0
votes
0answers
14 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
0
votes
0answers
16 views

Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
0
votes
0answers
12 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
0
votes
0answers
18 views

4th order method

I am asked to solve a ODE using the 4th order Runge-Kutta method, and then given the analytical answer, 'show the method is 4th order numerically' . What does the question 'show the method is 4th ...
1
vote
0answers
12 views

Finite Difference Discretization of Darcy's law and solving with Picard method

I am trying to discretize Darcy's Law using finite differences and then solving the resulting linear system of equations with the Picard method. So far only in 1D and the steady-state (no time ...
1
vote
2answers
40 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
1
vote
1answer
30 views

How can we apply the forward Euler method to $x''=x^2$?

If we want to apply the forward Euler method to $x''=x$ with $x(0)=0, x'(0)=1$, we can introduce a new function $$u:=\begin{bmatrix}x'\\x \end{bmatrix}$$ then $$u'=\begin{bmatrix}0&1\\1&0\\ ...
2
votes
0answers
34 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
0
votes
0answers
35 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
1
vote
1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
0
votes
0answers
19 views

Runge-Kutta for newton's law with dependency

I'm trying to determine the changes (position and velocity) on a mechanical system during a step of time. I have a mobile mass whose position (everything is 1D-only) is denoted $x(t)$, velocity $v(t)$ ...
0
votes
0answers
9 views

How to determine the upper bound of the global error when I calculate a ODE using Euler method

I have an ODE say, $u^\prime(t) = tan^{-1}u(t)$ $(0 \le t \le T)$, $u(0) = a$. Now I would like to decide the upper bound of the global error when using Euler method to solve this ODE. I know that the ...
1
vote
1answer
25 views

How to show that the one step method can't have consistency $p=3$?

I was looking at some exercises from last years' of my Intro to numerical math class, and found this: Consider the following explicit one step method: $$\psi^h x=x+h \gamma_1 f(x)+ h \gamma_2 ...
2
votes
2answers
71 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
1
vote
1answer
49 views

Which numerical method to use for ODE?

In practice what is the most common way to numerically estimate $y(t)$ (possibly using a series expansion) in the ODE with initial conditions, $$ y'(t) = f(t,y(t)), \qquad y(t_0)=y_0 $$ Wikipedia has ...
2
votes
0answers
56 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
0
votes
0answers
33 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
0
votes
0answers
15 views

Numerical solution of ODE

I have a general question about numerical solution of ODE. I want to solve a ODE on an interval where two solutions can exist and intersect. As far as I understand a numerical solution will give the ...
0
votes
1answer
24 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
0
votes
1answer
28 views

Is Cea's lemma sharp?

Given a problem in weak formulation $$ \begin{align} \text{find $u\in V$ s.th. for all $v\in V$} \\ a(u,v) = f(v) \end{align} $$ with bilinear form $a:V\times V\rightarrow\mathbb{R}$, bounded with ...
0
votes
0answers
21 views

Numerical solution to a coupled differentio-algebraic system of equations

$$\frac{\mathrm{d}X_1}{\mathrm{d}t} = P \times ( \frac{I_a^n}{K_i\times exp(I_a*m) + (I_a)^n} ) \times ( 1-( \frac{A.X_2 + B}{ K_o})^z)$$ $$X_1 = X_2 -[ P' \times \frac{I_a^n}{(Ki*exp(I_a * m) + ...
0
votes
0answers
30 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
1
vote
1answer
48 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
5
votes
0answers
161 views

Propper algorithm to integrate ODE's numerically.

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
4
votes
3answers
77 views

Robust Numerical ODE Solver?

I made a little explicit Runge-Kutta 4th order solver a few days ago, but when testing it against various 1st and 2nd order ODEs chosen at random (for example $d^{2}y/dt^{2} = -y \sin(y)$, ...
1
vote
0answers
30 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
0
votes
0answers
26 views

Predictor-Corrector

I'm constructing a Predictor-Corrector method in P(EC)^3E mode with a first order AB method as predictor and a fourth order AM method as corrector on the ODE $y'=-y$. By experiments on the value of ...
1
vote
2answers
46 views

Euler method application: step size

Suppose we have a system of ODE's: $a' = -a - 2b$ and $b' = 2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$. How can we find the maximum value of the step size such that the norm a solution of ...
1
vote
0answers
28 views

How to establish a lower bound on this difference operator?

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
1
vote
1answer
28 views

Difference between derivative and its approximation

If I define the approximation of the second derivative as $$\delta^2_xV_{i}=\dfrac{D^+_xV_{i}-D^-_xV_{i}}{(x_{i+1}-x_{i-1})/2}$$ where $$D^+_xV_{i}=\dfrac{V_{i+1}-V_i}{x_{i+1}-x_i}, ...
4
votes
1answer
101 views

Why is the numerical solution of this equation unstable? Is this equation stiff?

I am trying to solve the following equation with an explicit fourth-order using the Runge-Kutta method: $$y' = t(y - t \sin t)$$ with initial conditions $y(0) = 1$ over the interval $[0, 10]$. The ...
1
vote
0answers
17 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
0
votes
0answers
28 views

Order of **convergence** of a multistep method

State the Dahlquist equivalence theorem regarding convergence of a multistep method. The multistep method, with a real parameter $a$, $$y_{n+3} + (2a-3)(y_{n+2}-y_{n+1}) - y_n = ...
1
vote
1answer
107 views

Solving Differential Equations theoretically and using matlab

i am trying to solve the initial value and elliptic boundary value problems below. but now i need some help solving them using matlab. for the elliptic problem, any method is ok, but for the initial ...
0
votes
1answer
92 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
0
votes
0answers
30 views

Stiff differential equations without using Jacobian matrix

I want to solve a stiff system of differential equations. Its Jacobian matrix isn't constant and its determinant is close to zero so I cant inverse of it. Please tell me does exist a method that solve ...
0
votes
1answer
58 views

Stiff differential equation

I'm trying to solve a system of differential equations with Runge-Kutta method. When I use the step size $h=1$ my problem has true answer but when I use the smaller $h$ (for example $h = 0.1$) my ...
0
votes
0answers
32 views

Errors in numericaly solving hyperbolic PDE in matlab

I am a beginner for PDE and I want to solve a hyperbolic PDE using matlab's builtin function hyperbolic(). However I am facing some erros and I could not resolve them. Can someone suggest or comment ...
1
vote
1answer
37 views

Can Successive Over-Relaxation be used for Nonlinear Equations?

My question was whether or not successive over-relaxation (http://en.wikipedia.org/wiki/Successive_over-relaxation) could be used to find solutions to a nonlinear equation. In particular I am ...
1
vote
1answer
119 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
0
votes
1answer
51 views

How can Picard proved that his method was right?

In order to solve initial value problems .. We know that Picard's method is right , but i need to know how can Picard proved this ?
4
votes
1answer
66 views

Due to numerical inaccuracy, the solution of a boundary value problems becomes negative

I treat a toy example to get my point across. In reality I have to deal with a much more complex model. Let us consider a one dimensional boundary value problem using the bvp5c solver in Matlab. Two ...
4
votes
0answers
91 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
1
vote
0answers
74 views

Numeric solution of third order ODE

I need to solve the following third order (non-linear) ODE by numerical methods: \begin{equation}\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1. \end{equation} By assumption, the solution should approach $ ...
1
vote
0answers
23 views

numerically solve quadratic air drag in xy-plane

I am trying to find a reference on solving for the position of a point mass as a function time, subject to air drag( quadratic term only) in both the x and y directions. The equations that describe ...
1
vote
0answers
173 views

Change MATLAB code from Lax-Wendroff to Leapfrog

I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the $ U_j^{n-1}$ term, I just don't get the logic. Here's what ...
1
vote
0answers
23 views

Solving ODE numerically - getting local truncation error

Well I have NO idea how to do this or even where to start Compute the order of magnitude of the local truncation error of the following time integration scheme: $$y_{n+1} = y_{n-1} + 2h f(y_n)$$ H ...
3
votes
1answer
56 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
0
votes
0answers
54 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 ...
1
vote
0answers
42 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 ...