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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18 views

Sum of integrals is not the same as the integral of the sum

I came across this behaviour in integration (note I put b=2 in the end to make sure that MA is not trying to use complex b): <...
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0answers
22 views

Numerical solution for 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
77 views

Solving a function inside an integral

I have an integral given by, $$a = c z_s^{d+1}\int_0^1 dx x^d (1-c^2 z_s^{2d} x^{2d})^{-1/2}$$ where $d$ is just the dimension of my choice, say $d=3$, $a$ and $z_s$ are constants (can be between $0-...
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1answer
60 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
72 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
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1answer
172 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
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3answers
245 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 $$w'(x)=\frac{1}{x}\int_0^x \sin(w(t))dt,$$...
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2answers
116 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
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1answer
87 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
71 views

Getting two different results for the same computation

I have the following two codes: ...
2
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1answer
118 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
41 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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2answers
124 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 ...
3
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0answers
60 views

Why this change gives different results in Integrate?

I'm new in Mathematica and can't understand why changing a number from Real to Integer is giving different results in my equations: ...
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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}...
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1answer
53 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
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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^\...
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0answers
41 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
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1answer
147 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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1answer
165 views

Solving some integral equations [closed]

I couldn't solve the systems, I need plot graph of solution: ...
5
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1answer
622 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 ...
3
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2answers
221 views

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 $|\...
9
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2answers
357 views

Testing Turbulence Models

In the 90s, several turbulence models were proposed. Using Mathematica vv 10-12.1 we tested two models: 1) Spalart-Allmaras turbulence model (SA) for aerodynamic applications, published in AIAA ...
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1answer
50 views

How should one evaluate this integral [closed]

How does one go about evaluating such an integral where E and V are constants? $\int\sqrt{E-Vx^8}dx$
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1answer
50 views

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 ...
2
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1answer
2k views

How to solve a system of integral equations?

I have a system of two integral equations ...
2
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1answer
255 views

How do I solve an integral equation related to the Newtonian gravity?

I would like to prove that only the functions of the form $f(R)\propto 1/R^2$ satisfy the following integral equation (assume $r>r_0>0$): $$ \int\limits_{r-r_0}^{r+r_0}\left({r^2\over{2r_0^2}}-{...
2
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1answer
127 views

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] $ ...
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1answer
205 views

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 ...
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1answer
98 views

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 ...
3
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1answer
295 views

How to handle this constant-interval integral in ODE?

The problem is to solve a system of nonlinear equations with a definite integral. $Q_i(x), i=1,2,3, x \in [-\text{max},\text{max}]$ \begin{align*} a_iQ_i''(x)-b_i(\vec{Q})-d_i\int_{-\text{max}}^{\text{...
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0answers
46 views
1
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0answers
137 views

How to solve ode with convolution?

my question is as this and i want to get the solution of this ode x(t) $ ({M_{33}}{\rm{ + }}{{\rm{m}}_{33}})x''(t) + \int\limits_{ - \infty }^t {x'(tau){K_{33}}(t -...
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1answer
97 views

Double integration of an exponential function which includes a Csch function and complex numbers

Mathematica couldn't solve following using just integrate function. I checked on interpolating as well, that also turned to be a dead end. Can someone help me with evaluating this integral? If it ...
1
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0answers
58 views

How solve integral with three moment and delta function [closed]

I am trying solve this integral in mathematica: \begin{equation} \int \dfrac{d^3 p_3}{(2\pi)^3 2 E_3} \dfrac{d^3 p_4}{(2\pi)^3 2 E_4}(p_1 p_2)(p_3 p_4) \delta (p_3 + p_4 - p_1 - p_2). \end{equation} ...
3
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1answer
234 views

Inverse Laplace transform of this complicated function

I have been solving a coupled PDE system analytically and I need to find the inverse Laplace transform of $(1)$ and get $T(x,y)$. $s$ is the Laplace domain variable and $\alpha, \beta, \gamma, T_{fi}, ...
3
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1answer
190 views

Solving partial integro-differential equation

I want to solve a partial integro-differential equation of this form: using NDSolve, something like ...
1
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1answer
81 views

Solve an Integral-differential equation with DSolve

I do not understand why the following code does not solve the equation: ...
5
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1answer
74 views
0
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0answers
34 views

Help with NIntegrate with singularity and variable integration limit

I have a function of the type: $$ F(x,z)= \int_{-\infty}^{\infty} dx' \int^{z}_{-\infty} dz' f(z,z',x,x')\frac{\partial g(z',x')}{\partial z'}$$ where the integration order may be interchanged. I ...
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1answer
67 views

Definite Integral [closed]

I am not able to do the following integration. \begin{equation} \int_{0}^{a} x\sin^2 \left(\frac{n \pi x}{a} \right)dx \end{equation} It shows an error that "more input is needed". The mathematica ...
2
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2answers
140 views

Trouble second-order ordinary differential equations with integral terms

I am a beginner of mma. During the learning of differential equations, I encountered a very difficult problem, a second-order system of ordinary differential equations with integral terms. This ...
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2answers
93 views

Problem with FindRoot function

I have set up a code to solve simultaneous integral equations. These result essentially from taking integrals of the Fermi-Dirac distribution. Although there are no obvious problems with the code it ...
1
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1answer
133 views

Regression to fit data with an integral equation

I have a set data and I know that this data are best fitted by the following equation, assuming that f($\tau$) is a log normal distribution and $M_{eq}$ = 1.5: $M(t,\tau)= M_{eq} \int (1-e^{-t/\tau}) ...
1
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1answer
148 views

Solving IDE system using FDM based on NDSolve`FiniteDifferenceDerivative

I wanted to solve the following system of integro-differential equations (IDEs) using finite difference method (FDM). $y_1''(t)+t^2y_1(t)-y_2''(t)+\int\limits_0^t[(t-x)y_1(x)+y_2(x)]\mathrm{d}x=(2+t^...
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2answers
184 views

solve the iterative integral equatio [duplicate]

The following code is incorrect, please specify my mistakes ...
0
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1answer
74 views

Solve integral equation by iteration

For[{n = 1, y[0][x_] = 1}, n < 4, n++, y[n][x_] = 1 + Integrate[y[n - 1][t]^2 + t^2, {t, 0, x}]; Print[{n, y[n][t]}]] here how can I get table of values ...
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2answers
56 views

problem with constant variables

Variable x is supposed to be constant, but the program calculates for each variable x, 100 numbers of x. I want all my x variables to have the same definition, but the program constantly changes each ...

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