Tagged Questions
2
votes
0answers
41 views
Solution to second order nonlinear ODE
I need to find and exact solution for the following ODEs $$y''=-3y'+2y+2x+3,\qquad y(0)=2$$ $$y(1)=-4+5\exp\left(-3/2+\left(\sqrt{17}\right)/2\right)$$ and $$y''=2y^3-6y-2x^3;$$ $$1\leq x\leq2;$$ ...
0
votes
1answer
26 views
Euler's method for second order differential equation
Not really homework but sample exam.
The question is to use Euler's Method to approximate Y:
$Y''(t) = Y'(t) - 2Y(t)$
$Y'(0) = Y(0) = 1$
with $t_0 = 0$ and $h=0.2$
So what I did:
First ...
0
votes
1answer
19 views
Better than Runge-Kutta-Fehlberg 4(5) at high order?
I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
3
votes
2answers
45 views
How to verify the order of DOPRI Runge-Kutta method
I've written code in Fortran based on the RK8(7)-13 method by Dormand and Prince to solve the system $\mathbf{y}'=\mathbf{f}(t,\mathbf{y})$. The method is ...
0
votes
0answers
17 views
constructing linear multistep methods
hi im having abit of trouble with this question- could anyone offer any help?
the qeuestion is using order conditions, construct an explicit, zero stable, 2-step linear multistep method of order 2.
...
0
votes
0answers
34 views
How prove this equality stable in $\mu\le\frac{1}{4}$?
The difference equation
$$ u^{n+1}_{j}=u^{n}_{j}+\frac{3}{2}\mu(u^n_{j-1}-2u^n_{j}+u^n_{j+1})-\frac{1}{2}\mu(u^{n-1}_{j-1}-2u^{n-1}_{j}+u^{n-1}_{j+1}),$$
where $\mu:=\Delta t/(\Delta x)^2$
by ...
7
votes
0answers
49 views
How does one test a numerical ODE solver implementation? [migrated]
I'm about to start working on a software library of numerical ODE solvers, and I'm struggling with how to formulate tests for the solver implementations. My ambition is that the library, eventually, ...
2
votes
0answers
18 views
System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)
I have a large system (N>100) of equations
$\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$
where $\vec{P}$ is a vector of N functions of the variable t.
What is the correct ...
1
vote
1answer
42 views
Basis and weight functions
If we have a weight function $w(x) = e^{-x}$, then how can we find constant and linear polynomials $p0(x), p1(x)$ that are orthogonal on $[0,\infty)$ w.r.t $w(x)$?
And, if we let $f(x) = \cos x$, ...
0
votes
1answer
48 views
finite difference equations
i havent had a response to this question in a while, could someone please help me. Im struggling to understand the concepts of forward/backward/central differences on finite difference equations.
i ...
0
votes
1answer
102 views
Improved Euler method and local error
Given the differential equation $t'=g(x,t)$
Use the improved Euler method by analysing the local error and its stability for the equation $t'=\delta t$, where $\delta$ is a complex
...
1
vote
3answers
60 views
Numerical integration of nonlinear second order equation
I have (or, rather, a friend whom I'm trying to help has) a very messy differential equation, which I thought I'd try to solve numerically. However, I'm a little confounded as to what approach to use ...
0
votes
0answers
27 views
2nd order 2-step explicit Linear multistep method
I came across this question, i know how to carry out this question but my answer doesnt quite seem to correspond to the conditions.
the question states:
Construct a 2nd order 2-step explicit LMM ...
0
votes
0answers
30 views
linear multistep method- leapfrog
hi im having abit of trouble with this- im starting from the basics of Linera multistep methods and have come across this question:
write the leapfrog method down as a linear multistep method.
im ...
2
votes
1answer
100 views
butcher tableau runge kutta methods
Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me
Write the Butcher Tableau for the 1-stage $\theta$ method:
$$U^n ...
0
votes
1answer
23 views
explicit ODE IVPs
Ive had a go at this question, just need pointing in the right direction.
A linear scalar ODE of the form
$$ \frac{du}{dt}=:d_tu=:\dot{u}=5tu+\sin(t) $$
$$ u(0)=1 $$
can be solved explicitly. ...
0
votes
0answers
25 views
Truncation error and difference method
I am stuck on the following question. I am not sure of how to calculate the truncation error for the difference method
any help would be appreciated thank you!
0
votes
1answer
42 views
Modified Euler method
I am revising the modified euler method and would appreciate some help with this question:
The equation is $$y'=\frac{2}{x}y+x^2e^x, y(1)=0$$
Use modified euler method to calculate $y(1.1)$ taking ...
0
votes
2answers
82 views
Nonlinear DE and Numerical System
I'm trying to investigate nonlinear system numerical methods. For the nonlinear DE x' = 2t(1+x^2). Use the value tan(1) = 1.557407724654....
a) how to find the explicit solution $x(t)$ satisfying ...
1
vote
1answer
41 views
numerical methods sketches
Can someone show graphically in the tx-plane of illustrating the process of moving from (tk,xk) to (tk+1, xk+1) in
-Euler's method
-Improved Euler's method
and RK4?
I understand the formulas but ...
0
votes
2answers
87 views
Different methods and nonlinear systems
I'm trying to investigate nonlinear system numerical methods. So if we have a simple DE $x' = x$,
a) how to find the explicit solution $x(t)$ satisfying $x(0) = 1$?
b) how to use Euler's method to ...
0
votes
0answers
22 views
Numerical simulation of impulsive differential equations
I am interested to know discretization techniques of impulsive differential equations and its numerical simulation in MATLAB. I discretized the system of impulsive differential equations in uniform ...
1
vote
0answers
39 views
van der pol and liapunov
i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated.
a) show that the second-order differential equation for ...
0
votes
1answer
93 views
Nonlinear Second-order ODE BVP with 4 boundary conditions
My Lagrangian comes out in this form when I impose spherical symmetry:
$$ 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 $$
The following boundary conditions ...
1
vote
1answer
30 views
Derivative of solution of ODE
I have a set of nonlinear differential equations with parameters.
$$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$
where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in ...
0
votes
1answer
78 views
Approximate the solution to $y' = te^{3t} - 2y$ using Adams-Bashforth Three-step method
QUESTION
Consider the IVP
$y' = te^{3t} - 2y$ for $0 \le t \le 1$ with $y(0) = 0$
and actual solution
$$y(t) = \frac{1}{5}te^{3t} - \frac{1}{25}e^{3t} + \frac{1}{25}e^{-2t}$$
(a) Use the ...
2
votes
1answer
47 views
Integration a function with a polynomial for a denominator
QUESTION
The following differential equation describes the amount of $x$ of KOH after time $t$:
$$\frac{dx}{dt} = k \left(n_1 - \frac{x}{2}\right)^2 \left(n_2 - \frac{x}{2}\right)^2 \left(n_3 - ...
1
vote
1answer
44 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
94 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 ...
2
votes
1answer
71 views
Euler's Method on differential equation
For a differential equation, it is known that Euler’s Method leads to an underestimate when the curve is concave up, just as it will lead to an overestimate when the curve is concave down:
(from ...
1
vote
0answers
77 views
Solving ODE numerically with central difference quotient
I try to understand an old Fortran code that is not well documented. In this code the ODE
$$
\frac{dy}{dx} = -\frac{B(x - y)}{y}
$$
is solved numerically as an initial value problem from $x_0=0.99$, ...
0
votes
1answer
17 views
Conjuagate Gradient on Periodic BCs
I'm currently writing a CG solver. It works perfectly fine for Dirichlet boundary conditions, however, I also want it to work with periodic BCs.
The problem I'm solving is a 3D Poisson equation.
I ...
2
votes
1answer
149 views
Python numerical solution for a nonlinear second order ODE with two boundary conditions
I want to solve numerical the next equation, in Python
$$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$
it is a nonlinear second order $ODE$ with two $B.C$. ...
3
votes
2answers
151 views
Stiff differential equation where Runge-Kutta $4$th order method can be broken
Is there a stiff differential equation that cannot be solved by the Runge-Kutta 4th order method, but which has an analytical solution for testing?
0
votes
0answers
17 views
Numerical methods for ODE
(excuse my english in advance)
I have an ODE $$\dot{\alpha}(t,x)=V(t,\alpha (t,x),\lambda)$$
$$\alpha(0,x)=x$$ where the field $V$ is $C^{\infty}$ with respect to $t$,$x$ and a vector of parameters ...
0
votes
1answer
62 views
Runge-Kutta 4 explanation
I'm a game developer and I need to write a solar system simulation. Unfortunately I'm not very good at math and most importantly I haven't got to differential equations in my maths classes at school ...
1
vote
0answers
32 views
Stability Analysis, ODE, and Numerical Methods
Given the three equations, carry out stability analysis
This was an exam question I got wrong and still don't know how to do this question.
Is it possible someone can write the steps so that I can ...
2
votes
1answer
253 views
Runge-Kutta 4 - solving system of 6 differential equations (BVP)
I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
0
votes
1answer
22 views
How to show a method is 0-stable?
I want to show that the method $y_{j+1} = y_j + h\big( \theta f(t_j, y_j) + (1 - \theta) f(t_{j+1}, y_{j+1}) \big)$ is 0-stable.
I looked at the first page of this document to try to figure it out. I ...
0
votes
1answer
59 views
Analytic solution to explicit midpoint rule applied to an ODE
By solving a three-term recurrence relation, calculate analytically the sequence of values {$y_n:n=2,3,4,...$} that is generated by the explicit midpoint rule: $y_{n+2}=y_n+2hf(t_{n+1},y_{n+1})$ when ...
0
votes
1answer
60 views
Using centred difference formula to approximate differential equation
(Paper and pencil problem): Let $y_i=y(t_i)$ and $f_i=f(t_i)$, and show that by using the centered difference formula for $y^{\prime\prime}(t)$, we can compute approximations to $y_i$ by ...
0
votes
1answer
269 views
0
votes
0answers
95 views
Implementing the $\theta$-method for solving an IVP
I want to implement the $\theta$-method to solve an IVP in MATLAB. The $\theta$-method is:
$y_{j+1} = y_j + h[\theta f(t_j, y_j) + (1 - \theta)f(t_{j+1}, y_{j+1})]$ for $\theta \in [0, 1]$.
I want ...
1
vote
1answer
36 views
Differential equations notations confusion
Given these differential equations:
$\frac{d^2x}{dt^2} = 2\Omega\frac{dy}{dt}\sin(\lambda) - \frac{g}{L}x$
$\frac{d^2y}{dt^2} = -2\Omega\frac{dx}{dt}\sin(\lambda) - \frac{g}{L}y$
Now making the ...
2
votes
1answer
56 views
Runge Kutta with Impulse
I'm trying to study a simple predator-prey type ODE system of two variables, but I'd like to analyze the impulse response. Really, I have two versions of the same dynamical system, so two copies of ...
0
votes
0answers
89 views
Discretization of differential equation via FFT routine
I just have a question related to the following problem:
Find a discrete approximation to the differential equation $u^{\prime \prime} + 2u^{\prime} + 2u = 3\cos(6t)$ using Equation 3.12 for these ...
0
votes
2answers
52 views
Numerical Integration of ODEs - Consistency
EDIT: The answers below were helpful, but didn't get at the core of the problem, which I think has to do with the impulse aspect of things. I've since re-branded this question at Runge Kutta with ...
1
vote
1answer
70 views
Method of Undetermined Coefficients
I am trying to solve a problem using method of undetermined coefficients to derive a second order scheme for ux using three points, ...
0
votes
1answer
74 views
Finding the slope for Euler's method of evaulating differential equations
The change in the belocity of a body falling at a relatively slow speed over a short distance is given by $\frac{\mathrm{d}v}{\mathrm{d}t}= g - kv$, where $g$ is the acceleration due to gravity and ...
2
votes
2answers
39 views
Numerical Methods for ODEs Precision
I have come across the following statement : Higher order (Ode stepper) does not always mean high accuracy (from Numerical Recipies, third edition). Why so ?
Thank you in advance.
