Questions tagged [integral-equations]

Questions which deal with or solve for functions expressed in integral form, i.e., one or more unknown functions that appear under an integral sign.

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3
votes
2answers
323 views

Find area between three curves

I am trying to find the area between these three curves y=6/x, y=x+4 and y=x-4. Is there any good command to use in mathematica to make it simple?
0
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1answer
49 views

How can I change the dependent variables of a function?

What I want is to define a function which depends of some variables and then get the expression for that same function when it depends of different variables. My specific case is: (where ...
0
votes
1answer
143 views

Integrate real function returns complex function [closed]

I want to compute the integral $$ \int_0^c \exp(-cx+x^2) \mathrm{d}x, $$ where $c>0$ is an unknown constant. In Mathematica Version 12.2.0 ...
1
vote
1answer
92 views

Evaluating this generalised integral

I have the following integral $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\exp \left(a u^2+b v^2+c u v\right) \; dvdu,$$ which returns the following solution: $$\frac{2 \pi }{\sqrt{4 a b-c^2}...
2
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1answer
163 views

Integral of $r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)}$ with Mathematica

I'm trying to find the integral given below with Mathematica $\int_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)} \, dr$ However, ...
-1
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0answers
52 views

Gauss quadrature

I need help with this coding question. I'm not sure if I did it wrong or inputted the wrong thing Raw InputForm ...
7
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4answers
560 views

Numerical solution of Fredholm Equation

I would like to plot the solution of the Fredholm Equation $$f\left(x\right)+\frac{1}{\pi} \int_{-1}^{1} \frac{1}{1+\left(x-t\right)^2}f\left(t\right) dt=1, \ \ (|x|\leq 1)$$ I tried to use ...
1
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2answers
88 views

Solving Fredholm Equation with composite unknown function

I would like to numerically solve a Fredholm Equation where the unknown function is composite. For example, an equation like the one described in Solving Fredholm Equation of the second kind but ...
0
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0answers
37 views

Wolfram alpha query works online but not through mathematica?

Bit of a bizarre problem I'm having (well, more a question than a problem). I am trying to solve an integral, which I queried using the online wolfram alpha website. The input and output are ...
0
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1answer
53 views

Can I define a function by using `NIntegrate`?

Here I have a expression with formula like $$C\left(j \right)=\int_{0}^{2\pi}{f\left(k \right)e^{-kj}dk}$$Is it possible to define this function C[j_] with variable ...
0
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0answers
34 views

Solving coupled integro-differential equations using Laplace transform

I have two coupled integrodifferential equations, as shown in the image attached. I am trying to solve them using a Laplace transform, as follows ...
1
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1answer
65 views

Simplify not working

I am trying to simplify the following expression code: ...
2
votes
2answers
53 views

Assumptions for Definite Integral of Log[]

I have a given function: sol = t D[ri t/(v - b) - (1/v - 1/(v + b)) a/(b Sqrt[t]), t] // FullSimplify I now need to integrate this function across $v$ within a ...
2
votes
4answers
347 views

Solving Integral Equation -numerical solution

I am trying to find a (numerical) solution to the following integral equation: where $\epsilon$ is a real valued function and $\beta$ and $c_0$ are constants. MMA code: ...
0
votes
0answers
75 views

How to solve this equation with Duhamel integral?

How do I solve a differential equation with Duhamel's integral? I tried to solve it with NDSolve & LaplaceTransform But nothing works x[t] == -0.3375*!( ...
0
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0answers
73 views

Solving partial differential equations has integral terms

Recently, I met a partial differential equation with integral term. At first, I wanted to solve it directly through NDSolve, but found that it was not feasible because MMA could not directly deal with ...
3
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1answer
85 views

Trouble in a delay ordinary differential equation

I have a delayed partial differential equation to be solved because MMA cannot solve directly. I just used the method of this post Solve PDE with complicated coefficient non-linearity to transform it ...
5
votes
1answer
295 views

Solving integro-differential equation with DSolve in MMA 11.0

In this question DSolve(symbolic solution) gives wrong answer. I analyzed the 13 equations in which DSolve gives the same ...
0
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1answer
81 views

Solving integral equation with CDF

I would like to manipulate a functional equation where $F$ is a cumulative distribution function, which I would let Mathematica know. Using a simple example, consider $$(\rho + \lambda\overline{F}(w))...
3
votes
1answer
101 views

Not able to reproduce the result of this paper to test if a system is chaotic or nonchaotic

I am trying to reproduce the results of a paper A new test for chaos in deterministic systems by Georg A. Gottwald and Ian Melbourne. This paper talks of a simple 1-0 method to determine whether a ...
3
votes
1answer
76 views

Evaluating expected improvement in Mathematica

There is a concept in optimization called expected improvement where you sequentially search for the value of x that optimizes the problem. Here is a screen shot of the general idea of the derivation: ...
2
votes
1answer
244 views

NDSolve cannot solve an ODE system with functions given as integrals-

The problem is that my Mathematica programme cannot solve the differential algebraic equations system. To simplify the problem as much as possible I will consider only the differential equation (which ...
1
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0answers
110 views

Solving the following system of integrodifferential equations: speed up of the code

Consider a system of equations $$ \begin{cases}\frac{\partial f(E,t)}{\partial t}-H[T_{\gamma},f(E,t)]E\frac{\partial f(E,t)}{\partial E} -I[f(E,t),E,T_{\gamma}(t)]=0, \\ \frac{\partial T_{\gamma}}{\...
2
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0answers
54 views

How to get Integrate to determine units

In a similar fashion to the question asked here, I am having trouble trying to get Integrate to correctly determine the units of the integrand: Here I have plugged in specific values into the general ...
0
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2answers
78 views
2
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1answer
118 views

A two-dimensional non-linear integral equation

I'm trying to figure out the solution to the integral equation $$\frac{1}{2}xyf(x,y) - \int^1_0\int^1_0\left[\frac{\exp[1 - f(x,y)]}{(\exp[1 - f(x,y)] + \exp[1 - f(x',y')])^2}\right]dx'dy'=0$$ First, ...
0
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0answers
73 views

the number of equation does not match the number of variables

Excuse me , can you help me , please ? When i make run for this programme it give me a wrong that the number of equations does not match the number of variables ,,,, I try a lot on finding the wrong ...
0
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1answer
35 views

Need help with NIntegrate [closed]

Can I get some help with this: ...
5
votes
2answers
266 views

Numerical solution to an integro-differential equation

I would like to solve numerically the following integro-differential equation $$ \partial_t \rho(t,x) \,=\, \partial_x\big(f'(x)\,\rho(t,x)\big) \int_0^\infty f(\xi)\,\rho(t,\xi)\,d\xi \;+\\ +\; \...
13
votes
3answers
375 views

Solving the Lotka-McKendrick model with NDSolve

The Lotka-McKendrick model is a demographic model that represents the way a population changes over time due to fertility and mortality. For an age-specific population density $ u(a, t) $, and a total ...
3
votes
1answer
239 views
1
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0answers
60 views

NDsolve Boundary Condition is a Function of the Solution

I am trying to solve something like Fick's Law using NDSolve: $$\frac{\partial \varphi}{\partial t}=\frac{\partial^2 \varphi}{\partial r^2}+F(r,t)$$ Subject to a ...
3
votes
2answers
145 views

Solving an integral equation for upper boundary

I am reading a paper on High Harmonics Generation (HHG) and a Lewenstein model The paper is here. I would like to reproduce some results but I am stuck at the following problem. I have: $$p(\tau_b,\...
0
votes
1answer
69 views

Integrate over a set or over the domain of a distribution

If Z is a distribution, how do I express “integrate over its domain”? i.e. in handling symbolic calculation of an expectation of an unknown distribution, you'd normally put a Z under the the integral ...
1
vote
2answers
86 views

Differential equation with integral of its parameters

I wanted to solve a differential equation involving a term with integration over its parameters. I want to call the integral expression but I get the wrong result: ...
12
votes
1answer
254 views

High precision numerical solution of the nonlinear Volterra integral equation

Let consider nonlinear Volterra integral equation 5.1 from the paper An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order $$...
8
votes
3answers
276 views

Numerically solving an ODE whose right-hand side involves an integral

In a 1955 paper on surfaces of constant negative Gaussian curvature, Marc-Henri Amsler defines a function $w(x)$ by giving the ordinary differential equation $$w'(x)=\frac{1}{x}\int_0^x \sin(w(t))dt,$$...
1
vote
2answers
141 views

Solving integro-differential equation [closed]

I would like to solve the integro-differential equation of the form $$\left( -n \int_0^b b db + \frac{i \Lambda l_P^2}{9V_c}\frac{d}{db}+b^2+k\right)\psi(b)=0.$$ I followed the steps in Solve an ...
2
votes
1answer
101 views

How to find the expectation $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$?

I'm trying to find the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$ where $a$ and $b$ are constant values, $\mathcal{Q}$ is the Gaussian Q-function,...
0
votes
1answer
72 views

Gaussian Integral incorrect

I'm trying to learn about the Gaussian Normal distribution using Mathematica. So I defined my Gaussian like this ...
1
vote
2answers
129 views

Can't solve for y in a fairly simple work done by magnetic field equation

The equations below are essentially calculations for a railgun-type mechanism (here is a good image description). I also drew an image using my own variables as definitions: I assumed that ...
0
votes
1answer
51 views

Definite integral Closed-form or approximation\simplication or a plot diagram shows the variation based on $a$ and $b$

I am not professional in Mathematica and I am struggling to understand how the following function acts $f(x)=\int_{z=0}^\infty \int_{y=0}^{z} \frac{x}{ \sqrt{1-\left(\frac{z^2+y^2-x^2}{2 z y}\right)^2}...
1
vote
1answer
73 views

Integrating Superposition of several sinusoidal waves squared times its derivative

I'm currently finishing some research from a internship I did last summer and part of the project involves calculating the centroids of coronal holes from synoptic maps. If you aren't aware of what ...
15
votes
2answers
3k views

Solve an integral equation numerically

I am trying to find a numerical solution for an equation of the form: $$ f(t) = \int_{t_{min}}^{t} \mathrm{d}t' {\exp[2 (t^\prime - t)] E(t^\prime) f(t^\prime)} + \int_{t}^{0}{\mathrm{d}t' \exp(t^\...
2
votes
0answers
52 views

Solve Fredholm integral equation of the second kind containing Double Exponential Oscillatory

Given where $h =0.5$ and $\kappa = 1$. $G_F(s)$ is the Fourier cosine transform of $G(\lambda)$ defined as Then I want to solve the following Fredholm integral equation of the second kind for the ...
5
votes
1answer
164 views

Numerically solving a multivariable integro-differential equation

I have an integro-differential equation of the form, $$\small\frac{\partial f(x,t)}{\partial t}=\int_{-5}^5 |x-y|\,f\left(-\frac{x}{3}+\frac{4y}{3},t\right)\,f\left(\frac{2x}{3}+\frac{y}{3},t\right)\,...
-4
votes
1answer
174 views

Solving some integral equations [closed]

I couldn't solve the systems, I need plot graph of solution: ...
5
votes
1answer
652 views

Solving PDE involving Hilbert transform numerically

I'm trying to solve the following equation numerically: $$u_{tt}-\mathcal{H}(u_x)=A^2_{xx},$$ where $\mathcal{H}$ is the Hilbert transform and $A$ is a prescribed forcing function which we assume ...

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