# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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!
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### 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 ...
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### 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 ...
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### 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} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 ...
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### 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 ...
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I need to solve the following third order (non-linear) ODE by numerical methods: $$\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1.$$ By assumption, the solution should approach $... 0answers 23 views ### numerically solve quadratic air drag in xy-plane I am trying to find a reference on solving for the position of a point mass as a function time, subject to air drag( quadratic term only) in both the x and y directions. The equations that describe ... 0answers 173 views ### Change MATLAB code from Lax-Wendroff to Leapfrog I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the$ U_j^{n-1}$term, I just don't get the logic. Here's what ... 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 ... 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 ...
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 ...