Integral equation Definition and 109 Threads

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...

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...

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...

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

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...

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?

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...

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...

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...

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...

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

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!

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[...

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...

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...

fx = ∫ exp(1-x^2)^.5 limits are b,-b

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...

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...

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...

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...

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

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)...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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

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...

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.

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!

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?

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

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)...

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

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...

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\...

[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...