0
votes
0answers
12 views

Jacobi and Gauss Seidel Iteration for solution of ODEs

I have used the Jacobi and Gauss-Seidel iteration schemes for solution of the following ODE: $$y^{''}(x)-5y^{'}(x)+10y(x)=10x $$ I will outline my method below: Discretion the equation by ...
0
votes
0answers
17 views

Are there other well known oscillator systems besides Van der Pol oscillator? [on hold]

Is there any collections of oscillator systems similar to "matrix market"?
1
vote
0answers
23 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
0
votes
1answer
35 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...
1
vote
0answers
15 views

Is fourth order Runge-Kutta method validity

I wonder whether the fourth-order Runge-Kutta method is suitable for a second-order linear ODE with dissipative terms modelling free fall of an object through a viscous medium under the act of ...
0
votes
3answers
51 views

Improved Euler method for second order ODE

I am trying to solve the simple harmonic oscillator problem with various Euler methods. Having managed to solve it with simple and modified Euler methods now I am trying to solve it with the improved ...
1
vote
2answers
37 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
0
votes
0answers
17 views

IVP and Newton-Cotes

I need to derive a RK method for solving the following IVP and would appreciate some help: y'(t) = f(t,y); y(0) = y0 by using the 3-points open Newton-Cotes formula, and get the LTE as well. How ...
0
votes
0answers
30 views

Solve Poisson's equation

I want to solve Poissons equation $$ C=\nabla^2 v $$ where $C$ is a constant and v my variable. I want to solve over some 2D domain D with the boundary condition that v is zero on the edge. How does ...
0
votes
0answers
50 views

AUTO Software for ODE's: references or forums?

I'm learning the AUTO software that does numerical continuation of ODE's by following these two references: The official manual found here www.dam.brown.edu/.../auto07p.pdf Lecture notes found at ...
1
vote
1answer
55 views

differential system

We consider the differential system $$ \begin{cases} & y'(t)=a y(t)^3 + b z(t)\\ & z'(t)=c z(t)^5 - b y(t) \end{cases} $$ with $t>0$ $y(0)=y_0, z(0)=z_0,\quad a<0,\quad c<0,\quad ...
0
votes
0answers
16 views

Is multiple shooting method better that simple shooting and if so in which regard?

I'm working with shooting methods and cannot grok why one would want to use multiple shooting method then simple shooting method is ought (?) to give the same result? Is stability of the multiply ...
1
vote
1answer
67 views

Matlab ode45 numerical solution

I'm trying to solve a 2nd order differential equation, using the Runga Kutta's ode45 function in Matlab. It's for a bachelor project, where I'm trying to simulate the behavior of a spherical robot, ...
1
vote
1answer
53 views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
1
vote
1answer
44 views

Advanced Topic in Numerical solution of Differential Equations?

i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where ...
0
votes
1answer
33 views

What is application of Euler method?

What is application of Euler forward and backward method? What big O notation means (in the case of Euler method $O(h^2)$ or $O(h)$)?
0
votes
1answer
33 views

How to implement the adaptive Heun's method?

I'm trying to implement code for Heun's method function in python. But I'm also doing it in the adaptive style. Regularly for say rectangle method, if you do adaptive style, you compare the area from ...
0
votes
1answer
33 views

Advanced topic in numerical solution of differential equation

Investigate the stability of the PECE method where P=Predictor : $y_{n+1} = y_n + hf(y_n)$ C=Corrector: $y_{n+1} = y_n + h [(1-θ) f(y_n) + θ f(y_(n+1))], (0<θ<1) $ and E is the evaluation ...
0
votes
1answer
25 views

Error bound of the Euler method

I am self studying working through the book "A First Course in the Numerical Analysis of Differential Equations" and have come to a deadend on q 1.2. The linear system $y' = Ay, y(0) = y_0$, where ...
0
votes
0answers
20 views

Are there methods to solve coupled integral and integro-differential equations?

I have one fredholm integral equation $$ y(x)=f(x)+\int_0^1 K_1(x,g(x),t)y(x(t))dt$$ and an integro-differential equation $$ \frac{dg(x)}{dx}=h(x)+\int_0^1 K_2(x,y(x),t)g(x(t))dt$$. Are there any ...
0
votes
0answers
16 views

Stability Regions for Numerical Methods and the Model Equation…what does it tell us in general?

In pretty much any textbook I read about numerical methods for ODE's they discuss stability regions for certain numerical methods (Euler, Implicit Trapezoidal, Adams-Bashforth etc...) on one ODE: $y' ...
0
votes
0answers
20 views

Method of undetermined coefficients for finite difference approximations

I'm reading over my text, and the first mention of deriving the coefficients states: "Suppose we want a one-sided approximation to $u'(x)$ based on $u(x), u(x-h)$, and $u(x-2h)$ of the form: ...
0
votes
0answers
27 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
0
votes
2answers
36 views

Ordinary differential equations with signed first derivative

Consider the following coupled set of ordinary diferential equations: \begin{align} (K_{pa}+K_r)y_1(t)-K_ry_2(t)+C_0\operatorname{sign}(\dot{y}_1(t))\lvert\dot{y}_1(t)\rvert^\alpha &= ...
0
votes
0answers
40 views

Local truncation error

(a) Find the local truncation error for the Trapezoidal rule $$ Y_{n+1} -Y_n= h/2( F_{n+1} + 3F_n)$$ and hence find the order of the method. What do you expect would happen to the local errors ...
0
votes
1answer
78 views

Consider Van der Pol’s equation:

Consider Van der Pol’s equation: $$y′′−0.2(1−y^2)y′+y=0,\qquad y(0)=0.1,\ y′(0)=0.1$$ (i) Find the approximate solution for this problem using the Taylor series method. Your expansion should include ...
0
votes
1answer
41 views

Exact numerical solution to non-linear ODE

We're given the initial value problem below: \begin{align} y'(t)=4t\sqrt{y(t)}-\lambda (y(t)-(1+t^2)^2), \quad y(0) = a, \quad (a,\lambda)\in\mathbb{R}^2 \end{align} For which $\lambda$ and $a$ ...
0
votes
0answers
47 views

How can solve this differential equation (third equation )?

How can I solve this differential equation? $$ \frac{dy}{dx}=\sqrt{\frac{A}{y}+\frac{B}{y^2}+\frac{C}{y^4}+\frac{D}{y^5}+\frac{1}{(\frac{1}{y}+\frac{3}{y^2})^2}} $$ where $A,B,C,D$ are constants.
3
votes
0answers
24 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
2
votes
1answer
72 views

Differential equation with the Dirac Delta as the solution

I'm wondering if there exists a differential equation with that dirac delta as the solution. I can think of plenty of integral equations such that that's true, but I can't think of any differential ...
0
votes
1answer
87 views

Differential equation, why use x instead of f(x)

I'm struggling with understanding a simple ODE. Let's say $x(t) = \exp(-t)$ so $x'(t) = -\exp(-t)$. When using Euler integration to solve the equation numerically, starting from $x_0 = x(t)$, we take ...
1
vote
1answer
42 views

Approximating an IVP

I wish to solve the IVP: \begin{align} x(0) =& -1 \\ x' =& 1 + x^2 - t^3 \end{align} With a fourth order taylor series method, I solved the ODE on the interval [0, 2] and then made the ...
1
vote
1answer
45 views

Runge Kutta 2 problem

I have the following problem to solve: Estimate y(0.1) by rk2 with h=0.1. Where rk2 it's: I replace the second-order ODE by a system of two first-order ODEs: $ \ \ y_{1}{}'=y_{2}\\ \ \ ...
0
votes
0answers
72 views

Seemingly nonsensical differential equation approximation

This is homework, please no full solutions, I am just stuck on something super basic. Evaluate the local truncation error of the method $$y_{n+1}=4y_n-3y_{n-1}-2hf(t_{n-1},y_{n-1})$$where $h$ is ...
1
vote
2answers
76 views

Numerical solution of $y′′+2y=−x$?

How to solve $y′′+2y=−x$ differential equation numerically. $y′(1)=0$ and $y(0)=0$ ?
0
votes
0answers
46 views

two-point boundary value problem for elliptic equations (ODE)

we consider two-point boundary value problem $$Au=-au''+bu'+cu=f~~~~~~~~~~~~~~~~ in ~~\Omega=(0,1)$$ $$u(0)=u_0,u(1)=u_1$$ where $a=a(x)>0$, $b=b(x)\ne 0$ and $c=c(x) \ge 0$ We must prove ...
2
votes
2answers
71 views

Problems implementing Euler's Method on a second order ODE

I am trying to teach myself some numerical methods and having more issues than expected trying to solve $$y'' = -y$$ with initial conditions $y'(0) = 0$ and $y(0) = 1$. As I understand the problem, ...
1
vote
0answers
55 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
1
vote
0answers
26 views

Numerically integrate with non-linear steps?

Suppose I want to numerically solve an ODE, but I know a little about its behavior locally. Is there any way to use that information as part of an integrator? For instance, suppose I have: $$ax'' + ...
0
votes
0answers
38 views

Are pendulum system ODEs stiff?

I'm interested in writing a numerical integrator to solve the motion of systems of pendulums. For example, a simple case would be the double pendulum. The motion can be quite complex in general. Is ...
0
votes
0answers
26 views

Numerical problem with set of ODEs

I have a set of ODEs: $$\dot{x}_{i} = f_i(x_1, \ldots, x_N), ~ i \in \{1, 2, \ldots, N\}$$ This set of ODE has the following properties: $\displaystyle\sum_{i=1}^N f_i = 0 ~ \forall x_1, \ldots, ...
0
votes
1answer
80 views

Solving a nonlinear second order ODE numerically

I am struggling to solve the following equation numerically: $x'^2 - xx''=W(t)$, where W(t) is a sinusoidal function, only known by its samples (i.e. no analytic form is known). Up until now I tried ...
2
votes
1answer
45 views

Why are Galerkin methods/FEMs used for solving PDEs and rather not ODEs?

I have not yet understood Galerkin methods and in general not the structural differences between ODEs and PDEs (of course I know the basics but not why PDEs ist so much different except that they ...
1
vote
0answers
68 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
1
vote
1answer
52 views

Bounded second derivative of solution to first order differential equations

I am studying numeric solutions of differential equations, and part of my reading is found in Simmonds' book, Differential Equations with Applications and Historical Notes. Although the chapter on ...
3
votes
1answer
50 views

orthogonal functions with orthogonal first derivatives

Is there any set of functions $\phi_1(x) , \phi_2(x) , \ldots , \phi_n(x) , \ldots $ defined on $[a,b]$ such that \begin{eqnarray} &&\langle\phi_i, \phi_j\rangle = \int_a^b ...
1
vote
1answer
58 views

Discretize differential equation by finite differences. What is the matrix?

I have a differential equation: $$ -u''(x) + \sigma u'(x) = f(x), \quad 0<x<1$$ with boundary conditions $u(0) = \alpha$ and $u(1) = \beta$. I've discretized equation using symmetric (finite) ...
0
votes
0answers
66 views

(Numerically) Solve second order differential equation

I have a second order differential equation: $-u''(x) + \sigma u'(x) = e^x,\quad 0<x<1$, where $\sigma = 40$ and boundary conditions are $u(0) = 1/39, \ u(1)=e/39$. I have to solve it using ...
2
votes
2answers
99 views

Derive forward Euler method for two-variable function

I need to derive the forward Euler method for solving ODEs and I would like some comments on what I have so far; overdot denote the time derivative: $\dot x \overset{def}{=} dx/dt$. Say we have ...
3
votes
1answer
58 views

Proving conservation of mass for linear advection

Reading through some course notes about conservation of mass in linear advection approximation schemes, given $\phi(x, t) \in \mathbb{R}$ and is defined for $0 \leq x < 1$ with periodic boundary ...