The integral-equations tag has no wiki summary.
2
votes
1answer
42 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 ...
0
votes
0answers
38 views
+50
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 ...
2
votes
1answer
31 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
56 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 ...
0
votes
2answers
63 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 ...
0
votes
2answers
46 views
Proving an integral equation
for the following Question, i had to prove this :
that for every $$-1\le y \le 1 \\
\arcsin(y) + \arccos(y) = \frac{\pi}{2}$$
NOTE: this I've shown this using basic trigonometric id's
and ...
3
votes
3answers
94 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 ...
0
votes
0answers
25 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
67 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 ...
1
vote
1answer
50 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
0
votes
1answer
14 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 ...
2
votes
2answers
74 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?
1
vote
0answers
29 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 ...
1
vote
0answers
39 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
128 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
153 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
84 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
62 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 ...
0
votes
1answer
161 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
33 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 ...
0
votes
0answers
25 views
A question from Pipkin's Integral equations
I have the next question from Pipkin's:
question 9, page 10:
Find the values of $k$ for which the following equation has solutions that aren't identically zero. If $k\neq 0$, find representative ...
2
votes
0answers
35 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
68 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
111 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
$$
F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
5
votes
1answer
46 views
How to solve following differential equation?
$$
\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} .
$$
I don't know how to solve this type of equations. Also I don't ...
-2
votes
1answer
81 views
Can a function be only defined graphically ??
are there functions that can be only defined graphycally ?
for example the soultion of an integral equation
$$ f(s)= \int_{0}^{\infty}dxK(s,x)f(x) $$
if i find numerically a graph of the function $ ...
0
votes
0answers
31 views
Show a solution to $y(x)=g(x)+\int\limits_{0}^{x}k(x,t,y(t))dt$ exists under certain assumptions on $k(x,t,z)$ and $g(x)$.
I got this homework question that I am stuck on. Let $J = [0, a]$ (with $a > 0$ fixed). Let $g(x)$ be a function which is continuous at all $x \in J$ and let
$k(x, t, z)$ be a function which is ...
2
votes
2answers
66 views
Integral expansion help!
So I'm very close to finishing a proof of the exponential function in terms of differential equations. For this next step, I have to show the following.
For $n \ge 0$ define $E_n (t)$ recursively ...
6
votes
2answers
147 views
If $f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0$ and $f(0)=0$, then what is $f'(x)$?
$f\in C^{1}[0,\infty)$, $f(0)=0$ and
$$
f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0
$$
then $f'(x)=$ ?
I'v tried in the following ways. First, let $F(x)=\int_{0}^{x}f(t)dt$, then we are left to ...
2
votes
1answer
163 views
An inverse definite integral problem
I am seeking a function $f(x)$ that satisfies this condition:
$\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
0
votes
1answer
123 views
A Very Merry Math Problem [closed]
So what does this problem equate to? Props to the first math user to prove they know the answer, and have a merry time!
$${1\over 2 }{d(x)^2\over dx}\bigg)_0^m + \int(2re^{r^2})dr + \int(dy)$$
...
1
vote
2answers
196 views
Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$
Integral equation
$$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has:
a unique solution for $\lambda \neq \frac{4}{\pi +2}$;
a unique solution for $\lambda \neq ...
1
vote
2answers
54 views
Solving an equation with an integral
I need to solve the following equation for $v(x)$:
$$\int_0^tv(x)(x+1)dx=f(t)$$
I am given the function $f(t)$. I've done this so far:
If we derive both sides by $t$, we get $v(t)(t+1)=f'(t)$ and ...
0
votes
2answers
51 views
Solutions to $\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$
I have been looking at a problem requires the solution of an equation of the form:
$$\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$$
for integer values of $m$ ...
5
votes
2answers
177 views
A nonlinear “Fredholm” integral equation
Consider the integral equation
\begin{eqnarray*}
u \left( x \right) & = & \int_0^{\infty} u \left( t \right) u \left(
\frac{x}{t} \right) \mathrm{d} t
\end{eqnarray*}
where the objective ...
0
votes
1answer
53 views
Symbolic manipulations of integral equations
I was trying to learn about solving integral equations using symbolic algorithms. After a quick web search, I mostly found items like this Mathematica journal article that mostly focuses on how to use ...
2
votes
1answer
99 views
Does anyone know this functional integral equation?
$$\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]$$
I'd like to know if there is a solution for $f\colon(-1,1) \to\mathbb{C}$.
Of course if it ...
2
votes
2answers
58 views
Help in finding curve equation.
What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the ...
1
vote
1answer
83 views
How to solve integral equation $x(t)-\int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1.$
I was thinking about the problem that was as follows:
The integral equation $x(t)-\displaystyle \int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1,$ has
(a)no solution,
(b)a ...
0
votes
0answers
123 views
Relationship between integral equations and partial differential equations
In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem.
I've never ...
1
vote
0answers
63 views
Gelfand-Levitan-Marchenko equation
how can one solve the integral
$$ 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) $$
$ q(x) $ here is ...
0
votes
3answers
114 views
Find a $f(x) \not=0$ satisfies $\int_1^{\infty}(1-\frac{1}{x})f(x)dx=0$
can we find a function $f(x)\not=0$,such that
$$\int_1^{\infty}\left(1-\frac{1}{x}\right)f(x)dx=0$$
who can give an instance ?
thanks
0
votes
1answer
168 views
A integral equation generated from current density distribution in a wire
Consider a wire carrying a current $I$, I need to find the current density distribution in the wire of a cylinder shape. Let the density function be $j(x,y)$, in the circle $D:x^2+y^2<r^2$.
We ...
1
vote
0answers
50 views
Help solving this integral equation
I've got the following relation: For any $t_m \in (0, t_f)$,
$$I(k | t_0,t_f,x_0) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(k_m | t_0, t_m, x_0) I(k-k_m|t_m,t_f,x_m) dk_m dx_m $$
I want ...
0
votes
0answers
44 views
Solving x from$\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$?
I found something strange when I try to solve this equatiin of $x$:
$\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$,
where $t$ and $c_0$ are constants. $f(\tau)$ is a known polynomial ...
2
votes
1answer
131 views
Difficult integral equation with function
Suppose $f(x)\in C^1([0,1])$ and $f(0)=0$. Let
$$\phi(x)=
\begin{cases}
\int_0^x\frac{f(t)}{\sqrt{x-t}}dt &\quad\text{if}\quad x\in(0,1]\\
0&\quad\text{if}\quad x=0
\end{cases}
$$
(a) Prove ...
4
votes
1answer
79 views
Recurrence relation for a function with an integral of the function?
Pardon my lack of tex skills, but what is the recommended procedure in the following scenario:
$$g(f) = 1+\int_0^{1-f} g\left(\dfrac{f}{1-x}\right)\,dx$$
I am not sure how to proceed in such a ...
3
votes
1answer
439 views
How to solve Volterra's integral equation of second kind with numerical solution
The problem occurs to me when I tried to solve
\begin{align}E(x)=1+2(1-x)^2\int_{x}^{1}(1-t)E\left(\frac{x}{t}\right)dt\end{align}
with $E(1)=1$ and $\lim_{x\to 0^+}E(x) \to +\infty$.
I'd like to ...
2
votes
1answer
139 views
Numerical solution of fractional integro-diffrential equ. using collocation method?
problem comes from
"Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan"
...
2
votes
1answer
108 views
General question about solving equations involving a definite integral
Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...
