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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55 views

Why is my integral evaluating to 0? [closed]

I am trying to write a simple function, then integrate it without bounds. However, it just gives 0 which I know is incorrect.
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1answer
74 views

Different results in different versions of Mathematica (computing integrals)

I am working on computing the following integration: aa = N[2, 20] H = 4/160; X = Range[-aa, aa, H]; Integrate[400/(1 + 40*x^2), {x, X[[k]], X[[k + 1]] and ...
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1answer
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How to evaluate Sum with Singularities?

I derived equation of sum from the following problem, $\int_{a}^{b}\sum_{n=0}^{\infty} cos^n(x)dx$. Using the following definitions, $cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $(a+b)^n=\sum_{m=0}^{n}\binom ...
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1answer
49 views

Integral with exponential function

Could you please help me in solving the following integration. Thank you in advance. Integrate[((Sqrt[g*(x + h)/y])^V) * Exp[-u * Sqrt[g*(x + h)/y]/n] * Exp[-L*x], {x, 0, Infinity}]
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1answer
70 views

How to integrate exponential function over implicit function

The following integral can be solved in Mathematica: Integrate[Exp[a*(z - 1)], {x, y, z}\[Element]Sphere[],Assumptions -> a > 0 && a\[Element]Reals] ...
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1answer
179 views

How to increase the speed of Collocation method to solve System of differential equations

all. Kindly suggest me a ways to improve the speed of computations. I recently came to know about collocation methods and their application to find solution for system of differential equations. Here ...
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0answers
44 views

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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1answer
40 views

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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2answers
50 views

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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3answers
176 views

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: ...
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52 views

Problems with the output of the integral function

I write here because I've some problems when I want to integrate this quantity with respect to z: rH, a and z are real values, being TT4 a real value too and I especify that in the assumptions of the ...
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2answers
428 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?
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1answer
145 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 ...
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1answer
95 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}...
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1answer
70 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 ...
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2answers
95 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 ...
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4answers
581 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 ...
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0answers
41 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 ...
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43 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 ...
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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 ...
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1answer
66 views

Simplify not working

I am trying to simplify the following expression code: ...
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2answers
56 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 ...
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4answers
363 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: ...
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0answers
76 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*!( ...
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0answers
75 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 ...
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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 ...
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1answer
83 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))...
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1answer
108 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
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1answer
79 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: ...
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111 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}}{\...
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0answers
55 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 ...
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2answers
81 views

Computing a difficult integral which is taking too long

I am trying to solve the following code: ...
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1answer
119 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, ...
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83 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 ...
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1answer
35 views

Need help with NIntegrate [closed]

Can I get some help with this: ...
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0answers
61 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 ...
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3answers
379 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 ...
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2answers
272 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 \;+\\ +\; \...
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1answer
70 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 ...
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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: ...
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2answers
145 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 ...
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1answer
270 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 $$...
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1answer
81 views

Getting two different results for the same computation

I have the following two codes: ...
2
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1answer
103 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,...
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1answer
164 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, ...
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1answer
74 views

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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1answer
52 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}...
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1answer
75 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 ...
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1answer
175 views

Solving some integral equations [closed]

I couldn't solve the systems, I need plot graph of solution: ...
5
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1answer
172 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)\,...

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