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.
338
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NDSolve to solve 2D space 1D time partial integro-differential equations
I have a set of pdes in 2D space and 1D time, one of them is related to the integral in space by Reynolds function. The equations are similar to
$$\frac{\partial \phi}{\partial t}=-\frac{\partial \...
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1
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How to tell Mathematica to use only one of the interpolating functions?
This is a follow-up question to this one.
I am working on a project in which I have two systems of integro-differential equations. I asked this question because it seems ...
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2
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Error when integrating with constants?
I'm trying to integrate an ellipse: (x^2)/alpha^2) + ((y^2)/beta^2) = 1. I am not sure if I should have insert constants for alpha and beta or if there's a way to keep them there.
This is my code
<...
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3
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251
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How to solve this integro-dfferential equation
I am trying to solve a complicated system of integro-differential equations, but a MWE of my problem is the following:
...
5
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2
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486
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Find area between three curves
I am trying to find the area between these three curves y=6/x, y=x+4 and y=x-4. Is there any good command to use in mathematica to make it simple?
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Integrate real function returns complex function [closed]
I want to compute the integral
$$
\int_0^c \exp(-cx+x^2) \mathrm{d}x,
$$
where $c>0$ is an unknown constant. In Mathematica Version 12.2.0
...
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1
answer
121
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Evaluating this generalised integral
I have the following integral
$$\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}...
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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 ...
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2
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199
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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 ...
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4
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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 ...
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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 ...
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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
...
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216
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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 ...
1
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1
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91
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Simplify not working
I am trying to simplify the following expression
code:
...
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2
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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
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4
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559
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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:
...
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147
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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*!(
...
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193
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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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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 ...
0
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1
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161
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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
$$(\rho + \lambda\overline{F}(w))...
3
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1
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Not able to reproduce the result of this paper to test if a system is chaotic or nonchaotic
I am trying to reproduce the results of a paper A new test for chaos in deterministic systems by Georg A. Gottwald and Ian Melbourne. This paper talks of a simple 1-0 method to determine whether a ...
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Evaluating expected improvement in Mathematica
There is a concept in optimization called expected improvement where you sequentially search for the value of x that optimizes the problem. Here is a screen shot of the general idea of the derivation:
...
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Solving the following system of integrodifferential equations: speed up of the code
Consider a system of equations
$$
\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}}{\...
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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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2
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112
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Computing a difficult integral which is taking too long
I am trying to solve the following code:
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2
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1
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185
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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, ...
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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 ...
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Need help with NIntegrate [closed]
Can I get some help with this:
...
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148
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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 ...
18
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3
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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 ...
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Numerical solution to an integro-differential equation
I would like to solve numerically the following integro-differential equation
$$ \partial_t \rho(t,x) \,=\, \partial_x\big(f'(x)\,\rho(t,x)\big) \int_0^\infty f(\xi)\,\rho(t,\xi)\,d\xi \;+\\ +\; \...
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Integrate over a set or over the domain of a distribution
If Z is a distribution, how do I express “integrate over its domain”?
i.e. in handling symbolic calculation of an expectation of an unknown distribution, you'd normally put a Z under the the integral ...
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2
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125
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Differential equation with integral of its parameters
I wanted to solve a differential equation involving a term with integration over its parameters. I want to call the integral expression but I get the wrong result:
...
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2
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267
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Solving integro-differential equation [closed]
I would like to solve the integro-differential equation of the form
$$\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.$$
I followed the steps in Solve an ...
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High precision numerical solution of the nonlinear Volterra integral equation
Let consider nonlinear Volterra integral equation 5.1 from the paper An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order
$$...
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1
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110
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Getting two different results for the same computation
I have the following two codes:
...
2
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1
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297
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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, ...
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1
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254
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Gaussian Integral incorrect
I'm trying to learn about the Gaussian Normal distribution using Mathematica.
So I defined my Gaussian like this
...
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1
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54
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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}...
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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 ...
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1
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184
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Solving some integral equations [closed]
I couldn't solve the systems, I need plot graph of solution:
...
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1
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382
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Numerically solving a multivariable integro-differential equation
I have an integro-differential equation of the form,
$$\small\frac{\partial f(x,t)}{\partial t}=\int_{-5}^5 |x-y|\,f\left(-\frac{x}{3}+\frac{4y}{3},t\right)\,f\left(\frac{2x}{3}+\frac{y}{3},t\right)\,...
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How to solve this integral equation using Mathematica?
My question: How do I use Mathematica to find the solution to Case 2? As a cross check for the code, does it verify the solution in Case I?
Case 1: We will look at an easier problem first. Let $|\...
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How should one evaluate this integral [closed]
How does one go about evaluating such an integral where E and V are constants?
$\int\sqrt{E-Vx^8}dx$
0
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1
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125
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How to 3DPlot a triple integral in cartesian coordinates
I am trying to make a 3d graph of the triple integral in cartesian coordinates. I am positive that I have been filling out the correct parameters but I am unsure what could be wrong.
Additionally I ...
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2
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Testing Turbulence Models
In the 90s, several turbulence models were proposed. Using Mathematica vv 10-12.1 we tested two models:
Spalart-Allmaras turbulence model (SA) for aerodynamic applications,
published in AIAA Paper 92 ...
2
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1
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212
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How do I plot a definite integral on Mathematica?
I am not sure how to put this in good format, but I want to plot this on Mathematica:
$$
u(t) = \frac{1}{4\pi t} \int\mathrm dy\ \mathrm e^{-y^2/(4t)}\ y\ (1 - y^2)
$$
on the interval $ [a, b] $ ...
1
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1
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How can I always get the simplest solution from Integrate? [closed]
When I evaluate this.
Integrate[1/(x (1 + x^4)^(1/4)), x] // FullSimplify
In 11.3 I get
...
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636
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How to calculate integration with undefined functions?
There are two functions inside the integral symbol, one is related to the independent variable and the other is independent of the independent variable.
Both are the derivative and Tload is a ...
2
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Computing integral using trapezoid method and simpsons' rule for a given partition
I am given f[x_]=E(x^-x)-1 on interval [1,4] and partition ...