Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.
0
votes
1answer
18 views
Reflection Lines
I am analyzing the problem of G^1 continuity between patches. I have found the statement: if the reflection lines on a surface are C^0 then the surface will be G^1. I would like to know the proof of ...
0
votes
0answers
52 views
Exact versus numerical approximation of an integral
Why does the following approximation work very well for small theta's but not for large theta's?
$$
I = \int_0^\infty x (1 + ...
0
votes
1answer
23 views
explicit ODE IVPs
Ive had a go at this question, just need pointing in the right direction.
A linear scalar ODE of the form
$$ \frac{du}{dt}=:d_tu=:\dot{u}=5tu+\sin(t) $$
$$ u(0)=1 $$
can be solved explicitly. ...
0
votes
0answers
65 views
Finite difference methods for nonlinear problems
How can we use the nonlinear finite-difference method with $h = 0.25$ to approximate the solution to the BVP $y'' = 2y^3$, $-1 \leq x \leq 0$, $y(-1) = 0.5$, and $y(0) = \frac{1}{3}$?
So, I ...
2
votes
1answer
50 views
How to use undefined value in Composite Simpson's Rule
I have to use the Composite Simpson's Rule to approximate the integral $\int_0^1 t^2\cdot sin(\frac{1}{t}) dt$. I've used the Composite Simpson's Rule, but when I work through the algorithm, one step ...
1
vote
1answer
64 views
Runge Kutta Graphical Analysis
Can someone represent the 4th order Runge Kutta method graphically when iterating from (xk,tk) to (xk+1,tk+1)?
Thanks
0
votes
0answers
26 views
Truncation error and difference method
I am stuck on the following question. I am not sure of how to calculate the truncation error for the difference method
any help would be appreciated thank you!
0
votes
1answer
49 views
Modified Euler method
I am revising the modified euler method and would appreciate some help with this question:
The equation is $$y'=\frac{2}{x}y+x^2e^x, y(1)=0$$
Use modified euler method to calculate $y(1.1)$ taking ...
0
votes
2answers
86 views
Nonlinear DE and Numerical System
I'm trying to investigate nonlinear system numerical methods. For the nonlinear DE x' = 2t(1+x^2). Use the value tan(1) = 1.557407724654....
a) how to find the explicit solution $x(t)$ satisfying ...
1
vote
1answer
41 views
numerical methods sketches
Can someone show graphically in the tx-plane of illustrating the process of moving from (tk,xk) to (tk+1, xk+1) in
-Euler's method
-Improved Euler's method
and RK4?
I understand the formulas but ...
0
votes
0answers
21 views
Help with operation count for Choleski factorization and Guass elimination
I'm having a lot of trouble finding the multiplication and division operation count for Choleski factorization (without pivoting) and the Gauss elimination.
I had a go at the Choleski factorization, ...
0
votes
2answers
88 views
Different methods and nonlinear systems
I'm trying to investigate nonlinear system numerical methods. So if we have a simple DE $x' = x$,
a) how to find the explicit solution $x(t)$ satisfying $x(0) = 1$?
b) how to use Euler's method to ...
1
vote
0answers
21 views
show that the remainder for midpoint rule is $\frac{(b-a)^3}{24}f''(\xi)$ for some $\xi\in[a,b]$ using hermite interpolation
i saw some proof using taylor theorem but cannot find one using hermite interpolation.
for newton-cotes quadrature rule with $n$ which is even, we have $n+1$ degree of precision, instead of $n$ when ...
1
vote
1answer
53 views
Heavily stuck on Newton-Cotes integration
For an integral of form $\int_{-2}^2 x^2e^xdx$, calculate the
Newton-Cotes quadrature and estimate the error for:
$n=1$ (Trapezoid rule)
$n=2$ (Simpson's rule)
$n=3$ (3/8 rule)
...
0
votes
0answers
27 views
Numerical simulation of impulsive differential equations
I am interested to know discretization techniques of impulsive differential equations and its numerical simulation in MATLAB. I discretized the system of impulsive differential equations in uniform ...
0
votes
0answers
38 views
Numerical methods and illustration
Can someone provide a picture in the $t$-$x$ plane that illustrates the process of moving from $(t_k,x_k)$ to $(t_{k+1},x_{k+1})$ in Euler's method, Improved Euler's method and Runge-Kutta 4?
I had ...
0
votes
0answers
21 views
Implicit Runge Kutta method is well defined
I need a hint or two on how to show that the implicit Runge Kutta method is well defined.
As a hint we are told to show that the implicit problem has a solution if ...
1
vote
1answer
16 views
Linear equation with prescribed precision of result
Let $x$ be an unknown real vector of size n.
Suppose we can find n vectors $v_i$, and are given the values $x^Tv_i$.
Then we can simply solve for $x$ by Gauss or some other method and determine $x$ ...
1
vote
0answers
40 views
van der pol and liapunov
i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated.
a) show that the second-order differential equation for ...
0
votes
1answer
99 views
Nonlinear Second-order ODE BVP with 4 boundary conditions
My Lagrangian comes out in this form when I impose spherical symmetry:
$$ φ''(ρ)+{3\overρ} φ'(ρ)+{4μ^4\over M^2} φ(ρ)-{4μ^4\over M^4} φ^{3}(ρ)-{μ^4\over2M} ϵ=0 $$
The following boundary conditions ...
0
votes
1answer
108 views
Two Dimension Heat Equation ADI Local Truncation Error
Given a two dimensional heat equation $\displaystyle \frac{\partial u}{\partial t}=K(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})$ solved using ADI (the alternating-direction ...
6
votes
2answers
128 views
Newton's method, for which initial guesses it converges?
We've got a function: $ f: R \rightarrow R$ , $f(x) = x^3 - 9$.
Let $x^* $ be its root, which means $ f(x^*) = 0$. We want to find approximation for $x^*$ using a Newton's method.
There are two ...
1
vote
3answers
83 views
How to find the limit of a convergent matrix?
I'm trying to learn how to show a series of matrices is convergent and find the limit. However my numerical analysis books fail to mention this and I cannot find any relevant material online! Anyway ...
0
votes
1answer
33 views
$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$ ,$\theta=$?
This question is a follow up question to this answer.
In the equation:
$$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$
$a$ and $b$ are given. What is the best way to solve for ...
1
vote
4answers
67 views
Solving a set of 3 Nonlinear Equations
In the following 3 equations:
$$
k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1
$$
$$
2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2
$$
$$
k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3
$$
$c_1$, $c_2$ and $c_3$ are ...
1
vote
1answer
34 views
Derivative of solution of ODE
I have a set of nonlinear differential equations with parameters.
$$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$
where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in ...
0
votes
0answers
22 views
How to solve a simple B-spline with table function?
Determine the natural cubic B-spline for a function in a table form of:
$$x_i\|-1\|0\|1\|2 \\ y_i\|-3\|0\|1\|2$$
So I'm very new to cubic splines and am quite lost on what I should do. I ...
0
votes
0answers
34 views
Logarithmic accuracy
Does anyone know how the method of logarithmic accuracy works and what do I have to know about it (as far as applied Mathematics is concerned)? Any references, examples or guidelines would be ...
2
votes
0answers
44 views
Proving invertibility of matrices using banachs lemma
I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
4
votes
4answers
33 views
Generating a Monotonically Decreasing Sequence that adds to 1 for any length
I would like to generate a monotonically decreasing sequence whose elements will add to one, and generate this sequence for any order $L$. For instance, if $L=2$, then $\vec{s} = [ \frac{3}{4}, ...
0
votes
1answer
75 views
Nonlinear equation / iteration method
$$ 1 x^5-9.067 x^4+24.726 x^3-13.998 x^2-15.278 x+1.014 = 0 $$
Wolfram Result
but i need solve roots using "iteration method" (i don't know how it correctly called in English, but Newton's method is ...
0
votes
1answer
58 views
How can I minimize the following function
I have a simple function and want to minimize it using any method with precision of about 0.01. The domain of x is [-2,2]. The function is a simple $f(x) = x^2$ plus a triangle in the form of ...
0
votes
1answer
66 views
matrix with distinct bounded eigen values is bounded?
I am looking from the numerical methods perspective. I have a mapping $G$ that maps points in the numerical iteration to the new level. I would like to show its stability. For that I need to show that ...
0
votes
1answer
26 views
Simultaneous eigenfunction problems
I'm familiar with solving eigenfunction problems using finite difference methods and eigenvalue solver like Eigensystem[] in Mathematica. But now I've come across a problem where I have two ...
1
vote
1answer
144 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
3
votes
1answer
37 views
Numerical analysis - boundary value problem
So I need a push in the right direction with this question, so any help is much appreciated :)
Show that the solution to the boundary value problem
$-u''(x) + r(x)u(x) = f(x)$, $u(a)=\alpha$, ...
1
vote
1answer
203 views
Variational formulation of Robin boundary value problem for Poisson equation in finite element methods
So I am confused about some details of obtaining a variational formulation specifically for Poisson's equation. I am in a Scientific Computing class and we just started discussing FEM for Poisson's ...
0
votes
1answer
105 views
Approximate the solution to $y' = te^{3t} - 2y$ using Adams-Bashforth Three-step method
QUESTION
Consider the IVP
$y' = te^{3t} - 2y$ for $0 \le t \le 1$ with $y(0) = 0$
and actual solution
$$y(t) = \frac{1}{5}te^{3t} - \frac{1}{25}e^{3t} + \frac{1}{25}e^{-2t}$$
(a) Use the ...
2
votes
1answer
58 views
Integration a function with a polynomial for a denominator
QUESTION
The following differential equation describes the amount of $x$ of KOH after time $t$:
$$\frac{dx}{dt} = k \left(n_1 - \frac{x}{2}\right)^2 \left(n_2 - \frac{x}{2}\right)^2 \left(n_3 - ...
2
votes
2answers
163 views
Brent's algorithm
Use Brent's algorithm to find all real roots of the equation
$$9-\sqrt{99+2x-x^2}=\cos(2x),\\ x\in[-8,10]$$
I am having difficulty understanding Brent's algorithm. I looked at an example in ...
1
vote
1answer
45 views
backward euler method
Having abit of trouble, would be very grateful if someone could help.
Derive the backward Euler method for the problem:
$$\frac{du}{dt}=:d_tu=: \dot{u}=\lambda u $$
$$ u(0)=u_0$$
any help would ...
0
votes
1answer
85 views
finite differences for PDE's
I am having trouble with this question, much appreciated if anyone can help?
a) for the 2nd order wave equation:
\begin{align}
&\partial_{tt}u(x,t)-c^2\partial_{xx}u(x,t)=0 & (x,t) \in ...
1
vote
1answer
94 views
Nonlinear shooting example calculation
How can we use the nonlinear shooting method with $h = 0.25$ to approx the solution to $y'' = 2y^3$, $-1 \leq x\leq 0$, $y(-1) = 1/2$, and $y(0) = 1/3$.
I tried to convert this to a first order ...
0
votes
0answers
28 views
Sinusoidal interpolation
I am new to the concept of interpolation, and know only how to interpolate with a polynomial function. What if you suspect a sinusoid will be a more accurate fit? What method should be used?
0
votes
1answer
49 views
electrostatic potential and shooting method
Letting u be the electrostatic potential between 2 concentric metal spheres, with R1 < R2 and if we have a ODE:
d^2u/dr^2 + 2/r(du/dr_ = 0, for R1 <= r <=R2, with u(R1) = V1, and u(R2) = 0 ...
0
votes
1answer
132 views
linear shooting method and finite differences
how can we use the linear shooting method to approximate this solution below:
$$y'' + 4y = \cos(x), 0 \le x \le4, y(0) = 0, y(pi/4) = 0, h = \frac{\pi}{20}$$
My concern is with the RK-4 and setting ...
4
votes
0answers
191 views
How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?
I am trying to numerically solve the Possion PDE in cylindrical coordinate system.
$$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
2
votes
1answer
56 views
Don't understand how to start with this assignment question
I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
0
votes
1answer
40 views
Can anyone recommend me a good pdf link to learn B-spline with a lot of examples?
I really hope someone can recommend me a good link to study B-spline with a lot of examples that I could grasp the concept very easily! Thanks! :)
0
votes
1answer
19 views
Integrating during a least squares approximation, are these partial integrals correct?
I would like to find the least squares approximation:
$$g(\alpha_0,\alpha_1) = \int_{0}^{\pi/2} [\sin x-\alpha_0-\alpha_1x]^2dx$$
Taking the derivatives w.r.t. the alphas I end up with: ...
