An integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation. (Handbook of Mathematics - ...

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40
votes
2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
14
votes
2answers
455 views

if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant, show that $f(x)$ is constant; or ...
13
votes
3answers
704 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
13
votes
4answers
581 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
12
votes
0answers
1k views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
10
votes
2answers
4k views

Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow L^{...
9
votes
2answers
725 views

Solving for unknown functions

I am not a mathematician, so excuse if my question is silly or badly stated. I have the following problem. I have 2 conditions on two unknown continuously differentiable functions: $$A(t)=\frac{1}{B(...
9
votes
0answers
231 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
8
votes
2answers
3k views

Numerical solution of an integral equation

I have problems with a solution of an integral equation in MATLAB: all conditions are double-checked, but the answer is incorrect. Let me state the equation: $$ x(s) = g(s)+\int\limits_0^1x(t)f(t-s)\,...
7
votes
3answers
892 views

Prove there is a unique continuous function satisfying this integral equation

This is a question from an old real analysis qual: Prove that there is a unique continuous function $f:[0,1] \to \mathbb{R}$ such that $$f(x) = \cos x + \int_0^x f(y)e^{-y}dy$$ for $x \in [0,1]$ I ...
7
votes
2answers
309 views

A nonlinear “Fredholm” integral equation

Consider the integral equation \begin{eqnarray*} u \left( x \right) & = & \int_0^{\infty} u \left( t \right) u \left( \frac{x}{t} \right) \mathrm{d} t \end{eqnarray*} where the objective ...
7
votes
2answers
572 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: \...
7
votes
1answer
256 views

Fredholm Equations

I have the following problem to solve $$\phi (x)=\lambda \int_{o}^{\pi }\cos(2x+y)\phi (y) dy+ \sin x$$ following the instructions from the following link early to conclude that: $$\phi (x)=\lambda ...
7
votes
0answers
158 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) \...
6
votes
3answers
403 views

Continuous solutions of $f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy$ [duplicate]

Possible Duplicate: Eigenvalues of an operator Find all the functions $f \in C([0,\frac{\pi}{2}])$ which are solutions of $$ f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad \...
6
votes
2answers
116 views

Does this condition on the sum of a function and its integral imply that the function goes to 0?

Consider a bounded real-valued function $S:\mathbf{R}\to\mathbf{R}$ so that $$\lim_{x\to\infty} \left( S(x) + \int_1^x \frac{S(t)}{t}dt\right)$$ exists and is finite. Can one say that $\lim_{x\to\...
6
votes
2answers
263 views

Can we bound from above sub-solutions of Volterra integral equations? (Nonlinear Gronwall's Lemma)

Gronwall's lemma says the following. Assume that $v\in C^0([t_0, T])$ is a nonnegative function. If $u \in C^0([t_0, T])$ satisfies the integral inequality $$u(t) \le c + \int_{t_0}^t u(s)v(s)\, ds,\...
6
votes
1answer
323 views

Solving a homogeneous Fredholm equation of the 2nd kind whose kernel has simple poles in the domain of interest

Consider the Fredholm equation of the 2nd kind $$ f(s) = \lambda \int_{-\infty}^{\infty} f(s') \Big(\sum_{n=1}^{N} g_n(s) h_n(s') \Big) ds' , $$ with $f(s)$ an unknown function, $\lambda$ a constant, ...
6
votes
2answers
170 views

If $f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0$ and $f(0)=0$, then what is $f'(x)$?

$f\in C^{1}[0,\infty)$, $f(0)=0$ and $$ f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0 $$ then $f'(x)=$ ? I'v tried in the following ways. First, let $F(x)=\int_{0}^{x}f(t)dt$, then we are left to ...
6
votes
1answer
85 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
6
votes
1answer
206 views

Existence and Uniqueness of a solution

I'm stuck with this problem so hopefully somebody will help me :) Here you are: Let $K\in C([0,2])$ be positive, decreasing and such that $K(0)=1$. Prove that for every $h\in C([0,1])$ there exists a ...
6
votes
0answers
372 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
5
votes
4answers
90 views

Find all functions: $\left ( \int \frac{dx}{f(x)} \right )\left ( \int f(x)dx \right )=c$

Find all functions $f(x)$ so that: $$ \left ( \int \frac{dx}{f(x)} \right )\left ( \int f(x)dx \right )=c $$ where c is a constant. My attempt was to differentiate both sides but that appears to ...
5
votes
3answers
267 views

Integral equation with a constraint

I am stuck on the following problem: given the following Volterra integral inhomogeneous equation: $$\phi(x)=\exp(-x)+\lambda\int_0^x\frac{1}{x^2+t^2}\phi(t)dt$$ is it possible to solve it given the ...
5
votes
2answers
154 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for $...
5
votes
1answer
445 views

Integral equation solution (Fredholm, second type)

There is an equation $$ w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy $$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. ...
5
votes
1answer
109 views

How to solve following differential equation?

$$ \int \limits_{0}^{\infty}\sqrt{1 + y'^{2}(x)}dx = 2 \sqrt{x} + y \qquad (.1) $$ The solution is $$ 3y = x\sqrt{x} - 3\sqrt{x} . $$ I don't know how to solve this type of equations. Also I don't ...
5
votes
1answer
298 views

Find all continuous functions such that $f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$

From Spivak Find all continuous functions such that $$f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$$ $f(x)^2$ is clearly differentiable. I'd like to write $\dfrac{d}{dx}f(x)^2=2f(x)f'(x).$ ...
5
votes
1answer
129 views

Anomalous integral equation

I'm trying to solve the following equation: $$\int_0^{f(x)}f(t)dt=g(x)$$ Differentiating under integral I obtain: $$f[f(x)]\frac{d}{dx}f(x)=\frac{d}{dx}g(x)$$ I know the function $g(x)$. Is there a ...
5
votes
0answers
61 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
5
votes
0answers
118 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $...
5
votes
1answer
142 views

Existence of integral equation solution

I am trying to prove a differentiable solution in some open interval about the origin for the equation: $$u(x) + u(x)^2 + \int_0^x (1+\cos(x+u(y))) dy = 0$$ I have been trying to prove it as a ...
4
votes
2answers
79 views

Solve the integral equation $f(x) = x + \lambda \int_0^1 f(z)\,dz$

Find a closed-form solution for $f(x)$ in the following equation $$ f(x) = x + \lambda \int_0^1 f(z)\,dz $$ where $\lambda$ is a constant I tried integrating both sides from $0$ to $1$ but wasn't ...
4
votes
3answers
111 views

Find $f(x)=?$ functional equation

I would appreciate if somebody could help me with the following problem: Q: Find $f(x)$ ($f'(x)$: conti-function , $x \in\mathbb{R}$) $$f(x)=\sin ^2x+\int_{0}^{x}tf(t)dt$$
4
votes
1answer
270 views

If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$

If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$ solution:- $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$ $\int^x_0 f (t) dt =x+ \int^0_x t f (t) dt$ + $\int^...
4
votes
3answers
65 views

Fourier-like integral equation

(This question inspired by question A specific 1st order PDE which looks almost like a linear PDE.) Solve integral equation $$ g(x)=\int\limits_{-\infty}^\infty \rho(\omega)\left[e^{i\omega x} - \...
4
votes
2answers
40 views

Need solution to Volerra integro-diff equation

I need to solve a system of Volterra integro-diff equation of form $$ y(t) = x(t) - \int_{0}^{t} k(t-\tau) y'(\tau) \;\mathrm{d}\tau $$ where kernel is of form $$ k(t-\tau) = P(t)Q(\tau) $$ Is it ...
4
votes
2answers
2k views

How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)

I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\ $ in $\ L_2[0,1]\ $ space. And I want to understand how to solve it, not just obtain an answer.
4
votes
1answer
550 views

Finding eigenfunctions and eigenvalues

Let $K$ be the integral operator defined by $$ (Kf)(x)=\int_0^1 u(x)v(y)f(y) dy $$ for some continuous functions $u,v$ in the Hilbert space with inner product $\langle f,g \rangle = \int_0^1 f(x)^* ...
4
votes
1answer
156 views

Integral equation $u(t)=f(t)+a\int_0^t u(s)ds\quad t\geq 0$

Let $a\in\mathbb R$ and $f\colon [0,1]\to\mathbb R$ a continuous function. Solve the integral equation $$u(t)=f(t)+a\int_0^t u(s)\mathrm ds,\quad t\geq 0$$ and find an explicit formula for the ...
4
votes
1answer
313 views

Multiple Integral Equation

$$f(x) = 2a \int_{0}^{x}{f(t)\;dt} - \left(\frac{b^2}{2}\right)\int_{0}^{1}{|x-t|f(t)\;dt}$$ where $0<a<b$ My task is to solve for $f(x)$. I'm having difficulty solving this integral equation. ...
4
votes
2answers
2k views

Volterra integral equation of second type solve using resolvent kernel

Solve the integral equation $$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I did:...
4
votes
1answer
1k views

How to solve $t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$

I want to solve this equation. It reminds me something about Laplace transform. I am sure that I must use it order to solve it. $$t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$$ How to do it?
4
votes
1answer
78 views

Explicit solution for equation

The claim is that this equation has an explicit solution. $$\frac{\partial}{\partial t}c(x,t)=\frac{a}{\pi}\int_{\mathbb{R}}\frac{c(y,t)-c(x,t)}{(y-x)^2}dy.$$ What can one do to find this solution? ...
4
votes
1answer
473 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + \phi'(...
4
votes
3answers
252 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
4
votes
1answer
347 views

How can I solve this integral equation in terms of Hermite polynomials?

It must be proven that the solution of the integral equation $$f(x)=\int_{-\infty}^{+\infty} e^{-(x-t)^2} g(t)dt$$ is $$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$ ...
4
votes
2answers
70 views

Check whether a functional has an extremal or NOT

Find the extremal of the functional $$J(y)=\int_a^b F(x,y,y')\,dx$$where , $F(x,y,y')=y'+y$ , for admissible functions $y$. From Euler-Lagrange equation , $\displaystyle \frac{d}{dx}\left(\frac{\...
4
votes
1answer
232 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
4
votes
1answer
69 views

How to solve $xy=2\int_1^xy(t)dt+5$?

Could you please give me some hint how to solve this equation: $xy=2\int_1^xy(t)dt+5$. It is not known whether $y(x)$ is continuous or not, so I could not use Fundamental Theorem of Calculus for ...