CS计算机代考程序代写 scheme matlab algorithm 0/25 Questions Answered

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Numerical Methods in Engineering
(ENGR20005) Final Exam

Q1 Root finding
8 Points

The force exerted by a non-linear spring is given by the following

equation

where is the force, is the deflection and and are the

spring constants.

The figure above shows a body of mass initially at rest at a

height above the non-linear spring. When the body is released,

the spring compresses by a maximum of .

Q1.1
0.5 Points

Using the conservation of energy, show that

F = k d +1 k d2
3/2

F d k 1 k 2

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Hint: Equate the potential energy of the body with the potential

energy of the spring ( ).

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Q1.2
1.5 Points

Plot vs and estimate the value of in your plot.

Gven that:

Compare your estimated value of with MATLAB ‘fzero’ function.

Estimated value for

Enter your answer here

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Q1.3
0.5 Points

Rearrange the such that . Show that one possible

expression for is

0 = k d +1
2 k d −2

5/2 mg(h + d) = f(d).

未选择任何⽂件选择⽂件



f d d

k =1 40kg/s2

k =2 0.04kg/(s m )2 0.5

m = 0.5kg
g = 9.81m/s2

h = 0.1m

未选择任何⽂件选择⽂件



f d = g(d)
g(d)

Q1.4
3 Points

Plot vs and vs on the same graph. Using the one-point

iteration scheme, graphically show the convergence/divergences of

the method for the first 3 iterations with initial guess of =

0.20. Does the solution converge? Compare your results with

MATLAB ‘fzero’ function.

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Q1.5
2.5 Points

Now, use the Newton-Raphson method to solve for . Compare

your results with MATLAB ‘fzero’ function. Calculate for values of

ranging from 0.1 to 0.5. Plot against .

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Q2 Integration
8 Points

When modelling the flow past a slender object, you might have to

solve a certain type of problem known as an integral equation.

These are quite similar to the differential equations that you are

未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件

g(d) =
−mg + k d + k d1 2 3/2

mgh

g d d d

d guess

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

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

h d h

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

familiar with, but instead of differentials being the unknowns,

they’re now integrals.

Consider the following integral equation, which is classified as a

For this equation .

Q2.1
3 Points

Apply Gaussian quadrature to the integral equation above with the

point Gauss–Lobatto nodes and show that an algebraic

equation that describes the solution at the Gauss–Lobatto points

is given by

where and represents the Gauss–Lobatto quadrature

weight and node respectively.

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Q2.2
2 Points

In the case when , show that the resulting system of linear

equations is given by

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Fredholm equation of the second kind

ϕ(x) = 1 + (xt + x t )ϕ(t) dt∫
0

1
2 2

0 ≤ x ≤ 1

ϕ i

ϕ =i 1 + w [ +
2
1

ℓ=0

∑
M−1

ℓ
4

(1 + ξ )(1 + ξ )i ℓ
16

(1 + ξ ) (1 + ξ )i
2

ℓ

w j ξ j j
th

未选择任何⽂件选择⽂件



M = 4

Feel free use the MATLAB functions in lectures or workshops to

compute the Gauss–Lobatto nodes and weights.

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Q2.3
1 Point

Use MATLAB to solve the set of equations you derived in Question

2.2. Use any built-in MATLAB functions or operators to solve the

system of equations. Compare with the exact (analytical) solution

given by

Plot your approximate solution and the exact solution on the same

graph. Do they agree with each other?

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未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 =

⎣⎢
⎢⎢⎡

1.0000
0
0
0

0
0.9657

−0.1000
−0.1470

0
−0.1000
0.6676

−0.5197

0
−0.0294
−0.1039
0.8333 ⎦

⎥⎥⎥
⎤
⎣⎢
⎢⎢⎡
ϕ 0
ϕ 1
ϕ 2
ϕ 3⎦
⎥⎥⎥
⎤

⎣⎢
⎢⎢⎡

1
1
1
1⎦
⎤

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

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

ϕ(x) = 1 + x +
339
348

x
339
250 2

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

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

Q2.4
2 Points

Generalise your MATLAB program so that it is able to compute the

approximate solution for an arbitrary number of nodes, .

Compute your solution with and and compare

with the analytical solution

Does your solution get any more accurate as you increase from

to ?

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Q3 Regression
8 Points

You are given the following set of data points

5 16

10 25

15 32

20 33

25 38

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M = 10 M = 20

ϕ(x) = 1 + x +
339
348

x
339
250 2

2 20

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

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

x y

30 36

35 39

40 40

45 42

50 42

Q3.1
1 Point

Write a MATLAB code to plot the data set above

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Q3.2
2 Points

Write a MATLAB program that implements the linear least squares

regression algorithm to fit a straight line

through the set of data points. Plot this straight line along with the

original data points on the same graph for .

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x y

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

y = ax + b

0 ≤ x ≤ 50

未选择任何⽂件选择⽂件



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Type below the values of and from your linear regression

MATLAB program

Enter your answer here

Is the straight line a good fit though the data points? Write down

your observations below.

Enter your answer here

Q3.3
2 Points

Explain how you would fit the saturation-growth-rate equation

with a MATLAB code that implements the linear least squares

regression model.

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Q3.4
3 Points

Write a MATLAB code that that uses the linear least squares

regression model to fit the saturation-growth-rate equation above

to the data points given at the start of this question. Plot the



a b

y = c
d + x

未选择任何⽂件选择⽂件



saturation-growth-rate equation fit and the original data points on

the same graph for .

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What are the values of and that will give the best fit of the

saturation-growth-rate equation to the data points?

Enter your answer here

Q4 System of ODE
8 Points

The equations is a well known Predator-prey

model involving systems of non-linear ordinary differential

equations.

where is the number of prey and is the number of predators.

is the prey growth rate, is the predator death rate, and are the

rate characterising the effect of the predator-pray interaction on

prey death and predator growth respectively.

Q4.1
3 Points

The system of non-linear ODE can be solved iteratively

where

0 ≤ x ≤ 50
未选择任何⽂件选择⽂件



c d

Lotka − V olterra

=
dt

dx
ax − bxy

=
dt

dy
−cy + dxy

x y a

c b d

[J ][X] = −[C]

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Show that

and find the expression for when the equation in question 4 is

solved using the Crank-Nicholson time-stepping scheme.

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Q4.2
3 Points

Write a MATLAB script to solve the system of non-linear equations.

The given parameter values for the predator-prey simulation are

= 1.2,

= 0.6,

= 0.8,

= 0.3.

The initial conditions are = 2 and = 2. Find the values of

and for t = 0 to 40. Choose an appropriate and compare

your solution to MATLAB’s function

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Q4.3
2 Points

=[X] .[x − x n+1 n
y − y n+1 n

]

=[J] [ (a − by ) − 12Δt n+1
dy

2
Δt

n+1

− bx
2

Δt
n+1

(−c + dx ) − 1
2

Δt
n+1

]
[C]

未选择任何⽂件选择⽂件



x(0) y(0)
x y Δt

ode23s

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

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Plot (predator) against (prey). To observe what happens when

there is an imbalance of prey and predator in the ecosystem, plot

vs for the following initial conditions on the same graph.

2 2

4 2

6 2

8 2

Comment on your results.

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Q5 Boundary Value Problem
8 Points

Consider Poisson’s equation

in the domain

You are given the boundary conditions

y x

x(0) y(0)

未选择任何⽂件选择⽂件



+
∂x2
∂ ϕ2

=
∂y2
∂ ϕ2

g(x, y)

0 ≤ x ≤ 5, 0 ≤ y ≤ 5.

ϕ(x, 0) = 0

ϕ(x, 5) = 0

{

and also the source term

In this question, you are required to use numerical methods to

obtain an approximate solution for .

Q5.1
2 Points

Discretize

using the 2nd order central difference approximation (for both the

and derivatives). Write down all the necessary equations that you

will need in order to write a MATLAB program to approximate

without considering the boundary conditions (we will

incorporate the boundary onditions later in 5.2 below). For

simplicity, assume that the grid spacing in both the and

directions are equal, i.e. .

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Q5.2
2 Points

Write a short explanation on how you would incorporate all

boundary conditions in the equations that you that you derived in

Question 5.1. Where appropriate, apply the 2nd order central

ϕ(0, y) = {0
10

if 0 ≤ y < 2 if 2 ≤ y ≤ 5 (5, y) = ∂x ∂ϕ 0 g(x, y) = −50e .−((x−2.5) +(y−2.5) ) 2 2 ϕ(x, y) + ∂x2 ∂ ϕ2 = ∂y2 ∂ ϕ2 g(x, y) x y ϕ(x, y) x y Δx = Δy = Δ 未选择任何⽂件选择⽂件  未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 difference approximation at the boundaries as well and derive any necessary equations you need to write a MATLAB program to solve this problem. Upload PDF of your solution using link below Please select file(s) Select file(s) Q5.3 4 Points Write a MATLAB program to obtain approximate values of using an iterative scheme to solve the algebraic equations you derived in Questions 5.1 and 5.2. Plot the approximate using the function in MATLAB. Use a value of that will give you a good solution. Justify this value of . Upload PDF of your solution using link below Please select file(s) Select file(s) Upload your MATLAB code using the link below Please select file(s) Select file(s) Q6 ODE 8 Points The convection, diffusion equation is given by where is the convection velocity and is the viscosity. Q6.1 1 Point 未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件  ϕ(x, y) ϕ(x, y) contour() Δ Δ 未选择任何⽂件选择⽂件  未选择任何⽂件选择⽂件  = ∂t ∂ϕ −C + ∂x ∂ϕ ν ∂x2 ∂ ϕ2 U ν 未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 The finite difference discretisation of the interior grid points for the terms on the right-hand side can be expressed as Show that when both the convective term ( ) and the diffusive term ( ) are discretised using 2nd order central difference approximation. Upload PDF of your solution using link below Please select file(s) Select file(s) Q6.2 1 Point Show that when the convective term ( ) is discretised using 1st order forward difference approximation and the diffusive term ( ) is discretised using 2nd order central difference approximation. Upload PDF of your solution using link below Please select file(s) Select file(s) 未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 αϕ +i−1 βϕ +i γϕ =i+1 −C + ∂x ∂ϕ ν . ∂x2 ∂ ϕ2 α = − + 2Δ C Δ2 ν β = −2 Δ2 ν γ = + 2Δ C Δ2 ν C ∂x ∂ϕ ν ∂x2 ∂ ϕ2 未选择任何⽂件选择⽂件  α = Δ2 ν β = − − Δ C 2 Δ2 ν γ = + Δ C Δ2 ν C ∂x ∂ϕ ν ∂x2 ∂ ϕ2 未选择任何⽂件选择⽂件  未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 Q6.3 2 Points Equation 6 can be simplified to where is a linear operator which represents the discretised spatial derivatives. Write a MATLAB script that uses the RK4 method to solve this equation. Use 2nd order central difference approximation for both the convective and diffusive terms. Given that , and boundary conditions and . The initial condition . **NOTE: A time-varying boundary condition is applied at Upload your MATLAB code using the link below Please select file(s) Select file(s) Q6.4 2 Points Write a MATLAB script that uses the RK4 method to solve this equation. Now, use 1st order forward difference for the convective term and 2nd order central difference approximation for the diffusive term. Upload your MATLAB code using the link below Please select file(s) Select file(s) 未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 = ∂t ∂ϕ Lϕ L 0 < x < 1 ϕ(0, t) = cos(πt) ∣ =∂x ∂ϕ x=1,t 0 ϕ(x, 0) = cos(πx) ϕ(0, t) 未选择任何⽂件选择⽂件  未选择任何⽂件选择⽂件  未选择任何⽂件选择⽂件未选择任何⽂件选择⽂件 Q6.5 1 Point Plot vs at = 0, 1 and 2 using the scripts in 6.3 and 6.4. Use , , , Upload PDF of your solution using link below Please select file(s) Select file(s) Q6.6 1 Point Increase . Again, plot vs at = 0, 1 and 2 using the scripts in 6.3 and 6.4. Does both scripts give you a stable solution? Upload PDF of your solution using link below Please select file(s) Select file(s) Submit & View Submission  ϕ x t N =x 100 Δt = 0.001 C = 0.22 ν = 0.001  C = 0.5 ϕ x t  https://www.gradescope.com/courses/246904/assignments/1331184/submissions/82215596

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