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.

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  1. Arman777

    Python Solving an Integral equation with uncertainties

    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 *...
  2. A

    How can I simplify this integral equation with a complex numerator?

    Hello. I need help in simplifying this integral equation, i know the final result must be 2(1-x)^1/2 + C. I been stuck on this one for a while.
  3. P

    MHB Solving Integral Equation w/ Laplace Transform - Abdullah

    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...
  4. Adgorn

    Solution of a simple integral equation

    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...
  5. I

    Physical interpretation of this integral equation involving distance and time?

    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.
  6. A

    I Integral equation for large surfaces

    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??
  7. physics bob

    I Solving Quantum Mechanics Integral Equation: How to Get from (1) to (2)?

    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...
  8. B

    Quantum mechanical integral equation problem

    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]...
  9. B

    A Quantum mechanical integral equation problem

    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> ?
  10. Bishal Banjara

    How to reduce the integral equation for light deflection?

    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) $$ΔΦ≈\pi+4M/b$$...
  11. A

    How to solve this electrostatic potential integral equation with Python

    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 !
  12. JuanC97

    I How do you integrate eqns with indices?

    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...
  13. M

    Fredholm Integral Equation Numerically

    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...
  14. H

    Find f(x) which satisfies this integral function

    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...
  15. lightest

    Point charge with grounded conducting planes angled 120

    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...
  16. C

    How to calculate transmissivity of a set of reflective foils

    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...
  17. I

    I Solving this integral equation

    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...
  18. J

    'Unusual' mathematical step in integration

    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...
  19. kelvin490

    MATLAB Solving equation with integration using MATLAB

    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.
  20. Jianphys17

    I Fredholm integral equation with separable kernel

    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...
  21. S

    Integral Equation (or I think so) Calculus I problem

    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 $$ \left( \int_{1}^{x} f(t)dt \right) ' = f(x) $$...
  22. I

    Mathematica Numerical solution of integral equation with parameters

    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...
  23. Rectifier

    Solving Integral Equations: Find x from 1-x+ ∫^x_1 (sin t/t) dt

    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...
  24. Sunny29

    A How to Solve an Integral Equation Involving Exponential Functions?

    Please anyone can help solve this integral equation e^t+e^t ∫ (t, 0 ) e^(-τ) x f(τ) dτ
  25. T

    I Verifying derivative of multivariable integral equation

    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...
  26. G

    MHB Differential and integral equation.

    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)
  27. faradayscat

    Differential equations particular solution

    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 =...
  28. ognik

    Solving Integral in k-space: \frac{1}{4 \pi}|r-r'|

    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...
  29. S

    Lipschitz perturbations and Hammerstein integral equations

    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...
  30. H

    Solving integral equation with nystrom method

    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
  31. S

    MHB Solving an integral equation

    Have added attachment. Can anyone show me how to approach this problem? Thank you...
  32. Ganesh Ujwal

    MHB How to prove this integral equation?

    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 $$ \int {x\,f(\sin x)\,\,dx} $$...
  33. S

    MHB Integral equation by successive approximation

    if , then what will be . In fact I was solving the integral equation by the method of successive approximation.
  34. M

    Drawing a Graph using Integral Equation

    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...
  35. M

    Finding Volume with Rotational Solids: Cylindrical Shell Method

    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...
  36. C

    MHB Help solving/proofing an Integral Equation

    Need help showing that both sides of the following integral are equal. Any help would be greatly appreciated.
  37. B

    Writing y′′+P(x)y′+Q(x)y=g(x) as a Fredholm Integral Equation

    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'...
  38. P

    MHB Solving Integral Eq. using Fourier Transforms for Mohamed

    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} -...
  39. I

    Solving an Integral Equation on a Curve C

    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...
  40. S

    MHB Integral equation by successive approximation 2

    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...
  41. S

    MHB Integral equation by successive approximation

    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...
  42. N

    This integral equation from (intro) quantum mechanics confuses me

    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}...
  43. T

    Find the root of integral equation

    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...
  44. D

    MHB Integral equation Fredholm series

    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...
  45. D

    MHB Eigenvalue (and function) of integral equation

    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...
  46. J

    Integral equation with bounded unknown kernel

    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...
  47. U

    Solving an Integral Equation Involving x and 16

    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.