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.
54
questions
0
votes
1answer
42 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 :
$$
\forall x \in X, \...
-1
votes
1answer
85 views
Solve integral $ \int_{-\infty}^{\infty} e^{-x^2}dx$
i trying to solve this integral
$$\int_{-\infty}^{\infty} e^{-x^2}dx$$
I'm using this
CODE
...
1
vote
1answer
67 views
Gauss Integration of $\sqrt(x)$
I want to construct a gauss integration for the weight function $w(x) = x^{1/2}$ for
$$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$
Solving
\begin{align*}
a_{1}+a_{2} =& \int_{0}^{...
1
vote
0answers
52 views
Book recommendation on numerical methods for solving Integro-Differential equations
I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
6
votes
0answers
98 views
A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms
Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit.
I am studying the paper by Putnam ...
0
votes
0answers
12 views
Differential parameterized inequalities
Let $H$ be an Hamiltonian and denote $\vec{H}$ the associated Hamiltonian vector field.
I am interested in solving numerically the following problem
$$
\dot z(t) = \vec{H}(z(s),t_1,\ldots, t_p) \...
1
vote
2answers
103 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
1answer
49 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
...
1
vote
0answers
83 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(...
2
votes
2answers
245 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 ...
1
vote
0answers
77 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)=\...
0
votes
0answers
58 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 ...
4
votes
0answers
155 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:
$$f(u) = A - ...
5
votes
2answers
587 views
Unstructured mesh vs hybrid structured/unstructured for numerical simulations
While answering one of the questions on meshing process, I encountered a lack of understanding on my end for the comparison of the mesh quality.
First, consider an unstructured mesh created in GMSH ...
0
votes
1answer
561 views
Numerical packages to solve Volterra integral equations
I am looking for numerical packages (ideally Python) to solve second kind Volterra integral equations, such as
$$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$
or Volterra-Fredholm integral equations
$$u(x,t)=g(...
2
votes
1answer
136 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 $\vec{r}$ ...
1
vote
0answers
42 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 ...
5
votes
0answers
73 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 ...
1
vote
0answers
55 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 ...
1
vote
0answers
1k views
Solving an integral equation in Python
I have to solve the following equation for $x(i), 0 \leq i \leq 1$:
$$ y(i) = x(i)^{-a} \int_0^1 y(j)x(j) dj \left(\int_0^1 \mathcal A(j) x(j)^{1-a} dj\right)^{-1} \int_i^1 \left( \int_0^x x(j)^{1-a} ...
0
votes
1answer
162 views
Numerical integration of given points, simple/easy way
I have the x and the f(x) for a set of x. I don't know the function, actually. This is ...
0
votes
1answer
674 views
Solve integral equation for unknown constant
Consider the equations
$$\int_0^L \mathbf W(\mathbf u, s) \, \mathrm ds = \mathbf 0$$
where $0 \leq s \leq L$ and $\mathbf u$ is a vector of constants.
Numerically, what is the best way to ...
2
votes
1answer
143 views
Boundary elements method — calculation of solid angle
I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation
$$
\Delta u=-f\,,
$$
and its Green's function $G\left(\mathbf{x},\mathbf{x}'\right)$ with the property
$$
\Delta ...
3
votes
0answers
61 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 ...
3
votes
0answers
190 views
Numerically solving a system of partial integro-differential equations in Matlab
Given the following system of partial integro-differential equations
$$ \frac{dX(t)}{dt}=\Lambda-\mu X(t)-\beta X(t)Z(t),\\
\frac{\partial Y(t,\omega)}{\partial t}+\frac{\partial Y(t,\omega)}{\partial ...
1
vote
0answers
30 views
Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix
I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$.
I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions:
$$\lambda \psi(x)=\...
0
votes
0answers
62 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 ...
3
votes
1answer
517 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 ...
4
votes
0answers
149 views
numerical analysis of a partial integro-differential equation
I have to numerically solve a nonlinear partial integro-differential equation. This is my equation,
$$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
11
votes
1answer
8k views
Applying the Runge-Kutta method to second order ODEs
How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)?
So far I wrote simple ...
2
votes
1answer
314 views
Mathematica NIntegrate function in C++
I am working on computing a challenging integral. I am working with someone else who wrote some code in Mathematica to compute it. I do not have mathematica so I am trying to do the same thing in C++. ...
0
votes
0answers
81 views
Solving condensate density problem in MATLAB
I want to solve for $n_{0}$ for a fixed value of $n$, lets say $n=1$
$$ n= n_{0}+ \dfrac{1}{2}\int_{-1/2}^{1/2}dq\left(\dfrac{e_{q}+Un_{0}}{\hbar\omega}-1\right) $$
where $e_{q}=2[1-cos(2\pi q)] $ ...
0
votes
1answer
74 views
How to integrate Euler Bending Equation in C++? [closed]
I am trying to Draw shear force diagram and bending moment diagram of beams. In this, I need to integrate second order differential.
So, Anybody can suggest me, which numerical method should I use?
3
votes
1answer
145 views
Computing expectations
I want to compute the following conditional expectation
$E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$
where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
2
votes
2answers
502 views
Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?
Suppose I have a 1D advection equation in conservation (divergence) form
$\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
1
vote
0answers
85 views
Package for integration over non-rectangular region
I want to compute the expected value of a multivariate function f(x) wrt to dirichlet distribution. My problem is "penta-nomial" (i.e 5 variables) so calculating ...
1
vote
1answer
254 views
Solve a fourth order differential equation
I want to solve
$$
\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
$$
with $u(z,0) = 1+0.1e^{-\frac{z^2}{2}}$. I'd like to ...
2
votes
1answer
255 views
Solve a differential equation with finite difference method
I want to solve this equation
$$
-\frac{1}{2}f''(x)+2a\ f(x)^3 = f(x)\mu
$$
One exact solution (there are a lot of different kinds) of this equation is $f(x) = f_\infty \tanh(\sqrt{2a}f_\infty x) $ (...
5
votes
2answers
2k views
Line Integral Convolution (LIC) Requirements
I'm trying to plot some vector fields using LIC technique.
More specifically, I'm using the Python solution for this kind of plot.
Before applying that approach, I was plotting my vectors as quiver. ...
3
votes
0answers
186 views
Integration of nonlinear PIDE via spectral methods
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
8
votes
2answers
343 views
Periodic Green's functions in integral equation methods in different frequency regimes
I'm asking about the solution of the Helmholtz equation on a periodic domain with piecewise constant wavespeed in different frequency regimes. One possible approach is to solving this problem is to ...
0
votes
1answer
792 views
Matlab: Error in integral function
I want to compute an integral with the following code written in Matlab.
...
5
votes
2answers
152 views
Convergence issues for a non-linear system
I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like:
$$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \...
3
votes
1answer
393 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
votes
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) \...
10
votes
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
votes
1answer
229 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
votes
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
votes
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?
3
votes
2answers
118 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)\...
