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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3
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1answer
251 views

Volterra integral equation with variable boundaries

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy$$ I'm also Trying to solve this integral equation like she does Solving an integral equation with a separable kernel. and I also have some doubts ...
4
votes
3answers
102 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 , $x \in\mathbb{R}$) $$f(x)=\sin ^2x+\int_{0}^{x}tf(t)dt$$
2
votes
0answers
85 views

applications integrals

I have the following problem to solve: $\int_{0}^{1}K(x,y)\phi(y)dy$ where: $K(x,y)=x(1-y), 0\leq x\leq y\leq 1$ and $K(x,y)=y(1-x), 0\leq y\leq x\leq 1$ already tried using the methods suggested ...
0
votes
1answer
78 views

Reducing an integral equation to a differential one

In my course about differential equations I have the following problem: Find all the functions $f:\mathbb{R} \longrightarrow \mathbb{R}^+$ such that the area below the graphic of the function in an ...
-2
votes
2answers
123 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
7
votes
1answer
215 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 ...
-4
votes
2answers
300 views

Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
1
vote
1answer
490 views

Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
3
votes
2answers
177 views

Solve integral equation of second kind using Fredholm method

I need to solve this integral equation $$\phi (x)=(x^2-x^4)+ \lambda \int_{-1}^{1}(x^4+5x^3y)\phi (y)dy$$ Using the Fredholm theory of the intergalactic equations of second kind. I really don't ...
2
votes
0answers
83 views

What is the meaning of the definition below? Taken from a 1909 book on Integral Equations.

Definition: We say that the discontinuities of a function of $(x, y)$ are regularly distributed in $S$ or in $T$ if they all lie on a finite number of curves with continuously turning tangents, no one ...
2
votes
1answer
141 views

A problem related to an integral equation

I am stuck on the following problem that is as follows: The integral equation $\quad \varphi(x)-\lambda \displaystyle\int_{-1}^{1}\cos[\pi(x-t)]\varphi(t) dt= g(x)$ has 1.a unique ...
3
votes
2answers
496 views

What is a hypersingular integral kernel?

While reading literature about boundary element and finite element methods I have repeatedly seen that some integral kernels are singular and others are hypersingular. Could you explain what is the ...
3
votes
1answer
184 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. ...
3
votes
1answer
2k views

How to solve partial integro-differential equation?

Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$: $\partial_t u = \partial_{xx} u + f(u,\lambda)$ $\lambda = B\left(u_0 - \int_{x=0}^L ...
1
vote
2answers
793 views

Solving an integral equations using fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
3
votes
3answers
221 views

Let $f(x)=\sin(x)-\int_{0}^{x}{(x-u)f(u)du}$ where $f(x)$ is continuous. Find $f(x)$.

Let $$f(x)=\sin(x)-\int_{0}^{x}{(x-u)f(u)du}$$ where $f(x)$ is continuous. Find $f(x)$. Initially, I use FTC and obtain $f(x)=\sin(x)$ but in the question didn't mention $f$ is differentiable. Then ...
3
votes
0answers
79 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
1
vote
2answers
165 views

solve the following integration equation (integral equation/variation of calculus)

I'm still rusty on integral equations. I need to solve the following $$ f(x) = e^{-{\left | x \right |}} + \lambda \int_{-\infty}^{\infty} e^{-{\left | x - y \right |}} f(y)dy $$ where $f(x)$ is ...
2
votes
1answer
91 views

Need help solving an Integral Equation

Need help solving: $$ f(x) = x + \lambda \int_{0}^{1}y(x+y)f(y)dy $$ keeping terms through $\lambda^{2}$, (a) by using the Fredholm method (b) by using the Neumann method
4
votes
2answers
86 views

Integration solving problem

A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$ The author said that, equation (2) can be written from equation (1) by ...
0
votes
1answer
39 views

a problem on integral equation having no eigen value

show that the integral equation $$\phi(x)- \lambda\int^{\pi}_0 \sin x \sin 2t\phi(t)\,dt=0 , 0 \leq x \leq \pi$$ has no eigenvalue. can anyone help how can I able to solve this problem ...
1
vote
2answers
327 views

Solution of an integral equation $\phi(x)+\int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $

Solve the following integral equation: $\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $ I need to solve the integral equation above. Can anyone help me please?
2
votes
0answers
49 views

What kind of numerical methods are best applicable to this?

I'm wondering: what would be the best numerical method for solving a nonlinear integral equation of the form $$f(x) = a(x) + \int_{-A}^{A} K(x, t, f(t)) dt$$ where $f$ is the unknown function, a ...
5
votes
1answer
228 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, ...
2
votes
2answers
176 views

Uniqueness of solution to an integral equation on the half line

The equation in question is $$f(x)=\int_0^\infty f(y)(x+y)e^{-x^2/2-xy}\text{d}y$$ where $f: [0,\infty)\rightarrow[0,\infty)$. It is not hard to see $f(x)=Ce^{-x^2/2}$ solves the equation. However, ...
2
votes
1answer
416 views

Fourier transform using the convolution theorem

The function $f(t)$ satisfies the integral equation $f(t)+2\int_{-\infty}^{\infty}H(s)e^{-s}f(t-s)ds=H(t)e^{-t}$ and decays as t $\rightarrow_{-\infty}^{\infty}$ By taking the Fourier transform of ...
1
vote
3answers
88 views

Solve $y'-\int_0^xy(t)dt=2$

I have not idea how to approach this differential equation. $$y'-\int_0^xy(t)dt=2$$. Basically, I did, $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$ I am stuck. Thank You.
1
vote
1answer
529 views

Writing equivalent first order differential equation and initial condition

I have another homework question that I'm struggling a bit to understand exactly what I'm asked to do. I understand what an initial condition is, but I'm not quite sure how I specify such a ...
1
vote
1answer
440 views

Linear birth death process, probability of extinction by time t

I have a linear birth death process with birth rates $\lambda n$ and death rates $\mu n$ . Let r(t) be the probability of extinction by time t. If there is 1 individual alive at time 0 explain why ...
2
votes
1answer
46 views

Trying to show that equation has a single solution using Banach space Theorems

How do I show that $f(x) = \int_0^1 e^{-sx}\cos(\alpha f(s))~ds, $ $0\leq x\leq1$, $0\le\alpha\le1$ has a single solution. Using Banach space Theorems like Contraction mapping theorem? Thanks for ...
2
votes
0answers
117 views

Fredholm integral equation of the first kind

Can we solve the following specific integral equation: $$ \int_0^1v^n(1-v)^{x-1}K(v)dv=f(x) ,x\in[0,1) $$ If it is solvable, I wonder whether its solution can be represented in a closed form.
1
vote
1answer
156 views

Solve the integral equation

$$y(x) = 2 + \int_8^x (t-ty(t))dt$$ I am having a very hard time doing this problem. (i) Solve the separable differential equation $$y'(x) = x − xy(x)$$ to get $$y(x) = 1 + c \cdot e^{−x^2/2}$$ (ii) ...
3
votes
0answers
117 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
5
votes
1answer
78 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 ...
0
votes
2answers
87 views

$f\colon \mathbb R \rightarrow \mathbb R$ is a continuous function and $f(x)=\int_0^xf(y)~dy.$

I faced the problem that says: If $f\colon \mathbb R \rightarrow \mathbb R$ is a continuous function and $f(x)=\int_0^xf(y)~dy.$ Then which of the following option is correct? $1.f(x)=e^x$ ...
-2
votes
1answer
87 views

Can a function be only defined graphically ??

are there functions that can be only defined graphycally ? for example the soultion of an integral equation $$ f(s)= \int_{0}^{\infty}dxK(s,x)f(x) $$ if i find numerically a graph of the function $ ...
2
votes
2answers
68 views

Integral expansion help!

So I'm very close to finishing a proof of the exponential function in terms of differential equations. For this next step, I have to show the following. For $n \ge 0$ define $E_n (t)$ recursively ...
6
votes
2answers
163 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 ...
2
votes
1answer
340 views

An inverse definite integral problem

I am seeking a function $f(x)$ that satisfies this condition: $\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
3
votes
2answers
393 views

Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

Integral equation $$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has: a unique solution for $\lambda \neq \frac{4}{\pi +2}$; a unique solution for $\lambda \neq ...
1
vote
2answers
85 views

Solving an equation with an integral

I need to solve the following equation for $v(x)$: $$\int_0^tv(x)(x+1)dx=f(t)$$ I am given the function $f(t)$. I've done this so far: If we derive both sides by $t$, we get $v(t)(t+1)=f'(t)$ and ...
0
votes
2answers
59 views

Solutions to $\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$

I have been looking at a problem requires the solution of an equation of the form: $$\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$$ for integer values of $m$ ...
6
votes
2answers
247 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 ...
0
votes
1answer
95 views

Symbolic manipulations of integral equations

I was trying to learn about solving integral equations using symbolic algorithms. After a quick web search, I mostly found items like this Mathematica journal article that mostly focuses on how to use ...
2
votes
1answer
128 views

Does anyone know this functional integral equation?

$$\sqrt{2}f(x) =\lim_{\delta \to 0^{+}}\left[x-i\delta-\int_{-1}^{1} \frac{|f(y)|^2}{y-i\delta-x}dy\right]$$ I'd like to know if there is a solution for $f\colon(-1,1) \to\mathbb{C}$. Of course if it ...
1
vote
1answer
108 views

How to solve integral equation $x(t)-\int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1.$

I was thinking about the problem that was as follows: The integral equation $x(t)-\displaystyle \int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1,$ has (a)no solution, (b)a ...
0
votes
0answers
320 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
1
vote
1answer
242 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2) $$ -y''(x)+q(x)y(x)=0 $$ (3) $$ y(0)=0=y(\infty) $$ $ q(x) $ here is ...
0
votes
3answers
120 views

Find a $f(x) \not=0$ satisfies $\int_1^{\infty}(1-\frac{1}{x})f(x)dx=0$

can we find a function $f(x)\not=0$,such that $$\int_1^{\infty}\left(1-\frac{1}{x}\right)f(x)dx=0$$ who can give an instance ? thanks
0
votes
1answer
505 views

A integral equation generated from current density distribution in a wire

Consider a wire carrying a current $I$, I need to find the current density distribution in the wire of a cylinder shape. Let the density function be $j(x,y)$, in the circle $D:x^2+y^2<r^2$. We ...