# Questions tagged [numerical-methods]

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

9,607 questions
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### Multi-Gaussian Integrals with Heaviside for cosmic connectivity

Context I would like to predict the connectivity of the so-called cosmic web in arbitrary dimensions. This is the cosmic web (in a hydrodynamical simulation) The little wiggly things are galaxies (...
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### I really don't understand “The Shooting Method”

I am attempting to learn from a textbook that has the following question: The boundary-value problem $$y'' = 4(y-x), \qquad 0 \leq x \leq 1, \qquad y(0)=0, \, \, \, y(1)=2$$ has the ...
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### prove $||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
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### Euler's method to solve a differential equation

Apply Euler's method on the initial value problem $y'(t)=y(t)$ with $y(0)=1$ (in the interval $[0,1]$) and equidistant grid $I_h$, $h=\frac1n$. Give the approximation $y_h$ explicitly. This question ...
1answer
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### Is it a valid claim that ODEs are easier to solve numerically than PDEs?

My final project in my Partial Differential Equations class involved studying one non-linear PDE in depth. In reading about my equation, I've realized that PDEs of 3 spatial variables can be re-...
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### zero flux boundary condition

I need to proof horizontal divergence with the zero flux boundary conditions such that \begin{equation} \boldsymbol{\nabla} \cdot \mathbf{u} = 0 \end{equation} where $\mathbf{u}=(u,v,w)^T$ and each ...
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### Test if a function is continuous or has at least one discontinuous vertical asymptote between an interval

Imagine evaluating a function with little intervals incrementally across a graph and testing by using the end points of the each interval (and maybe a midpoint), whether the function is continuous for ...
1answer
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### How to solve a non linear ODE with Newton's method?

Im trying to solve this ODE using the Newton method (for non-linear equations using Jacobian Matrix): $u(0) = u(L) = 0$ (let's take $L=9)$ $T=500$ $K= 5 \times 10^9$ $w = 100$ $u''$ and $u'(x)$ ...
2answers
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### fixpoint iteration to solve $y'(t)=y(t), y(0)=1$

Solve the initial value problem $y'(t)=y(t)$, $y(0)=1$ on the interval $[0,1]$ with a fixpoint iteration of the operator $T: Y\to Y, (Ty)(t):=y_0+\int_0^t f(s,y(s))\, ds$. Begin with $y_0(t)=0$ and ...
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### Is that a Neumann condition or a Dirichlet?

In a such problem: $\ u''''=f$ with boundary conditions: $\ u'(0)=u''(0)=u'(1)=u'''(1)=0$ Is $\ u'(0)=0$ and $\ u''(0)=0$ Neumann conditions or Dirichlet conditions ?
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### How to solve a matrix dominated by zeros?

I am trying to solve a matrix of this form: Is there a known algorithm or a method to solve this kind of matrices more efficiently than a normal Gauß elimination method? I input the diagonals as ...
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### Finding the real roots of a univariate polynomial on the interval [0,1]

I have numerous, univariate polynomials with degree in excess of 100 and with very, very large coefficients (Here's an example coefficient ...
1answer
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### Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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### Matlab Code for Robin and Neumann boundary for ode. [on hold]

please can someone help me out. i need the matlab code for Robin and Neumann boundary condition for ode not pde.
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### Motivation of Milstein scheme

Milstein scheme is motivated by improving the convergence speed sigma terms of Euler scheme. where sigma and mu is globally Lipschitz continuous$$t_i=\frac{T}{n}i\quad dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$$...
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### How to use Finite Difference Method solving ODE with Boundary Value Problems?

Using these formulas, it is clear how to solve the problem: For node 1, we have the boundary value on the left side, for ex. u(0) = 0 and for node 2, we use the formula replacing u'' with u_i-1 = (...
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### PYTHON RK2 (Midpoint Method)

Good evening, I am writing code for a Numerical Analysis Project, but I am having difficulty iterating the RK2 (Midpoint Method) Correctly. I am using Python to do it, could anyone take a look at my ...
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### How to implement Finite Difference Method ODE Boundary Value Problem in Python?

I implemented the Finite Differences Method for an ODE with Boundary Value Problem. Here is the approximations I used for the FDM: And here is the balk problem: with u(0) = u(L) = 0 (attached on ...
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