PDE

Question 1

General Form of PDE

Answer 1

F(x,y,…,u,ux,uy,…,uxx,uyy,…) = 0 where ux = du/dx and uxx = d^2u/dx^2 etc

Question 2

Order of a PDE

Answer 2

The order of the highest derivative it contains

Question 3

Homogeneous Linear PDE

Answer 3

F(x,y,…Cu,Cux,…,Cuxx) = CF(x,y,…,u,ux,…,uxx,…)

Question 4

Linear PDE

Answer 4

If F is linear in u, ux, uy, …, uxx, uyy with coefficients depending only on the independent variables x,y

Question 5

Laplace’s Equation

Answer 5

∇^2 = d^2 / dx^2 + d^2 / dy^2 + d^2 / dz^2

Question 6

Poisson’s Equation

Answer 6

∇^2u = f(x,y,z)

Question 7

Wave Equation in 1D

Answer 7

d^2u / dt^2 -c^2 (d^2u / dx^2 + d^2u / dy^2) = 0

Question 8

Heat Equation

Answer 8

dT / dt -K∇^2T = 0 where K = k/cp

Question 9

Heat Equation for a uniform rod along x axis

Answer 9

ut - Kuxx = 0

Question 10

Solution for the wave equation for a string with fixed ends u(0,t) = u(l,t) = 0

Answer 10

Of the form u(x,t) = v(x)q(t)

Question 11

Solution for a circular membrane

Answer 11

Of the form u(r,Φ,t) = v(r,Φ)q(t)

Question 12

General form of a power series

Answer 12

R(z) = ∞ Σ n=0 anz^n

Question 13

Bessel Function of the first kind of order z

Answer 13

Jo(z) = ∞ Σ n=0 (-1)^k (z/2)^2k / (k!)^2

Question 14

Bessel Function of the first kind of order m

Answer 14

Jm(z) = (z/2)^m ∞ Σ n=0 (-1)^k (z/2)^2k / k!(m+k)!

Question 15

General solution for q(t) in 1D heat equation

Answer 15

q(t) = Ce^-λt

Question 16

General solution for v(x) in 1D heat equation

Answer 16

v(x) = Acos (√λ/K x ) + Bsin (√λ/K x)

Question 17

u(x,t) for heat eqation using superposition principal

Answer 17

u(x,t) =∞ Σ n=1 Bn sin (nπx / l ) e^(-n^2π^2 / l^2)Kt

Question 18

Superposition Principle

Answer 18

If u1, u2, … are solutions of the homogeneous equation, then cu1 + cu2 + … is also a solution

Question 19

Fourier Series General Form

Answer 19

a0/2 + ∞ Σ n=1 (ancos(nx) + bnsin(nx)) where an = 1/π π ∫ -π f(x) cos(nx)dx, bn = 1/π π ∫ -π f(x) sin(nx)dx

Question 20

Convergence Theorem

Answer 20

Let f be a piecewise smooth function on (-π, π), and let all discontinuities be finite jumps. Then the Fourier Series converges to f(x) for all x where f is continuous

Question 21

Fourier Series for Even Functions

Answer 21

f(x) = ao/2 + ∞ Σ n=1 an.cos(nx) since the bn term is 0 for even funcitons

Question 22

Fourier Series for Odd Functions

Answer 22

f(x) = ∞ Σ n=1 bn.sin(nx), since an term is 0 for odd functions

Question 23

Complex Form of Fourier Series

Answer 23

f(x) = ∞ Σ n=-∞ cne^inx

Question 24

Integration by Parts

Answer 24

∫ udv = uv - ∫ vdu

Question 25

Definition of Hyperbolic Functions

Answer 25

coshx = 1/2 (e^x + e^-x), sinhx = 1/2 (e^x - e^-x)

Question 26

Fourier Series for f(x) on the interval [-l, l]

Answer 26

f(x) = a0/2 + ∞ Σ n=1 (an.cos(nπx / l) + bn.sin(πnx / l)

Question 27

Solution of wave equation for boundary conditions u(0,t) = 0, ut(x,0) = 0

Answer 27

u(x,t) = (Acos.wnt + Bsin.wnt)sin(nπx / l) where wn = nπc / l, A,B arbitrary constants

Question 28

Laplace's Equation in Polar Coordinates

Answer 28

1/r d/dr (r du/dr) + 1/r^2 d^2u / dΦ^2 = 0

Question 29

Poisson's Integral

Answer 29

1/2π( π ∫ -π (a^2 -r^2)f(φ)dφ / d^2 - 2ar.cos(φ-

Question 30

Fourier Transform

Answer 30

F(p) = 1/√2π ∞ ∫ -∞ e^ipx f(x)dx

Question 31

Inverse Fourier Transform

Answer 31

f(x) = 1/√2π ∞ ∫ -∞ e^-ipx F(p)dp

Question 32

Dirichlet Integral

Answer 32

∞ ∫ 0 (sinPs / s )ds = ∞ ∫ 0 (sin.s / s)ds = π/2