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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2
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38 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
2
votes
0answers
20 views

Check whether the extremal has weak minima or weak maxima

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its weak maximum on all its extremals weak minimum on all its extremals weak maximum on some, but not on all of its extremals weak ...
3
votes
1answer
48 views

Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
0
votes
0answers
35 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
0
votes
1answer
43 views

Solving an equation by Laplace transform

Consider the following equation: $$ y^{\prime\prime}(x) +x = \int _0 ^x (x-u)y(u)du \qquad y(0)=0 \quad y^{\prime}(0)=1$$ I solved it by Laplace transform and got $-\sinh x$ as a solution. It is ...
2
votes
1answer
47 views

Fredholm integral equation [closed]

How can I solve the following fredholm integral equation $$ψ(x)=x+λ\int_{0}^{2π}|x-t|ψ(t)dt$$ The kernel contains absolute value
2
votes
0answers
184 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 ...
3
votes
0answers
55 views

Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is ...
3
votes
1answer
49 views

Integral $\int_\tau^\infty e^{\frac{-g_m}{\bar\gamma_m}}\frac{dg_m}{1+Pg_m}$ [closed]

$$I=\int_\tau^\infty e^{\frac{-g_m}{\bar\gamma_m}}\frac{dg_m}{1+Pg_m}$$ As you know exponential integral define in [0 inf], but I want to calculate it in [thu inf]. I'm really appreciating everyone ...
0
votes
1answer
16 views

Solution to Fredholm equation of the second type with symmetric Gaussian kernal

Is something known about the solution to Fredholm equations of the 2nd type of the following form: $\displaystyle f(x) = g(x) + \int_{-k}^k f(y) h(x-y) dy$ where $f: [-k, k] \to \mathbb{R}$, $g(x)$ ...
1
vote
1answer
39 views

solution of a volterra equation

I have to find solution of followong volterra equation $y(x)= x - \int_{0}^{x}(x-t) y(t) dt$ with $y(0) = 0$. My attempt: I differentiated the above and got $y^\prime = 1 - \int_{0}^{x} y(t) dt ...
-1
votes
1answer
70 views

Eigenvalue problem in functional analysis?

How can I find the eigenvalues and eigenvectors of \begin{align} Ay(p):=\int_{0}^{\infty} k^2 \cos(pk)y(k)dk \end{align} $A$ is a Hilbert-Schmidt operator. Well actually, i came across this in ...
1
vote
1answer
61 views

How we can show integral equation has a unique solution and how we can solve it?

My question is on the title, e.g. how we can show that this integral equation has a unique solution and how we can estimate what the approximate answer near $0.1$ can be? $$ x(t) + 0.1 \int_0^1 ...
0
votes
0answers
25 views

Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
0
votes
1answer
50 views

Integral Equation and Fourier Analysis

I am trying to solve the following equation \begin{align*} F(\omega) G(\omega)= 2 \pi \delta(\omega)-2\pi \delta^{(2)}(\omega) \end{align*} where $F(\omega)$ and $G(\omega)$ are Fourier transforms ...
0
votes
0answers
20 views

Solution of an integral equation

I want to know the minimum condition under which the following integral equation has a solution $$x(t) = x(0) + \int_0^tF(x(t),t)$$ has a solution. Is measurablity of $F$ enough ?
0
votes
0answers
21 views

Solution of a Volerra type equation

If $y(t)=1+\int_0^{t} y(v)e^{-(t+v)}dv$ then $y(1)=$ (a) 0 (b) 1 (c) 2 (d) 3 It is a Volterra equation. To solve it we apply successive approximation method or Resolvent kernel method. But we ...
1
vote
1answer
42 views

Solving $ y(x)-2-\int_0^x e^{y(t)-t} \, dt = 0 $

So, I'm trying to solve the following differential equation methodically: $$ y(x)-2-\int_0^x e^{y(t)-t} \, dt = 0 $$ I rearranged the equation a bit and differentiated both sides and got: $$ e^x ...
1
vote
0answers
29 views

Properties of solutions of system of integral equation.

Assume $g:[0,\infty) \to \mathbb R$ to be continuous and $$\int_{0}^{\infty} s|g(s)| \,\mathbb ds< \infty .$$ I want to find $\alpha>0$ such that the system of integral equations ...
2
votes
1answer
33 views

Uniqueness of a positive solution to an integral equation

This is a followup to another question I asked recently. This is a slight modification to that question. In the fluid mechanics of pipe flow, it is sometimes stated that the velocity profile $u(r)$ ...
1
vote
1answer
23 views

Proof of the uniqueness of the solution to an integral equation

In the fluid mechanics of pipe flow, it is sometimes stated that the velocity profile $u(r)$ which corresponds to a kinetic energy coefficient of 1 is always uniform, so $u(r) =$ some constant. ...
3
votes
2answers
85 views

Solving recursive integral equation from Markov transition probability

How do I solve something like: $$f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty e^{\frac{-(y - x/2)^2}{2}}f(y)\:\mathrm{d}y$$ for $f(x)$? Is there also a general formula that this falls under? ...
5
votes
0answers
67 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, ...
2
votes
1answer
77 views

Solve Integral equation with convolution

I have to solve the following integral equation \begin{align*} \int_{-\infty}^\infty e^{-y^2} \log \left( \int_{-\infty}^\infty e^{-(y-x-t)^2} f(t) dt\right) dy=-cx^2 \end{align*} where $c$ is some ...
1
vote
0answers
52 views

How would I find this constant?

I have this equation, and I'm not sure how to solve for the constant $\nu$, since everything else is known: $$\begin{equation} a + \sqrt{a_i + 4 b_i \nu} + \sum^N_{j=1} (\sqrt{a_j + 4 b_j \nu}) ...
2
votes
2answers
55 views

Solving the integral equation $y(x) = 3 + 2\int_1^x t y(t) dt $ by reducing it to a differential equation

Solve the integral equation $$y(x) = 3 + 2\int_1^x t \ y(t) \ dt $$ First I solved for the integral equation. Then I'm told to differentiate and I get $${dy \over dx} = 2 x y(x) $$ Then I ...
6
votes
4answers
80 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 ...
0
votes
0answers
15 views

Multiple integral of iterated kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...
3
votes
2answers
42 views

Relation on $\int_1^x\exp{t^2}dt$

Could you give me some leads to show the following relation : $$\forall x>0, \int_1^x\exp{t^2}dt = \frac{1}{2x}\exp{x^2} + \frac{1}{4x^3}\exp{x^2} - \frac{3}{4}\mathbb{e}+ \frac{3}{4} \int_1^x ...
2
votes
1answer
45 views

Integral form of this IVP

How do I show that the following initial value problem $$ xu''+u'+xu=0,\quad u(0)=1,\quad u'(0)=0 $$ has the following integral form: $$ u(x)=1+\int_{0}^{x} t\ln(t/x)u(t)\,dt $$ I am stuck because if ...
0
votes
0answers
33 views

Some linear integral equation

Please help me with the following problem: Let $\gamma\in (0,1)$ and $a<0<b$, $-a<b$, and $x\geq0$. Solve the following equation $$f(x)=\frac{\gamma}{b-a}\int_{\max(a+x,0)}^{b+x}f(y)dy$$ I ...
0
votes
1answer
40 views

Books for these topics.

I have an lecturership exam in India and in the syllabus there are few topics under the tags "Calculus of variations" and "Linear integral equations", and if please if someone could tell me which ...
1
vote
1answer
39 views

Solution of integral equation

If $x$ is a real-valued, differentiable function of $t$, what is, and how do I find the solution of $$\int_a^b x(t) \frac{dx(t)}{dt} dt$$
2
votes
1answer
77 views

Solve $\int_0^T f(t) dt =1$ for T.

I have to solve this equation for a physics problem and I don't know where to start: $$\int_0^T f(t) dt =1 \quad\text{and}\quad f(T)=C$$ Where $T>0$, $C>0$ and $f(t)>0$ we can suppose that ...
2
votes
1answer
71 views

Help me identify these sorts of equations

$$\int^x_0f(t)\,dt = xe^{2x}+\int^x_0e^{-t}f(t)\,dt$$ Assume $f$ is continous, solve for $f$. NB! I'm in my first calculus course so nothing too advanced please. While searching for a name for ...
3
votes
0answers
57 views

Integral equation $f(x) - \int_0^x f(t)dt = 0$

I'd like to know the solution (if solvable) to the following integral equation $$f(x) - \int_0^x f(t)dt = 0$$ Also I'd like to know what is the required mathematical background to be able to find a ...
1
vote
2answers
43 views

How to solve the integral equation?

How to solve the integral equation $$ \int_{-20}^{x} \left| \left| \left| \left| \left| \left| \left| \left| t \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 ...
4
votes
1answer
219 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 _+ ...
1
vote
1answer
44 views

$A$ is a symmetric operator ? Please criticize my proof.

Let $A:L^2([0,1])\to L^2([0,1])$ given by $$ Af(t)=\int_0^1K(s,t)f(s)ds, $$ where $K$ is a mensurable square integrable operator, i.e $\int_0^1\int_0^1|K(s,t)|^2\,dsdt<\infty$. $A$ is acompact ...
0
votes
0answers
47 views

Rayleigh-Ritz method to solve the P.D.E.

How we take an approximate solution from boundary conditions to find the solution of a partial differential equation by Rayleigh-Ritz method?
1
vote
1answer
32 views

Prove a certain integral expression of Bessel type for the Bessel function of the first kind

I know that $$ \frac{1}{2\pi}\int_0^{2\pi}e^{i\,z\,\cos\theta}d\theta=J_0(z) $$ where $J_n(z)$ denotes the Bessel function of the first kind of integral order. My question is - how do I show that ...
1
vote
0answers
38 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...
0
votes
0answers
18 views

$\text{Im} \lbrace\iint_{-\infty}^{\infty}f(x,y)g^\ast(x)g(y)\,dx\,dy \rbrace=0 \Rightarrow f(x,y)=a(x)\delta(y-x)$?

Suppose $f(z_1,z_2):\mathbb{R}^2\rightarrow\mathbb{C}$, $g(z):\mathbb{R}\rightarrow\mathbb{C}$, $a(z):\mathbb{R}\rightarrow\mathbb{R}$. Suppose also that $$\text{Im} \left ...
1
vote
0answers
139 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
0
votes
0answers
15 views

What is this form of integral equation called and what are the classes of solutions?

This is probably a quick one for someone familiar with integral equations: I have an equation of the form $\int_{a}^{b}du \, g(u,v) \, f(u)=c$ with $a$, $b$, and $c$ constant; and $f(u)$ a known ...
0
votes
0answers
14 views

uniqueness of solution for a type of integral equations

I have an integral equation that goes $f(x)=G(\int k(x,y)f(y)dy)$ where $x$ and $y$ are real numbers, $k(x,y)>0$, $G(\cdot)\in [0,1], G'(\cdot )<0$ I'm wondering can we say anything about ...
2
votes
1answer
61 views

A question about $f(x)\equiv 0$

If $f(x) \in C(-\infty+\infty)$, $\;g(x)=f(x) \int_0^x f(t)\,dt\,$ and $\;g(x)$ is monotone-decreasing in $(-\infty,+\infty),$ Prove:$f(x)\equiv 0$. It is easy to get $g(0)=0$,and I'm thinking about ...
-1
votes
1answer
45 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $? [closed]

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?
0
votes
0answers
51 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
27 views

Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel [duplicate]

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...