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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37 views

Solution of a partial integro-differential equation

I want to find a solution for the following equation $$ \partial_y u(x,y) + \partial_x u(x,y) + \int_{0}^{x} r(x-x') u(x',y) dx' $$ $$ u(x,0)=0 \quad u(0,y)=\delta(y) $$ in $(x,y) \in ...
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1answer
45 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
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23 views

How did Poisson discover his integral formula?

I am quite curious about the history behind it. His derivation should be different from those on today's textbooks.
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2answers
50 views

Numerical solution of the Volterra equation with an exponential factor

Given : $$u(x)=x+2 \int_0^x e^{x-t}u(t)dt$$ Solve the Volterra Equation numerically using Trapezoidal Rule in $(0,5)$ choosing $n=8$ and compare with the exact values. The Exact Solution I ...
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0answers
70 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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0answers
42 views

Can my wrong derivation of the Gamma function be fixed?

I found the following simple but wrong derivation of the Gamma function: We start from the definition of the exponential function $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} \\ \Rightarrow 1 = ...
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1answer
56 views

Show that $\int_0^1 \phi^2(x)dx$ does not exist where $\phi(x)=x^{x-1}$

iI am currently studying integral equations from the book "Integral Equations" by Harry Hochstadt. In its second exercise (page $42$) it is asked to (Q.No $2$) show that $\displaystyle \int_0^1 ...
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56 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
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1answer
56 views

Solving the following Volterra Integral Equation?

How do I solve the following Volterra, non-homogeneous, $1st$ kind Integral Equation : $$ \dfrac{x^2}{2}=\int_0^x (1-x^2+t^2)u(t) dt$$ I know I cannot apply Laplace Transform because the kernel is ...
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0answers
18 views

Why is this restriction necessary?

I have to prove that if $\omega$, $\phi$ are 2 solutions to the equation $$\alpha(t)=\int_{0}^{t}f[s,\alpha(s)]ds$$ That then for $t\geq 0$ $$\phi(t) - \omega(t) = ...
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0answers
42 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
1answer
130 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
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1answer
114 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 ...
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0answers
45 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, ...
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1answer
47 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 ...
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1answer
101 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
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0answers
218 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
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0answers
73 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
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1answer
57 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 ...
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1answer
34 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 $f: [-k, k] \to \mathbb{R}$, $g(x)$ ...
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1answer
50 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 ...
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1answer
83 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 ...
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1answer
113 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 ...
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0answers
49 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 ...
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1answer
53 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 ...
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0answers
23 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 ?
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1answer
45 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 ...
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0answers
48 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
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1answer
42 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 $u(r)$ ...
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1answer
41 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
117 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
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0answers
98 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
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1answer
105 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
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0answers
56 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
70 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 ...
5
votes
4answers
84 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
28 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
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2answers
47 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
53 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 ...
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0answers
40 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 ...
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1answer
46 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 ...
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1answer
47 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
83 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
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1answer
75 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
68 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
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2answers
50 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
225 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
47 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 ...
1
vote
1answer
42 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
43 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 ...