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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21 views

Using Pymc3+Theano+astropy for Bayesian inference with integral expressions

This is my first time using Pymc3 or Theano, so I apologize if this question is straightforward. I'm interested in using Bayesian inference to see how effective the (non) observation of something can ...
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99 views

Efficient multidimensional numerical integration in R and C++

I'm trying to perform a 4-dimensional numerical integration in R using a function I wrote in C++ code which is then sourced in <...
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1answer
46 views

Compute efficiently a 1D function relying on a 2D convolution

Let $X = [0,1]$, $h$ the Gaussian function (i.e. $\forall x \in X, h(x) = e^{-\frac{x}{2}}$) and $p \in L^2(X^2)$ I would like to compute numerically the following function : $$ \forall x \in X, \...
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94 views

Solve integral $ \int_{-\infty}^{\infty} e^{-x^2}dx$

i trying to solve this integral $$\int_{-\infty}^{\infty} e^{-x^2}dx$$ I'm using this CODE ...
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1answer
67 views

Gauss Integration of $\sqrt(x)$

I want to construct a gauss integration for the weight function $w(x) = x^{1/2}$ for $$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$ Solving \begin{align*} a_{1}+a_{2} =& \int_{0}^{...
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64 views

Book recommendation on numerical methods for solving Integro-Differential equations

I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
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105 views

A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms

Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit. I am studying the paper by Putnam ...
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2answers
322 views

Convolution in Python

I have an integral of a convolution between two functions. How can I calculate this in Python? It is a continuum convolution.
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1answer
64 views

Best ways to avoid singularities in kernels when solving integral equations numerically

Take a Fredholm integral equation $$ u(x) - \lambda \int_{-1}^{1} K(x,y)u(y) \, \mathrm{d} y = f(x) $$ and discretize it via (say) Gaussian quadrature with nodes $\{x_j\}$ and weights $\{w_j\}$ to get ...
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142 views

How to minimize a integral function using a constant step gradient method in Python?

I am developing a practical work of the following system of ode \begin{align}x'(t) &= k_1h(t) - (k_2+k_3)x(t)\\ y'(t) &= k_3x(t)\end{align} and $z(t) = (1-k_4)(x(t)+y(t))+k_4h(t)$, where $h(...
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2answers
460 views

Solving a 1D diffusion equation with linear and nonlinear source terms

I would like to numerically solve the following equation: $$\frac{\partial \rho (z,t)}{\partial t} = B(N_D \rho (z,t) + \rho(z,t)^2) + D \frac{\partial^2 \rho (z,t)}{\partial z^2}$$ with the boundary ...
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106 views

Calculate integrals using numpy.fft

Good evening, I would like to understand why I do not get the correct result: I assume that I know my function on discrete data points and expand it as a discrete Fourier transform: $\text{sin}(x)=\...
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59 views

What are the differences between these different forms of equation?

What are the differences between Conservative differential form, Non-conservative differential form, Conservative Integral form and Non-conservative integral form of differential equations? I know ...
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189 views

MATLAB: Compute the Schwarz-Christoffel transformation symbolically

Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane. The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - ...
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2answers
702 views

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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1answer
727 views

Numerical packages to solve Volterra integral equations

I am looking for numerical packages (ideally Python) to solve second kind Volterra integral equations, such as $$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$ or Volterra-Fredholm integral equations $$u(x,t)=g(...
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1answer
142 views

Calculation of the EFIE integral

I need help computing the following integral: $$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$ in this integral $\vec{r}$ ...
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44 views

Stability of different quadrature rules in 1st-kind Volterra integral equation

I am dealing with a integral equation $$ f'(t) = -\int_0^t K(s) f(t-s)\quad t\in [0,t_\max] \tag{1}$$ in which $f(t)$ and $f'(t)$ are known, well-behaved functions of $t$ and $K(t)$ is the unknown. In ...
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75 views

Complex Integral Equation Solution in MATLAB

I need to solve an integral equation in the form: $$A(z)+\int\limits^{z_2}_{z_1}B(z') \frac{z^N}{z^N-z'^N} \frac{e^{i\beta}}{|z|}\mathrm{d}z'=0 $$ where $A(z)$ distribution is known and we are ...
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58 views

Computing dilogarithm

I'm measuring the integral of a quantity which, mathematically, requires the computation of a dilogarithm function. $$\operatorname{Li}_2(be^{ax})$$ where $b$ and $a$ (are real and) can be positive ...
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1k views

Solving an integral equation in Python

I have to solve the following equation for $x(i), 0 \leq i \leq 1$: $$ y(i) = x(i)^{-a} \int_0^1 y(j)x(j) dj \left(\int_0^1 \mathcal A(j) x(j)^{1-a} dj\right)^{-1} \int_i^1 \left( \int_0^x x(j)^{1-a} ...
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1answer
284 views

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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1answer
825 views

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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1answer
177 views

Boundary elements method -- calculation of solid angle

I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation $$ \Delta u=-f\,, $$ and its Green's function $G\left(\mathbf{x},\mathbf{x}'\right)$ with the property $$ \Delta ...
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66 views

Broadening spectral data by using FFT's

I obtain numerical discrete data of the form $$ S_{raw}(\omega) = \sum_{j}w_{j} \delta(\omega-\omega_{j}) $$ to compare the result with experimental data the delta peaks need to be broadened ...
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194 views

Numerically solving a system of partial integro-differential equations in Matlab

Given the following system of partial integro-differential equations $$ \frac{dX(t)}{dt}=\Lambda-\mu X(t)-\beta X(t)Z(t),\\ \frac{\partial Y(t,\omega)}{\partial t}+\frac{\partial Y(t,\omega)}{\partial ...
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30 views

Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix

I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$. I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions: $$\lambda \psi(x)=\...
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63 views

Quick scheme for separable first-order ODE

I'm trying to integrate an incredibly simple ODE: $$ y'(x) = -f(y),\quad y(0) = y_0 \ , $$ from $x=0$ to $x=1$. This is a decay type of equation, $f$ is the (variable) decay rate and $y$ is the ...
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1answer
534 views

Integration including bessel function

I’m trying to evaluate an improper integral of a 0th order Bessel function of the first kind using Matlab: v = integral(@(x)besselj(0, x), 0, Inf) which returns ...
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150 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
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1answer
9k views

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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1answer
333 views

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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82 views

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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1answer
77 views

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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1answer
150 views

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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2answers
582 views

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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91 views

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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1answer
269 views

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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1answer
258 views

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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2answers
2k views

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. ...
3
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0answers
188 views

Integration of nonlinear PIDE via spectral methods

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 $\...
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2answers
375 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 ...
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1answer
808 views

Matlab: Error in integral function

I want to compute an integral with the following code written in Matlab. ...
5
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2answers
161 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 \...
3
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1answer
406 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 ...
3
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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) \...
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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 ...
2
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1answer
242 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 ...
9
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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}...
3
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0answers
51 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?