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. A

    Numerical solution for an integral equation?

    Hello, I have been encountered an integral equation that I need to solve\evaluate numericly and I didn't find anything like it in my search yet. The equation: \frac{d^2 F(t)}{dt^2}=const*\int_{t'}\frac{sin(F(t)-F(t'))}{(t-t')^{2k}}dt' If it helps there is a specific case, when K=1...
  2. J

    An integral equation Abel and L operator?

    an integral equation Abel and L operator?? 1. The Abel operator The general Abel integral equation \begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered} has the solution \begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi...
  3. LeonhardEuler

    Is there a general method for solving Fredholm integral equations?

    Hello Everyone. An interesting equation has come in my thesis research, and I was wondering whether anyone had any useful information about it. It is this equation: \int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)K(x_1,...x_n,s_1...s_n)dx_1...dx_n=C K is a known function of the x's and s's. C is...
  4. I

    Laplace transform solve integral equation

    Homework Statement Homework Equations The Attempt at a Solution The answer is y(t) = t^{4}+\frac{t^{6}}{30} Don't know what to do next any advices please
  5. M

    Solving an integral equation by iteration

    Hello guys! I was given a Volterra integral equation y(x)=1/2*x^2+integral(0--->x) [t(t-x)y(t)]dt to solve using iteration. I have no idea how and where to start... The full problem goes as follows: Show that the solution y(x) of y''+xy=1, y(0)=y'(0)=1 also satisfies the integral...
  6. M

    Can Fredholm Integral Equations Be Solved Without Fourier Transforms?

    Is there any way to solve Fredholm integral equation without using Fourier transform. \varphi(t)=f(t)+\lambda\int^b_aK(t,s)\varphi(s)ds?
  7. O

    The picture in the addition, you see one integral equation and I have

    The picture in the addition, you see one integral equation and I have moved forward to solve it but when I tried to use patial integrant ( uv-∫v.du) I couldn't go further, Is there anyone to help me solving that equation.Actually, I stuck while putting the boundries (from 0 to ∞ ) because it...
  8. G

    Volterra Integral Equation as a Generalisation of Picard Theorem

    Hi Physics Forums, this is my first post here thanks in advance for any help hopefully I'll be able to return the favor. Homework Statement As a generalisation of the Picard Theorem: An integral equation of the form: y(x) = f(x) + \int_0^xK(x,x')y(x')dx' (0 \leq x \leq b) where...
  9. T

    Solution of Laplace Integral Equation Using Convolution Theorem

    Homework Statement By taking the Laplace transform and using the convolution theorem, obtain the solution of the integral equationHomework Equations f(t) = sin t + ∫e^(t-u)*f(u) du integral is from 0 to tThe Attempt at a Solution I used the following site as a reference for how to construct the...
  10. A

    Solving Integral Equation for y(\pi/3)

    Homework Statement Find y(\pi/3) in the following integral equation: $$y(x)=e^{-\int_{1}^{x}\frac{y(t)}{\sin^2(t)}dt}$$ Homework Equations The Attempt at a Solution Differentiating both sides gives a differential equation with the general solution y(x)=(-\cot(x)+c)^{-1}, and since...
  11. D

    How can I solve for the form of f in this integral equation with a>b?

    Any advice on how to approach a problem like this either numerically or analytically? I'm looking to find the form of f where b>a. \frac{1}{f(r)+1} = \int_r^b f(x) dx + \int_a^r \frac{x^2}{r^2}f(x)dx
  12. D

    Multiplication in a Definite Integral Equation

    If f(x)=\int_0^\infty g(x) dx and I wanted to multiply the integral by, say, a, what would I multiply the left side by? In other words, ? \times f(x) = a \int_0^\infty g(x) dx Thanks in advance!
  13. V

    How to Simplify This Complex Double Integral Equation?

    Homework Statement Help needed in simplifying this one $$\left[\int\limits_0^{Inf} {\int\limits_0^{Inf} {{e^{ - \alpha X - \beta Y}} \cdot F(X + Y + c)} } \cdot {X^d} \cdot {Y^e} \cdot dX \cdot dY\right]$$ is equal to $$\left[\sum\limits_{i = 0}^d {d{C_k} \cdot } {( - 1)^{d - i}} \cdot \left[...
  14. V

    How can I solve this tough integral equation for a dipole magnetic field?

    Hi, I have an integral equation for which I'd appreciate any tips! The equation is: B^2(\vec{r}) = \int_{\vec{r}}^{\infty} \left[ (\vec{B}(\vec{s}) \cdot \nabla) \vec{B}(\vec{s}) \right] \cdot \vec{ds} The path of integration can be chosen arbitrarily, starting at some point r, and ending at...
  15. M

    Efficient methods for solving homogeneous fredholm integral equation

    I have an integral equation of the form y(x) = \int_0^{2\pi} dx'~K(x,x';\omega,\alpha)y(x'), where K(x,x';\omega,\alpha) is a real, non-symmetric, non-separable kernel that depends on the parameters \omega and \alpha, as well as some others that I won't need to vary as much. The kernel is...
  16. A

    Is this a non linear integral equation ?

    fx = ∫ exp(1-x^2)^.5 limits are b,-b
  17. S

    Solving integral equation with double ingegrals and singularities

    Hello I need help to solve the following integral equation: f(x,y,w)=137.03.*y.^2./((0.238.*exp(0.067.*y.^2)+1).*(w-5.26.*x.*y-2.63).*(w-5.26.*x.*y+2.63))=1+8478./(10828-w.^2-1.13.*j.*w) xmin=-1, xmax.=1, ymin=0, ymax=inf (nad can be taken 500 because the function decreases rapidly) I want...
  18. J

    How to Solve the Logistic Equation for Elk Population Growth?

    Homework Statement This a problem that I didn't get completely right after a test, so I wouldn't mind figuring out what were my errors. A state game commission releases 40 elk into a game refuge. After 5 years, the elk population is 104. The commission believes that the environment can...
  19. D

    Integral equation with a derivative of the function inside the integral

    f(x) = 2\int_{0}^{t} sin(8u)f'(t-u) du + 8sin(8t) , t\geq 0 is this problem solvable? I've never seen an integral equation like this with an f'(t-u) i tried to solve it us the convolution theorem and laplace transforms but ended up with s^{2} F(s) + 64F(s)- 16(F(s) - f(0)) =64...
  20. A

    Recurrence relation for an integral equation

    Hi guys, would appreciate any help with this problem. It's from a graduate school entrance examination which I am practicing. problem statement When n is a natural number, the indefinite integral I_n is defined as I_n=\int\frac{1}{(x^2+a^2)^n}dx Here, a is a constant real number, and...
  21. R

    Integral equation with endpoint singularity

    I am trying to solve this integral equation numerically. The kernel has a singularity at the endpoint 1. Any suggestions?? f(s) = \int_0^1 \frac{1+st}{(1-st)^3} f(t) dt
  22. M

    An Integral Equation with the Convolution Theorem for Fourier Transforms

    The problem: Solve the integral equation \int\stackrel{\infty}{-\infty}exp(-abs(x-y))u(y)dy+u=f(x) for -\infty<x<\infty. The solutions say "Use the convolution theorem to find u(x)=f(x)-\frac{4}{3}\intf(t)exp(-3abs(x-t))dt." The Convolution Theorem in my book states "If the functions f(x)...
  23. D

    Find all functions of this integral equation, very tricky

    Homework Statement \int{f(x)dx}\int{\frac{1}{f(x)}dx} = -1 Homework Equations The Attempt at a Solution At first glance, I thought ex is a solution. But I'm not convinced yet because I didn't formally go through it. When I did it turns into some nasty integral. Here's a couple things I've...
  24. L

    Solving Diffuse Light Simulation with Double Integral Equation?

    Recently, I've been working on a program to simulate diffuse light, and I've hit a snag. I need to solve (at least so that a computer can compute L(x) quickly) something of the form: L(x)=T(x)+c\int_0^{l_2}\int_0^{l_1} W(x,u_1,u_2) L(u_1) du_1 du_2 W and T are pretty well behaved, and...
  25. P

    Can You Integrate x/sqrt(x(1-x))?

    sorry, is it possible to take integral from this: x/sqrt(x(1-x)) Need help! Thanks!
  26. J

    Capacitance matrix and integral equation method

    Dear all, I'm having some trouble calculating the capacitance matrix, as outlined below. So first of all, the lecture notes I'm using use an integral equation method (method of moments) to determine the capacitance of two infinitely long and thin parallel strips in vacuum, at a distance d of...
  27. M

    From integral equation to normal equation

    Consider \frac{d^{2}y}{dx^{2}}=k--------------[0] and \frac{dy}{dx}=y_{a} Then y_{f}-y_{i}=\frac{k}{2}(x_{f}^{2}-x_{i}^{2})-----------[1] y_{af}-y_{ai}=k(x_{f}-x_{i})-------------[2] is equation 1 and 2 correct ? if no, then what is the correct solution if yes...
  28. F

    Integral equation to solve with La Place transformation

    Homework Statement y(t)+2\int_0^tcos(t-\tau)y(\tau)d\tau = 9e^{2t} Solve this integral equation by using the La Place transform. Homework Equations NoneThe Attempt at a Solution I tried differentiation the whole equation: y'(t) + 2cos(t-\tau)y(\tau) = 18e^{2t} with boundaries 0 and t...
  29. Z

    Proving Integral Equation: f(x)= O(x)

    let be the functions f(x) and g(x) defined by an integral equation g(x)= \int_{0}^{\infty}dy K(yx)f(y)dy then i want to prove that for example f(x)= O(x) then using a change of varialbe yx=t i manage to put g(x) \le \frac{C}{x^{2}} \int_{0}^{\infty}dtK(t)t if the last...
  30. M

    Solving Integral Equation with Laplace Transform

    Homework Statement y(x) = e^{3x} + \int_0^{x}\hspace{1mm}y(t)\hspace{1mm}e^{x-t}\hspace{1mm}dt Homework Equations (f*g)(x) = \int_0^{x}\hspace{1mm}y(t)\hspace{1mm}g(x-t)\hspace{1mm}dt L(f*g;s) = L(f;s)L(g;s) I will use Y(s) to denote L(y;s) The Attempt at a Solution I tried...
  31. M

    Asymptotics of a linear delay integral equation

    Hello all, During the course of a calculation I was doing for my research, I derived a delay integral equation of the form g(x) = \int_0^1 dy K(y,x)g(x-y) where K(x,y) is a known, but somewhat ugly, kernel that has a (1-y^2)^{-1/2} singularity, but is integrable such that \int_0^1 dy...
  32. N

    Solution of an integral equation

    Homework Statement Given \frac{1}{| \int\ f(\ x)\ g(\ x)\ d\ x\ |}=\int \frac{\ f(\ x)}{\ g(\ x)}\ d\ x Does the above put any condition on f(x) and g(x)? Homework EquationsThe Attempt at a SolutionThe | | in the denominator reminds me of Darboux inequality...In fact it looks impossible to...
  33. N

    How to invert an integral equation

    Homework Statement Suppose we have a physical quantity f(r) depending on another quantity q(r). f(r) is known at all points. If the following relationship holds: Homework Equations f(r)=\int_{\Omega}q(r-r')dr' where \Omega is a bounded volume, is there any possibility to...
  34. B

    Solving Integral Equation: x(t) = -8?

    Homework Statement I integrated dx/dt = x^2+(1/81) Homework Equations My result; 9(arctan(9x))= t+C needed to be solved for initial condition x(0)=-8 and fit into x(t) = format The Attempt at a Solution I cannot figure out why the computer is not accepting my solution: arctan...
  35. Y

    Integral equation. Please check

    I have recently been playing with integrals and I still do not fully understand them. Usually the best way for me to learn is to play with values and figure it out on my own, so I would like you (physics forum) to check my work so far. After a bit of time I got this...
  36. S

    Testing Weak Singularity of Integral Equations

    Homework Statement If we want to show whether a kernel is weakly singular or not, what do we do? eg. consider: a) \int_0^x sin(x-s)y(s)ds b) \int_{-3}^3 \frac{y(s)}{x-s}ds Homework Equations A discontinuous kernel k(x; s) is weakly singular (at x = s) if k is continu- ous...
  37. S

    Solving Integral Equation: sin(x)+∫_0^π sin(x-t)y(t)dt

    Homework Statement solve the following integral equation y(x)=sin(x)+\int_0^\pi sin(x-t)y(t)dt Homework Equations The Attempt at a Solution If the limits of integration were from 0 to x then i could solve this using Laplace transfroms because the definition of the convolution...
  38. Z

    Integral equation for Xi-function

    think i have discovered an integral equation for the Xi-function \Xi (z)= A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x+z) \frac{dx}{x} with \Phi(u) = \sum_{n=1}^{\infty}(2\pi ^{2} n^{4}e^{9u}-3\pi n^{2}e^{5u} )exp(-\pi n^{2}e^{4u}) and 'A' is a Real constant.
  39. S

    Solving Nonlinear Integral Equation with Newton Method

    Homework Statement If I have a non linear integral equation of the form: y(s)+\int^x_0{K(x,s,y(s)}ds=f(x) and i want to find a way to solve this numerically using the Newton method Homework Equations The Attempt at a Solution after discretizing, and using the quadrature...
  40. B

    Integral equation with unknown kernel?

    Hi, all, I would to solve an integral equation, here is the form f(x)=\int_{x}^{R}K(x,t)g(t)dt f(x) and g(t) are known function, R is an constant, how to compute the unknown Kernel K(x,t)? Thanks a lot
  41. T

    Solving Integral Equation U(t)=Int_0^inf K(x)dx

    How do you solve an integral equation of the form U(t) = \int_0^\infty U(t-x)K(x)dx ? where K(x) is a given function. One can guess a solution of the form U(t) = e^(a*t), but is there a method for obtaining the general solution? edit: for some reason the TeX looks like a black...
  42. M

    What type of integral equation is this and how can I solve it?

    I would ask help to solve the following integral equation. Any suggestion? Any references to look at? f(x)=int_{0}^{x} (A*exp{t-x}+B)f(t)g(t/x)*(1/x)dt A, B known constant Thanks a lot.
  43. T

    Solve Integral Equation | G(x) and F(x)

    Hello, Could you please help me to solve this equation: G(x)= INTEGRAL[F(t)dt] lower integral limit: x-m, upper integral limit: x, m is a constant, G(x) is a known function, F(x) is unknown and should be found. Thank you very much!
  44. M

    Discussing Differential & Integral Equations on this Forum

    This forum discuss Differential Equations - ODE, PDE, DDE, SDE, DAE I know the abbreviation ODE and PDE. But what are DDE and DAE ? SDE must stand for system of DE. Do you also discuss Integral Equation in this forum?
  45. C

    Problem Finding Integral equation for 2*Cos( a*x) *Cos(b/x)

    Hi I'm struggling to create an integral equation for Integral 2 * Cos( a*x) *Cos(b/x) dx Has anybody seen a potential solution for this. Mathematica online can solve each part individually however it is unable to create the integral for the entire equation. Thanks
  46. B

    Laplace transforms; Abel's integral equation

    Using Laplace transforms, find the solution of Abel's integral equation: \int^{x}_{0}\frac{f(u)}{\sqrt{x-u}}du = 1 + x + x^2 I recognized that the integral is a Laplace convolution, leading to: (f*g)(x) = 1+x+x^2 where g(x)=x^{-1/2} So: L(f*g)=L(1)+L(x)+L(x^2)...
  47. M

    Solve Integral Equation: Mathematica Function?

    Hi can anybody suggest my how to solve an integral equation of this type integral(a,b)( c * f(T) dT) ==d where a,b,c,d are known constants? Does Mathematica own a dedicated function for solving integral equations? Thanks
  48. F

    Solving Integral Equation for u(x)

    Homework Statement Solve for u(x): 0 = e^{2\int u(x) dx} + u(x) e^{\int u(x) dx} - a(x) Homework Equations The Attempt at a Solution I tried using the quadratic formula, e^{\int u(x) dx} = \frac{-u(x) \pm \sqrt{u^2(x) + 4a(x)}}{2} , converting to log notation and...
  49. P

    Integral Equation: Solving Methods & Resources

    Hi, I came across an integral equation of the Form x(t)=\int_\mathbb{R_+}{ds\, x(s)K(s,t)} wher K is some real function. (x is also real). Actually I will later need the higher dimensional case x(t_1,\dots,t_n)=\int_\mathbb{R_+^n}{ds_1,\dots,ds_n\...
  50. quasar987

    Existence of solution to integral equation

    [SOLVED] Existence of solution to integral equation Homework Statement There's k:[0,1]²-->R square integrable and the operator T from L²([0,1],R) to L²([0,1],R) defined by T(u)(x)=\int_{0}^{1}k(x,y)u(y)dy (a) Show that T is linear and continuous. (b) If ||k||_2 < 1, show that for any f in...
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