Final Review

Question 1

What is an ordinary differential equation (ODE)?

Answer 1

An equation involving derivatives of a function with respect to a single independent variable

Question 2

What distinguishes an ODE from a PDE?

Answer 2

ODEs involve one independent variable, PDEs involve multiple

Question 3

What is an initial value problem (IVP)?

Answer 3

An ODE with specified values of the function and its derivatives at a starting point

Question 4

Why convert higher-order ODEs into first-order systems?

Answer 4

Numerical methods require systems of first-order equations

Question 5

How do you convert a 2nd order ODE to a system?

Answer 5

Define new variables for each derivative (e.g., y1=y, y2=y′)

Question 6

What is the Euler method formula?

Answer 6

yᵢ₊₁ = yᵢ + h f(tᵢ, yᵢ)

Question 7

What does Euler’s method approximate geometrically?

Answer 7

The tangent line (linear approximation)

Question 8

What is the order of error for Euler’s method?

Answer 8

O(h²) local truncation error

Question 9

Why is Euler method inaccurate?

Answer 9

It uses only slope at the start of interval

Question 10

What happens if step size h decreases?

Answer 10

Accuracy increases, computational cost increases

Question 11

Why use Runge-Kutta instead of Euler?

Answer 11

Higher accuracy without needing derivatives

Question 12

What is RK2 based on conceptually?

Answer 12

Averaging slopes

Question 13

What is RK4 known for?

Answer 13

High accuracy using weighted average of 4 slopes

Question 14

What is a key advantage of RK methods?

Answer 14

Better stability and accuracy than Euler

Question 15

What is the forward difference formula?

Answer 15

f′(x) ≈ (f(x+h) − f(x))/h

Question 16

What is the error order of forward difference?

Answer 16

O(h)

Question 17

What improves accuracy in differentiation?

Answer 17

Using central difference

Question 18

What is central difference formula?

Answer 18

f′(x) ≈ (f(x+h) − f(x−h)) / (2h)

Question 19

Why is central difference better?

Answer 19

Error is O(h²), more accurate

Question 20

What causes numerical differentiation errors?

Answer 20

Truncation + round-off error

Question 21

Why use numerical integration?

Answer 21

When integrals are too complex analytically

Question 22

What is trapezoidal rule formula?

Answer 22

Area approximated using straight-line segments

Question 23

What is Simpson’s 1/3 rule?

Answer 23

Uses quadratic interpolation over intervals

Question 24

What requirement does Simpson’s rule have?

Answer 24

Number of intervals must be even

Question 25

What is Gauss quadrature advantage?

Answer 25

High accuracy with fewer evaluation points

Question 26

What does Gauss quadrature do?

Answer 26

Uses optimal points and weights instead of equal spacing

Question 27

What is least squares regression?

Answer 27

Minimizes sum of squared errors between data and model

Question 28

What is the regression model for a line?

Answer 28

y = ax + b

Question 29

What is error in regression?

Answer 29

eᵢ = yᵢ − (axᵢ + b)

Question 30

Why square errors?

Answer 30

Avoid cancellation and penalize large errors

Question 31

What does R² represent?

Answer 31

Goodness of fit (how much variance is explained)

Question 32

What does R² = 1 mean?

Answer 32

Perfect fit

Question 33

What does R² = 0 mean?

Answer 33

Model explains none of the variation

Question 34

What is a partial differential equation (PDE)?

Answer 34

Equation involving partial derivatives of a multivariable function

Question 35

What are the three types of PDEs?

Answer 35

Parabolic, Elliptic, Hyperbolic

Question 36

How are PDEs classified?

Answer 36

Using discriminant: B² − 4AC

Question 37

When is a PDE parabolic?

Answer 37

B² − 4AC = 0

Question 38

When is a PDE elliptic?

Answer 38

B² − 4AC < 0

Question 39

When is a PDE hyperbolic?

Answer 39

B² − 4AC > 0

Question 40

What does an elliptic PDE represent physically?

Answer 40

Steady-state behavior

Question 41

What is Laplace’s equation?

Answer 41

∇²u = 0

Question 42

What does Laplace’s equation imply?

Answer 42

No sources or sinks

Question 43

What is Poisson’s equation?

Answer 43

∇²u = f(x,y)

Question 44

What does Poisson’s equation represent?

Answer 44

Internal generation (e.g., heat)

Question 45

What type of problems use parabolic PDEs?

Answer 45

Diffusion/heat transfer

Question 46

What is the key feature of parabolic PDEs?

Answer 46

Time-dependent evolution to steady state

Question 47

What do hyperbolic PDEs model?

Answer 47

Wave propagation

Question 48

What is a key characteristic?

Answer 48

Information travels as waves

Question 49

What is finite difference method?

Answer 49

Approximates derivatives using discrete points

Question 50

What is central difference for second derivative?

Answer 50

(uᵢ₊₁ − 2uᵢ + uᵢ₋₁)/Δx²

Question 51

What does discretization do?

Answer 51

Converts PDE into algebraic equations

Question 52

Why use a grid?

Answer 52

To approximate solution over domain

Question 53

What does steady state mean?

Answer 53

No change with time

Question 54

What does transient mean?

Answer 54

Changes with time

Question 55

What does “no sources/sinks” mean?

Answer 55

No internal generation

Question 56

What does diffusion represent?

Answer 56

Movement from high to low concentration

Question 57

What does advection represent?

Answer 57

Transport due to flow

Question 58

Why is smaller step size not always better?

Answer 58

Round-off error increases

Question 59

Why is Euler unstable sometimes?

Answer 59

Errors accumulate quickly

Question 60

What happens if Simpson’s rule used with odd intervals?

Answer 60

Invalid / inaccurate

Question 61

Why prefer RK4 in practice?

Answer 61

Best balance of accuracy and cost

Question 62

What is the biggest source of numerical error?

Answer 62

Truncation + rounding