Partial Differential Equations and Complex Variables

Transcription

1 EE-22, Spring 29 p. /25 EE 22 Partial Differential Equations and Complex Variables Ray-Kuang Lee Institute of Photonics Technologies, Department of Electrical Engineering and Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan Course info: rklee

2 EE-22, Spring 29 p. 2/25 EE 22 Time: M7M8R6 (3:2PM-5:PM, Monday; 2:AM-3:PM, Thursday) Teaching Method: in-class lectures with examples. I would try to write in the black board, not the slides. Evaluation:. Four Homeworks, 4%; 2. Midterm 3%; Tentatively scheduled on 4/27, covering Ch.2 of the textbook. 3. Final exam 3%: Tentatively scheduled on 6/5, covering Ch.3 - Ch.8 of the textbook. 4. Bonus: just rise your hand in the classroom, 2%.

3 EE-22, Spring 29 p. 3/25 Textbook and Reference Books [Note]: Class handouts; Prof. S.D. Yang s note: sdyang/courses/pde.htm [Textbook]: E. Kreyszig, "Advanced Engineering Mathematics", 9th Ed., John Wiley & Sons, Inc., (26). [Ref.]: Stanley J. Farlow, "Partial Differential Equations for Scientists and Engineers", Dover Publications, (993); (for PDE, but optional). [Ref.num]: Matthew P. Coleman, "An Introduction to Partial Differential Equations with MATLAB", Chapman & Hall/Crc Applied Mathematics & Nonlinear Science (24); (optional).

4 EE-22, Spring 29 p. 4/25 Syllabus: for PDE. Introduction to PDE and Complex variables, (2/23, 2/26). 2. Diffusion-type problems: [Textbook] Ch.2, [Ref.] Ch.2. Derivation of the Heat equation, (3/2). Boundary conditions for Diffusion-type problems, (3/5). Separation of variables, (3/9). Solving nonhomogeneous PDEs, (3/2). Integral transforms, (3/6, 3/9). The Fourier transform, (3/23). The Laplace Transform, (3/26). 3. Hyperbolic-type problems: [Textbook] Ch.2, [Ref.] Ch.3. -D Wave equation, (4/2, 4/6). D Alembert solution of the Wave equation, (4/9). Sturm-Liouville problems, (4/3). 2-D Wave equation in Cartesian and polar coordinates, (4/6, 4/2). Laplace s equation in Cartesian, polar, and spherical coordinates, (4/23).

5 EE-22, Spring 29 p. 5/25 Syllabus: for Complex variables. Midterm, (4/27). 2. Introduction to Numerical PDE (4/3): [Ref.num]. 3. Complex variables: [Textbook]Ch.3-Ch.8. Complex numbers and functions, (5/4). Cauchy-Riemann equations, (5/7, 5/). Complex integration, (5/4, 5/8). Complex power & Taylor series, (5/2, 5/25). Laurent series & residue, (5/28, 6/, 6/4). Conformal mapping, (6/8, 6/). Applications: real integrals by residual integration, potential theory, (6/5, 6/8). 4. Final exam, (6/5).

6 EE-22, Spring 29 p. 6/25 Related courses. Applied Mathematics (Phys.), 2. Complex Analysis (Math.), 3. Numerical Mehtods for Parital Differential Equations (Math.), 4. Numerical Analysis (EE), 5. Computational Methods for Optoelectronics (IPT), 6....

7 EE-22, Spring 29 p. 7/25 Partial Differential Equations A(x,y) 2 u x + B(x,y) 2 u 2 x y + u u C(x,y) 2 = f(x,y,u, y2 x, u y ),

8 EE-22, Spring 29 p. 8/25 Vector calculus: scalar and vector fields scalar fields: Ψ, f, V, ρ vector fields: A, F, E, H, D, B, J

9 EE-22, Spring 29 p. 9/25 Vector calculus: Gradient For the measure of steepness of a line, slope. the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. f(x, y, z) = f xî + f y ĵ + f ˆk, z f(ρ, θ, z) = f ρ êρ + ρ f(r, θ, φ) = f r êr + r f θ êθ + f z êz, f θ êθ + r sin θ f φêφ, in Cartesian coordinates in cylindrical coordinates in spherical coordinates

10 EE-22, Spring 29 p. /25 Maxwell s equations with total charge and current Gauss s law for the electric field: E = ρ ǫ S E d A = Q ǫ, Gauss s law for magnetism: B = S B d A =, (83-879) Faraday s law of induction: E = κ t B C E d l = κ t Φ B, Ampére s circuital law: B = κµ (J + ǫ E) t C B d l = κµ (I + ǫ t Φ E)

11 EE-22, Spring 29 p. /25 Wave equations For a source-free medium, ρ = J =, 2 ( E) = µ ǫ t 2 E, ( E) 2 2 E = µ ǫ t 2 E. When E =, one has wave equation, 2 E = µ ǫ 2 t 2 E which has following expression of the solutions, in D, with v 2 = µ ǫ = c 2. E = ˆx[f + (z vt) + f (z + vt)], plane wave solutions: E + = E cos(kz ωt), where ω k = c.

12 Cavity modes EE-22, Spring 29 p. 2/25

13 EE-22, Spring 29 p. 3/25 More PDEs Diffusion equation: Schrödinger equation: i 2 Ψ(x,t) = t 2m 2 A(x,t) = κ t x 2A(x,t) Nonlinear Schrödinger equation: i 2 Ψ(x,t) = t 2m 2 x2ψ(x,t) + V (x)ψ(x,t) 2 x 2Ψ(x,t)+V (x)ψ(x,t)+γ Ψ(x,t) 2 Ψ(x,t)

14 EE-22, Spring 29 p. 4/25 Diffusion equation z U(z, t) = i 2 2 U(z, t) t2 Z Y X.75 Intensity [a.u.] Time 5 5 Distance

15 EE-22, Spring 29 p. 5/25 Laplacian eq. in a disk Eigenmodes of Laplacian equations, [ 2 x x 2 ]u(x, y) = f(x, y). Mode λ =. Mode 3 λ = Mode 6 λ = Mode λ =

16 EE-22, Spring 29 p. 6/25 FFT method for wave equation u tt = u xx + u yy, < x, y <, t >, u = on the boundary t = t = t = t =.5.5

17 EE-22, Spring 29 p. 7/25 Mach-Zehnder structure by BeamProp

18 EE-22, Spring 29 p. 8/25 Soliton collisions U(t =, x) = sech(x + x ) + sech(x x )

19 EE-22, Spring 29 p. 9/25 FDTD: example From:

20 EE-22, Spring 29 p. 2/25 FDTD: example From:

21 EE-22, Spring 29 p. 2/25 Metallic Waveguide Examples in "Field and Wave Electromagnetics," 2nd ed., by David K. Cheng, pp ; simulated by ToyFDTD

22 EE-22, Spring 29 p. 22/25 Fields profile in 2D, H x (x,z), H z (x,z), and E y (x,z) g933326,

23 EE-22, Spring 29 p. 23/25 Optimization of SHG pulse A z B z = η A 2 T + iξ 2 A T 2 iρ A B, = η B 2 T + iξ 2 A 2 T 2 i kb iρ A 2,

24 EE-22, Spring 29 p. 24/25. Office hours: 3:-5:PM, Thursday at Room 523, EECS bldg I should reply every Website: For more information and course slides: rklee 4. TA hours: at Room 52, EECS bldg. (2 2 hours per week to be confirmed) (a) I-Hong Chen, 2nd-year PhD student, ehome4829@gmail.com (b) Chih-Yao Chen, 2nd-year Master student, hihittttt@yahoo.com.tw

25 EE-22, Spring 29 p. 25/25 Enjoy this Course!! 2> ω a > Frequency (ωd/2πc) x j M Γ X M -5 5 x i