# 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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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: $\text{sin}(x)=\... 0answers 57 views ### What are the differences between these different forms of equation? What are the differences between Conservative differential form, Non-conservative differential form, Conservative Integral form and Non-conservative integral form of differential equations? I know ... 0answers 90 views ### MATLAB: Compute the Schwarz-Christoffel transformation symbolically Suppose we have a conformal mapping from the unit disk in the$\omega$plane onto the exterior of a polygon in the$z$plane. The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - ... 2answers 394 views ### Unstructured mesh vs hybrid structured/unstructured for numerical simulations While answering one of the questions on meshing process, I encountered a lack of understanding on my end for the comparison of the mesh quality. First, consider an unstructured mesh created in GMSH ... 1answer 317 views ### Numerical packages to solve Volterra integral equations I am looking for numerical packages (ideally Python) to solve second kind Volterra integral equations, such as$$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$or Volterra-Fredholm integral equations$$u(x,t)=g(... 1answer 108 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$\vec{r}$... 0answers 37 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 ... 0answers 67 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 ... 0answers 49 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 ... 0answers 880 views ### Solving an integral equation in Python I have to solve the following equation for$x(i), 0 \leq i \leq 1$: $$y(i) = x(i)^{-a} \int_0^1 y(j)x(j) dj \left(\int_0^1 \mathcal A(j) x(j)^{1-a} dj\right)^{-1} \int_i^1 \left( \int_0^x x(j)^{1-a} ... 1answer 92 views ### Numerical integration of given points, simple/easy way I have the x and the f(x) for a set of x. I don't know the function, actually. This is ... 1answer 357 views ### Solve integral equation for unknown constant Consider the equations$$\int_0^L \mathbf W(\mathbf u, s) \, \mathrm ds = \mathbf 0$$where 0 \leq s \leq L and \mathbf u is a vector of constants. Numerically, what is the best way to ... 1answer 120 views ### Boundary elements method — calculation of solid angle I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation$$ \Delta u=-f\,, $$and its Green's function G\left(\mathbf{x},\mathbf{x}'\right) with the property$$ \Delta ... 0answers 59 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 ... 0answers 187 views ### Numerically solving a system of partial integro-differential equations in Matlab Given the following system of partial integro-differential equations $$\frac{dX(t)}{dt}=\Lambda-\mu X(t)-\beta X(t)Z(t),\\ \frac{\partial Y(t,\omega)}{\partial t}+\frac{\partial Y(t,\omega)}{\partial ... 0answers 29 views ### Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix I have a real symmetric function C(x,y) defined on x,y\in[0,\infty), i.e. C(x,y)=C(y,x). I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions:$$\lambda \psi(x)=\... 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 ... 1answer 489 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 ... 0answers 145 views ### numerical analysis of a partial integro-differential equation I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\... 1answer 6k views ### Applying the Runge-Kutta method to second order ODEs How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)? So far I wrote simple ... 1answer 281 views ### Mathematica NIntegrate function in C++ I am working on computing a challenging integral. I am working with someone else who wrote some code in Mathematica to compute it. I do not have mathematica so I am trying to do the same thing in C++. ... 0answers 77 views ### Solving condensate density problem in MATLAB I want to solve for n_{0} for a fixed value of n, lets say n=1$$ n= n_{0}+ \dfrac{1}{2}\int_{-1/2}^{1/2}dq\left(\dfrac{e_{q}+Un_{0}}{\hbar\omega}-1\right) $$where e_{q}=2[1-cos(2\pi q)] ... 1answer 73 views ### How to integrate Euler Bending Equation in C++? [closed] I am trying to Draw shear force diagram and bending moment diagram of beams. In this, I need to integrate second order differential. So, Anybody can suggest me, which numerical method should I use? 1answer 137 views ### Computing expectations I want to compute the following conditional expectation E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t] where \log A_{t}=\rho \log A_{t-1} + e_{t} and e_{t} is IID N~(0,\sigma_e) and \eta_{t} is ... 2answers 347 views ### Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space? Suppose I have a 1D advection equation in conservation (divergence) form \partial_t u(x,t) = -\partial_x [v(x)u(x,t)], where u is a conserved quantity in space, and v gives the velocity of the ... 0answers 83 views ### Package for integration over non-rectangular region I want to compute the expected value of a multivariate function f(x) wrt to dirichlet distribution. My problem is "penta-nomial" (i.e 5 variables) so calculating ... 1answer 245 views ### Solve a fourth order differential equation I want to solve$$ \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 $$with u(z,0) = 1+0.1e^{-\frac{z^2}{2}}. I'd like to ... 1answer 250 views ### Solve a differential equation with finite difference method I want to solve this equation$$ -\frac{1}{2}f''(x)+2a\ f(x)^3 = f(x)\mu $$One exact solution (there are a lot of different kinds) of this equation is f(x) = f_\infty \tanh(\sqrt{2a}f_\infty x) (... 2answers 2k views ### Line Integral Convolution (LIC) Requirements I'm trying to plot some vector fields using LIC technique. More specifically, I'm using the Python solution for this kind of plot. Before applying that approach, I was plotting my vectors as quiver. ... 0answers 177 views ### Integration of nonlinear PIDE via spectral methods At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction \... 2answers 308 views ### Periodic Green's functions in integral equation methods in different frequency regimes I'm asking about the solution of the Helmholtz equation on a periodic domain with piecewise constant wavespeed in different frequency regimes. One possible approach is to solving this problem is to ... 1answer 764 views ### Matlab: Error in integral function I want to compute an integral with the following code written in Matlab. ... 2answers 149 views ### Convergence issues for a non-linear system I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like:$$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \... 1answer 377 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 213 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 113 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 266 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 316 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 ...