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.
61
questions
0
votes
0
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70
views
It is possible to solve integro-differential equations using in Fenics?
I am interested in solve the following integro-differential equation:
\begin{align}
\frac{\partial{\rho(\theta, t)}}{\partial{t}} = D \frac{\partial{\rho(\theta, t)}}{\partial{\theta^2}} - \beta \...
- 111
3
votes
0
answers
119
views
Numerical integration of a rapidly varying complex exponential
I have a function $f : \mathbb{R}^2 \mapsto \mathbb{R}^+$ and I wish to numerically evaluate the integral below over a finite domain $\Omega \subset \mathbb{R}^2$
$$
I = \int_\Omega e^{i k \cdot f(\...
- 131
0
votes
1
answer
155
views
Solving a partial integro-differential equation numerically
I am trying to find the solutions for a probability density $p(x,t)$, governed by,
$$
\frac{\partial p(x,t)}{\partial t} =\int_{-\infty}^\infty dx' \; \Lambda(x-x')\frac{\partial^2 p(x',t)}{\partial x'...
- 201
2
votes
0
answers
127
views
Numerical solution to integro-differential equation
The time dynamics of an atom interacting with a reservoir of spectral density $J(\omega)$ are obtained by solving the following integro-differential equation:
$$ \frac{\mathrm{d}c(t)}{\mathrm{d}t} = - ...
- 21
1
vote
1
answer
357
views
Huygens Fresnel Diffraction integral using dblquad in python
I am attempting to create a python function to assist in calculating the following numerical integration of the Huygens Fresnel integral in the form of ...
1
vote
1
answer
65
views
Nonlinear integral equation on the half line
I want to numerically solve the following equation for $\phi$ on $\mathbb{R}_+^{*2}$:
$$ \partial_t \phi (w, t) = \int_0^{+ \infty} k(\alpha w + \beta w', w') \phi(\alpha w + \beta w', t) \phi(w', t) ~...
- 433
2
votes
1
answer
313
views
2nd order differential equation coupled to integro-differential equation in python
I'm trying to solve the following equations numerically in python
$$\begin{align}
12\pi\int_0^\infty drf(r)\phi(r)r^4&=E\\
f(r)-\frac{1}{2\mu}\bigg(\frac{d^2\phi(r)}{dr^2}+\frac{2}{r}\frac{d\phi(r)...
- 112
0
votes
0
answers
154
views
Finite difference solver for the 2D Poisson's equation with an integral boundary condition
I wanted to attempt an implementation of a finite-difference-based solver for the 2D elctrostatic Poisson equation when metallic objects are present. Also, I hope to take as input, the location of ...
1
vote
0
answers
238
views
Efficient multidimensional numerical integration in R and C++
I'm trying to perform a 4-dimensional numerical integration in R using a function I wrote in C++ code which is then sourced in <...
- 119
0
votes
1
answer
57
views
Compute efficiently a 1D function relying on a 2D convolution
Let $X = [0,1]$, $h$ the Gaussian function (i.e. $\forall x \in X, h(x) = e^{-\frac{x}{2}}$) and $p \in L^2(X^2)$
I would like to compute numerically the following function :
$$
\forall x \in X, \...
- 11
-1
votes
1
answer
102
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
...
1
vote
1
answer
72
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}^{...
user37062
1
vote
0
answers
94
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 ...
- 287
6
votes
0
answers
109
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 ...
- 161
1
vote
2
answers
666
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.
2
votes
1
answer
96
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
...
- 2,125
1
vote
0
answers
255
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(...
- 111
2
votes
2
answers
741
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 ...
- 29
1
vote
0
answers
233
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)=\...
- 153
4
votes
0
answers
244
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:
$$f(u) = A - ...
- 73
6
votes
2
answers
863
views
Unstructured mesh vs hybrid structured/unstructured for numerical simulations
While answering one of the questions on meshing process, I encountered a lack of understanding on my end for the comparison of the mesh quality.
First, consider an unstructured mesh created in GMSH ...
- 8,592
0
votes
1
answer
1k
views
Numerical packages to solve Volterra integral equations
I am looking for numerical packages (ideally Python) to solve second kind Volterra integral equations, such as
$$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$
or Volterra-Fredholm integral equations
$$u(x,t)=g(...
- 1
2
votes
1
answer
199
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 $\vec{r}$ ...
- 21
1
vote
0
answers
53
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 ...
- 725
5
votes
0
answers
82
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 ...
- 91
1
vote
0
answers
68
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 ...
- 183
1
vote
0
answers
2k
views
Solving an integral equation in Python
I have to solve the following equation for $x(i), 0 \leq i \leq 1$:
$$ y(i) = x(i)^{-a} \int_0^1 y(j)x(j) dj \left(\int_0^1 \mathcal A(j) x(j)^{1-a} dj\right)^{-1} \int_i^1 \left( \int_0^x x(j)^{1-a} ...
- 119
0
votes
1
answer
541
views
Numerical integration of given points, simple/easy way
I have the x and the f(x) for a set of x. I don't know the function, actually. This is ...
- 103
0
votes
1
answer
860
views
Solve integral equation for unknown constant
Consider the equations
$$\int_0^L \mathbf W(\mathbf u, s) \, \mathrm ds = \mathbf 0$$
where $0 \leq s \leq L$ and $\mathbf u$ is a vector of constants.
Numerically, what is the best way to ...
- 187
2
votes
1
answer
227
views
Boundary elements method -- calculation of solid angle
I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation
$$
\Delta u=-f\,,
$$
and its Green's function $G\left(\mathbf{x},\mathbf{x}'\right)$ with the property
$$
\Delta ...
- 297
3
votes
0
answers
91
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 ...
- 31
4
votes
0
answers
214
views
Numerically solving a system of partial integro-differential equations in Matlab
Given the following system of partial integro-differential equations
$$ \frac{dX(t)}{dt}=\Lambda-\mu X(t)-\beta X(t)Z(t),\\
\frac{\partial Y(t,\omega)}{\partial t}+\frac{\partial Y(t,\omega)}{\partial ...
- 41
1
vote
0
answers
33
views
Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix
I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$.
I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions:
$$\lambda \psi(x)=\...
0
votes
0
answers
65
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 ...
- 136
3
votes
1
answer
568
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 ...
- 33
5
votes
0
answers
159
views
numerical analysis of a partial integro-differential equation
I have to numerically solve a nonlinear partial integro-differential equation. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
11
votes
1
answer
10k
views
Applying the Runge-Kutta method to second order ODEs
How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)?
So far I wrote simple ...
- 113
2
votes
1
answer
364
views
Mathematica NIntegrate function in C++
I am working on computing a challenging integral. I am working with someone else who wrote some code in Mathematica to compute it. I do not have mathematica so I am trying to do the same thing in C++. ...
- 123
0
votes
0
answers
84
views
Solving condensate density problem in MATLAB
I want to solve for $n_{0}$ for a fixed value of $n$, lets say $n=1$
$$ n= n_{0}+ \dfrac{1}{2}\int_{-1/2}^{1/2}dq\left(\dfrac{e_{q}+Un_{0}}{\hbar\omega}-1\right) $$
where $e_{q}=2[1-cos(2\pi q)] $ ...
- 41
0
votes
1
answer
80
views
How to integrate Euler Bending Equation in C++? [closed]
I am trying to Draw shear force diagram and bending moment diagram of beams. In this, I need to integrate second order differential.
So, Anybody can suggest me, which numerical method should I use?
3
votes
1
answer
157
views
Computing expectations
I want to compute the following conditional expectation
$E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$
where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
- 235
2
votes
2
answers
757
views
Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?
Suppose I have a 1D advection equation in conservation (divergence) form
$\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
1
vote
0
answers
108
views
Package for integration over non-rectangular region
I want to compute the expected value of a multivariate function f(x) wrt to dirichlet distribution. My problem is "penta-nomial" (i.e 5 variables) so calculating ...
- 111
1
vote
1
answer
316
views
Solve a fourth order differential equation
I want to solve
$$
\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
$$
with $u(z,0) = 1+0.1e^{-\frac{z^2}{2}}$. I'd like to ...
- 159
2
votes
1
answer
327
views
Solve a differential equation with finite difference method
I want to solve this equation
$$
-\frac{1}{2}f''(x)+2a\ f(x)^3 = f(x)\mu
$$
One exact solution (there are a lot of different kinds) of this equation is $f(x) = f_\infty \tanh(\sqrt{2a}f_\infty x) $ (...
- 159
5
votes
2
answers
3k
views
Line Integral Convolution (LIC) Requirements
I'm trying to plot some vector fields using LIC technique.
More specifically, I'm using the Python solution for this kind of plot.
Before applying that approach, I was plotting my vectors as quiver. ...
- 223
3
votes
0
answers
194
views
Integration of nonlinear PIDE via spectral methods
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
- 203
8
votes
2
answers
393
views
Periodic Green's functions in integral equation methods in different frequency regimes
I'm asking about the solution of the Helmholtz equation on a periodic domain with piecewise constant wavespeed in different frequency regimes. One possible approach is to solving this problem is to ...
- 4,470
0
votes
1
answer
833
views
Matlab: Error in integral function
I want to compute an integral with the following code written in Matlab.
...
- 3
5
votes
2
answers
176
views
Convergence issues for a non-linear system
I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like:
$$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \...
- 247