What is Integral equation: Definition and 109 Discussions
In mathematics, integral equations are equations in which an unknown function appears under an integral sign.
There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwell's equations.
I have some variables that are uncertain, these are
w_m = u.ufloat(0.1430, 0.0011)
z_rec = u.ufloat(1089.92, 0.25)
theta_srec = u.ufloat(0.0104110, 0.0000031)
r_srec = u.ufloat(144.43, 0.26)
and some constant values
c = 299792.458 # speed of light in [km/s]
N_eff = 3.046
w_r = 2.469 *...
We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $.
In this case...
I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But...
I am able to solve the problem however if x was position and t was time how is this problem interpreted?
I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.
We often neglect the terms of a surface integral ##\int_v(\nabla•A)dv=\int_s(A•ds)## for ##s## to be very large or ##v## to be very large,
What is actually the reason behind this to neglect??
The book on quantum mechanics that I was reading says:
d<x>/dt = d/dt ∫∞-∞ |ψ(x,t)|2 dx
=iħ/2m ∫∞-∞ x∂/∂x [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (1)
=-∫∞-∞ [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (2)
I want to know how to get from (1) to (2)
The book says you use integration by part:
∫abfdg/dx dx = [fg]ab - ∫abdf/df dg dx
I chose f...
Homework Statement
The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h]
in here, x0, p0 and h are constants, so,
Homework Equations
what are the <x>; expetation value, and <P>;expectation value of momentum ?
The Attempt at a Solution ,
[/B]...
The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h]
in here, x0, p0 and h are constants, so, what are the <x> and <P> ?
1. At pg.212, Hartle book (2003) writes equation 9.81 as an approximation of 9.80, directly. 2. $$ΔΦ=\int_0^{w_1}\frac{(1+\frac{M}{b}w)}{(1+\frac{2M}{b}w-w^2)^\frac{1}{2}}dw$$ equation(9.80)
$$ΔΦ≈\pi+4M/b$$...
Hi!
I would like solve this kind of relation:
\phi = \int_0^r \phi (r') 4 \pi r'dr'
But I don't know how to proceed...
Can you advise me ?
Thank's in advance !
Hello, I just want to clarify some things with a simple exercise: I have the equation ## \frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0## and I want to integrate it once assuming that ## f=f(A^1,A^2,...,A^n)=f(A^\rho) ##.
I think the solution should be ## \frac{\partial f}{\partial...
Homework Statement
A specific problem of the Fredholm integral equation is given as
$$\phi(x) = \pi x^2+\int_0^\pi3(0.5\sin(3x)-tx^2)\phi(t)\,dt$$
and the exact solution is ##\phi(x) = \sin 3x##.
Homework Equations
Nothing comes to mind.
The Attempt at a Solution
I'm unsure how to approach...
Homework Statement
find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)##
Homework EquationsThe Attempt at a Solution
to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get integral...
Homework Statement
The problem states:
"A point charge q is located at a fixed point P on the internal angle bisector of a 120 degree dihedral angle between two grounded conducting planes. Find the electric potential along the bisector."
Homework Equations
ΔV = 0
with Dirichlet boundary...
Suppose I have an infinitely thin foil that absorbs 1% of incoming radiation. It therefore emits 0.5% of the incoming radiation in both directions perpendicular to the plane of the foil. That is, transmission is 0.5%. How to calculate transmission for a series of such foils? There will be a lot...
I have the following expression :
$$ y_{E} = \int_{0}^{\infty} 0.5 * [E_{1}(µ(E)*r) - E_{1}(\frac{µ(E)*r}{cos \alpha})] * f(r) dr $$
where :
- $y_{E}$ has been measured for some E (something like 5 different $E_{i}$, to give you an idea)
- µ(E) is retrieved from a table in the litterature...
I'm following a derivation in my lecture notes of total average particle number in an ideal classical gas (statistical physics approach). I follow it to the line (though the specific terms don't matter):
\left<N\right> = e^{\mu/\tau} \frac{\pi}{2} \int_0^\infty \left(n \,dn \,e^{- \frac{\hbar^2...
I would like to solve an equation below using MATLAB:
All the parameters except p are known, so I only need to solve for p. However since I need to consider the sign of the integrand and there is an absolute value sign in it I don't know how to solve it. Could anyone please help? Thank you.
Hi at all
On my math methods book, i came across the following Fredholm integ eq with separable ker:
1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi
With integral ends(0,pi/2)
I do not know how to proceed, for the solution...
Homework Statement
Find a continuous funciton ##f## such that
$$
f(x) = 1+ \dfrac{1}{x} \int_{1}^{x} f(t)dt
$$
I think I solved it but I would like to see if it's right.
Well, first of all, by the fundamental theorem of calculus I know that
$$
\left( \int_{1}^{x} f(t)dt \right) ' = f(x)
$$...
Hello! Could you tell me about how to take the next numerical calculation in mathematica? (perhaps there are special packages).
I have an expression (in reality slightly more complex):
## V=x^2 + \int_a^b x \sqrt{x^2-m^2} \left(\text Log \left(e^{-\left(\beta...
The problem
I want to find ##x## which solves ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0 ##
The attempt
##\int^x_1 \frac{\sin t}{t} \ dt = x -1 ## I see that the answer is ##x=1## but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any...
I had posted a question earlier which this is related to, but a different equation.
$$\frac{d}{dt} \int_0^t H(t,s)ds = H(t,t) + \int_0^t \frac{\partial H}{\partial t}(t,s)ds$$
This was another formula needed in a proof however I don't see how this one holds either. I tried following a proof of...
Hello! How do I solve the following integral and differential equations
1. $x^2f''(x)-2xf'(x) = x$
2. $\int_0^{x} (1+f(x))\;{dx} = x$
I'm supposed to http://mathhelpboards.com/linear-abstract-algebra-14/linear-subspaces-17957.html all the polynomials that satisfy this, but I can't. (Shake)
Homework Statement
Particular solution of
y" - y' - 2y = e^(2x)
Homework Equations
None
The Attempt at a Solution
This makes no sense to me, why do I have to use the solution of the form
y(t) = cxe^(2x)
For the problem above, but when I switch the signs and it becomes
y" - y' + 2y =...
Homework Statement
Hi - this exercise appears in a section on Fourier Transforms.
Show ## \int e^{ik \cdot (r - r')} \frac{d^3 k}{(2 \pi)^3 \vec{k}^2} = \frac{1}{4 \pi}|r-r'| ##
Hint: use spherical coords in k-space.
Homework Equations
I am not sure of the coord. transform, but from the...
Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...
Hello everyone!
I am building set of Fortran code to solve integral equation. I have read "Numerical recipe" and heard about "Nystrom method". But there's no sample problem, I found it difficult to understand. Can anyone explain "Nystrom method" for me with a simple problem?
Many thanks
How to prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$
To prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$ is true, first I started calculating the integral of the left indefinitely
$$ \int {x\,f(\sin x)\,\,dx} $$...
Homework Statement
Ok guys, its been a while since I have done this one so I don't even know where to begin. Here is the problem I have to complete:
The power (in watts) from an engine is represented by the equation below, where t is the time in seconds.
P = 30t^2.2 + 3t
1) Draw a table...
Homework Statement
The volume of the solid obtained by rotating the region bounded by
x=6y^2 http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmmi10/alpha/144/char3B.png y=1 [PLAIN]http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmmi10/alpha/144/char3B.png...
Is it possible to convert a general linear second order boundary value ode
y'' + P(x)y' + Q(x)y = g(x), y(a) = y_a, y(b) = y_b
to a Fredholm integral equation, explicitly determining the Kernel in the process, without removing the y' term? (Here is an example of doing it without the y'...
The first thing you need to do is take the Fourier Transform of both sides of the equation.
$\displaystyle \begin{align*} \mathcal{F} \left\{ f(t) + 3\int_{-\infty}^{\infty}{f(t-u)\,\mathrm{e}^{-2u} \, \mathrm{H}(u)\,\mathrm{d}u} \right\} &= \mathcal{F} \left\{ 15\mathrm{e}^{-2t^2} -...
Homework Statement
Let the curve C be given by ##\vec{r}(t)=3t^{2}\hat{\imath}-\sqrt{t}\hat{\jmath}## between ##0 \leq t \leq 4##. Calculate ##\int_{C} xy^{2}dx+(x+y)dy##.
Homework Equations
The Attempt at a Solution
First find the derivative of r...
I have to solve the integral equation y(x)= -1+\int_0^x(y(t)-sin(t))dt by the method of successive approximation taking y_0(x)=-1.
Sol: After simplification the given equation we have
y(x)=-2+cos(x)+\int_0^x y(t)dt . So comparing it with y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt we have...
I have to solve the integral equation y(x)=1+2\int_0^x(t+y(t))dt by the method of successive approximation taking y_0(x)=1.
Sol: After simplification the given equation we have
y(x)=1+x^2+2\int_0^xy(t)dt. So comparing it with y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt we have
f(x)=1+x^2...
Homework Statement
Hi.
As you all know, ##<F> = <F>^*##, where ##F## is an observable linked with the operator ##\hat{F}##. This means ## <F> = \int \Psi^* \hat{F} \Psi d\tau = <F>^* = [\int \Psi^* (\hat{F} \Psi) d \tau]^* = \int \Psi (\hat{F} \Psi)^* d \tau \Rightarrow \int \Psi (\hat{F}...
Homework Statement
Hi everyone.
I have encountered a weird equation while doing some research and I have no idea how to solve it.
The equation goes like this
∫ dR / (1+ c*r) ^ (a/r) = d, limits of integration are from 0 to Rmax,
where Rmax ^2 = [u]^2 - α^2, where u is a constant value...
Given
\[
f(x) = 1 + \lambda\int_0^1(xy + x^3y^2)f(y)dy.
\]
To use the Fredholm series, we need to find \(D(\lambda)\) which is \(1 - \text{order }\lambda + \cdots\). I have already calculated the orders and of order 3 and 4 the terms are 0 so the terms greater than 3 are all zero since we have...
Given
\[
f(x) = \lambda\int_0^1xy^2f(y)dy
\]
I am trying to determine the eigenvalues and eigenfunction. I know that the \(\frac{1}{\lambda}\) are the eigenvalues.
We can write \(f(x) = xA\) and \(A = \lambda\int_0^1y^2f(y)dy\).
\[
A\Bigg(1 - \lambda\int_0^1y^3dy\Bigg) = 0\quad (*)
\]
So is...
I need to solve an integral equation of the form
$$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$
where
- f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$
- K: [0,1] x R -> [0,1] is the unknown kernel
I am looking for a solution other than...
Homework Statement
If \displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n} = 6 and a+b=16, then find a and b.
The Attempt at a Solution
\displaystyle \int^b_a \dfrac{dx}{1 + (16/x - 1)^n} = 6
I tried substitution but it did not work.