# Questions tagged [integral-equations]

Questions regarding the numerical solution and analysis of equations that feature an integral transform on the unknown function. Problems with integral-equation formulations, their discretization, calculation and usage of Green's functions, eigenvalue analysis of the integral operators, and software recommendations should be marked with this tag.

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### Unstructured mesh vs hybrid structured/unstructured for numerical simulations

While answering one of the questions on meshing process, I encountered a lack of understanding on my end for the comparison of the mesh quality. First, consider an unstructured mesh created in GMSH ...
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### Numerical integration of given points, simple/easy way

I have the x and the f(x) for a set of x. I don't know the function, actually. This is ...
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### Solve integral equation for unknown constant

Consider the equations $$\int_0^L \mathbf W(\mathbf u, s) \, \mathrm ds = \mathbf 0$$ where $0 \leq s \leq L$ and $\mathbf u$ is a vector of constants. Numerically, what is the best way to ...
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### Applying the Runge-Kutta method to second order ODEs

How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)? So far I wrote simple ...
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### Mathematica NIntegrate function in C++

I am working on computing a challenging integral. I am working with someone else who wrote some code in Mathematica to compute it. I do not have mathematica so I am trying to do the same thing in C++. ...
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### Solving condensate density problem in MATLAB

I want to solve for $n_{0}$ for a fixed value of $n$, lets say $n=1$ $$n= n_{0}+ \dfrac{1}{2}\int_{-1/2}^{1/2}dq\left(\dfrac{e_{q}+Un_{0}}{\hbar\omega}-1\right)$$ where $e_{q}=2[1-cos(2\pi q)]$ ...
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### How to integrate Euler Bending Equation in C++? [closed]

I am trying to Draw shear force diagram and bending moment diagram of beams. In this, I need to integrate second order differential. So, Anybody can suggest me, which numerical method should I use?
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### Computing expectations

I want to compute the following conditional expectation $E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$ where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
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### Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form $\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$ where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
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### Package for integration over non-rectangular region

I want to compute the expected value of a multivariate function f(x) wrt to dirichlet distribution. My problem is "penta-nomial" (i.e 5 variables) so calculating ...
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### Solve a fourth order differential equation

I want to solve $$\frac{\partial^2}{\partial t^2}u(z,t) + a\frac{\partial^2}{\partial z^2}u(z,t) + k\frac{\partial^4}{\partial z^4}u(z,t) = 0$$ with $u(z,0) = 1+0.1e^{-\frac{z^2}{2}}$. I'd like to ...
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### Solve a differential equation with finite difference method

I want to solve this equation $$-\frac{1}{2}f''(x)+2a\ f(x)^3 = f(x)\mu$$ One exact solution (there are a lot of different kinds) of this equation is $f(x) = f_\infty \tanh(\sqrt{2a}f_\infty x)$ (...
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### Line Integral Convolution (LIC) Requirements

I'm trying to plot some vector fields using LIC technique. More specifically, I'm using the Python solution for this kind of plot. Before applying that approach, I was plotting my vectors as quiver. ...
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At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction \... 2answers 328 views ### Periodic Green's functions in integral equation methods in different frequency regimes I'm asking about the solution of the Helmholtz equation on a periodic domain with piecewise constant wavespeed in different frequency regimes. One possible approach is to solving this problem is to ... 1answer 778 views ### Matlab: Error in integral function I want to compute an integral with the following code written in Matlab. ... 2answers 152 views ### Convergence issues for a non-linear system I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like: $$\vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \... 1answer 383 views ### Finite difference equations versus boundary integral equations for elliptic pdes In certain cases, boundary integral methods are preferred for elliptic partial differential equations as opposed to finite difference methods. For instance, for solving the Poisson equation in a ... 0answers 92 views ### analytic or numeric integral of diverging function I'm trying to carry out the following integral numerically$$\int_{r_\mathrm{in}}^{r_\mathrm{out}} \Sigma\left(r'\right) \frac{r'}{r} \left( \frac{1}{r-r'}\, E(L) + \frac{1}{r+r'}\, K(L) \right) \... 2answers 1k views ### An Octree Code in Fortran I am new to scientific computing. I am looking for a Fortran ( preferably f90) implementation of an Octree. My problem requires an Octree which divides my domain until there aren't more than some N ... 1answer 221 views ### Flux calculation - discretization of solid angle I am currently tasked with calculating the total flux of photons or irradiance from a flat emitter ('pixel'). Previously we measured the Luminance head-on (90 degree from the emitting surface) and ... 1answer 1k views ### Numerical integration for modelling curve for superconductors (Python) I am a physicist who is trying to model the current-voltage characteristics of a superconductor-superconductor junction. The equation for this model is: \begin{align} I(V) = \frac{1}{eR_{\mathrm{n-n}... 0answers 48 views ### Lax equivalence theorem for integro-differential equation Can the Lax equivalence theorem (http://en.wikipedia.org/wiki/Lax_equivalence_theorem) be applied to the discretization of integro-differential equations, or does a similar theorem exist for them? 2answers 115 views ### Representing an integral as a special function In my research I have come across the following integral \begin{equation} f = \int_0^{2\pi} \text{d}\theta \exp\left\{\frac{3}{2}(h_1 \cos^2\theta + h_2 \sin^2\theta + 2 h_{12} \sin\theta \cos\theta)\... 2answers 290 views ### What does fundamental solutions stand for in boundary element method? I gain some introductory knowledge from the materials I read. I feel Ok with the numerical implementation part of boundary element method when the integral equation has been formulated. But the ... 2answers 317 views ### Numerical solution of fractional integro-diffrential equ. using collocation method? problem comes from "Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan"D^...
In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields ...