n=1 n=1 a n cos( n x L ), b n sin( n x L ), a n = 2 L b n = 2 L Z L D Alembert s solution to the wave equation 0 Z L 0 f(x) cos( n x L ) dx, f(x) sin( n x L ) dx. PDE: u tt = c 2 u xx, 1 <x<1, t>0 IC: u(x, 0) = f(x), u t (x, 0) = g(x). SOLUTION: u(x, t) = 1 2 [f(x + ct)+f(x ct)] + R 1 x+ct 2c x ct g(s)ds Sturm-Liouville Eigenvalue Problems ODE: [p(x)y 0 ] 0 q(x)y + r(x)y =0, a < x < b. BC: 1 y(a)+ 2 y 0 (a) = 0, 1y(b)+ 2 y 0 (b) = 0. Hypothesis: p, p 0, q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for x 2 [a, b] > > 0. Properties (1) The di erential operator Ly =[p(x)y 0 ] 0 q(x)y is symmetric in the sense that (f, Lg) =(Lf, g) for all f,g satisfying the BC, where (f,g) = R b a f(x)g(x) dx. (2) All eigenvalues are real and can be ordered as 1 < 2 < < n < with n!1as n!1, and each eigenvalue admits a unique (up to a scalar factor) eigenfunction n. (3) Orthogonality: ( m,r n )= R b a m(x) n (x)r(x) dx = 0 if m 6= n. (4) Expansion: If f(x) :[a, b]! R is square integrable, then R 1X b a f(x) n(x)r(x) dx f(x) = n=1 c n n (x), a<x<b,c n = R b a 2 n(x)r(x) dx,n=1, 2,... 1
5 Grades: Final exam: 50%. Students must get at least 35% on the final exam to pass the course. Two in-class midterm exams (each 20%): 40%. There will be no make-up midterms. If you cannot make it to any of the midterms (for a legitimate reason), you must inform me at least two days before the test date. Homework (including Matlab assignments): 10%. Assignment should be submitted at the beginning of the class on the day they are due. No late submission or electronic submission will be accepted. You must submit your assignment at the section you are registered in. The handed in assignment must be your own work.
6 Midterm dates (two in-class midterms): Friday, February 15 Wednesday, March 20
7 Matlab: Some homework include Matlab assignments. The scripts will be provided. However, the assignments may require some modification of the scripts. Matlab is free for all UBC students. See UBC IT services.
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