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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1answer
56 views

Solution of differential equation by conversion to an integral equation [on hold]

Let us say we have the following initial value problem : $$ y''+g(t,y)=0$$ such that $y(0)=y_0$ And $y'(0)=z_0$ . And $g$ is a continuous function in the domain $D$ contains the point $(0,y_0)$. ...
0
votes
1answer
16 views

Evaluation of non-solvable/solvable x(?)

Here goes the question Assuming "x" is a real number,such that the equation is given $(x+\frac {1} {x})^2=3$,to evaluate $x^3+\frac {1} {x^3}$ And here goes my working,taking 2+ hours,only to ...
1
vote
1answer
25 views

Uniqueness of homogeneous Fredholm equation of the first kind

Suppose $K(x,t)$ is known and $$ \int f(x)K(x,t)dx=0 $$ Are there some known sufficient and \ or necessary conditions on $K(x,t)$ such that the only solution is $f(x)=0$ a.s.? ($f$ can be in a space ...
0
votes
0answers
9 views

How to solve for the prior probability distribution in this integral equation?

I've obtained the Bayesian posterior probability for a problem and found it to be equal to $$ z = \frac{\int_{0}^{1} p^{h'} p^{h} (1 - p)^t\Pr(p)\,dp}{\int_{0}^{1}\hspace{1.35em}p^h (1 - p)^t ...
4
votes
2answers
101 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 ...
2
votes
0answers
40 views

Solution of Abel type integral equation

I would like to know when (for what functions $f$) and how I can find integrable solution of equation \begin{align} f(x)=\int_x^{\infty}\frac{u(y)}{\sqrt{y-x}} \ dy, \end{align} where $u$ is unknown ...
1
vote
0answers
10 views

How to numerically solve an integral equation with a Cauchy principle kernel?

Consider such a Fredholm equation of $f(x)$: $$ f(x) = g(x) + \lim_{\epsilon \rightarrow 0^+ }\int_{-\infty}^{+\infty} \frac{d y V(x-y)}{a^2+ i \epsilon - y^2} f(y) .$$ Here $V(y)$ is a nice ...
2
votes
2answers
46 views

Find continuous $f$ with period $1$ such that $f(x) =\int_0^1 f(x-t)f(t) dt$

The problem is to find all $f : \mathbb{R} \to \mathbb{C}$ that is continuous and has a period of $1$ (not necessarily smallest period) such that the following equality holds: $$f(x) = \int_0^1 ...
1
vote
0answers
36 views

numerical analysis of partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
1
vote
1answer
43 views

Homogeneous Fredholm Integral Equation

I'm having problem obtaining the solution of the homogeneous Fredholm Integral Equation of the 2nd kind, with separable kernel. I always get a zero if I use the normal method i was taught for the ...
0
votes
1answer
38 views

Fredholm integral equations

I'm having problem obtaining the solution of the homogeneous Fredholm integral equation of the 2nd kind with a separable kernel. I always get a zero if I use the normal method I was taught for the non ...
1
vote
0answers
46 views

Association of PDE's with Integral Equations?

We know the following associations : Volterra Integral Equations $\leftrightarrow$ Initial Value Problems Fredholm Integral Equations $\leftrightarrow$ Boundary Value Problems My questions are : ...
1
vote
0answers
28 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 ...
1
vote
1answer
44 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
0
votes
0answers
21 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.
1
vote
2answers
45 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 ...
2
votes
0answers
42 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 ...
1
vote
0answers
36 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 = ...
0
votes
1answer
51 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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votes
0answers
46 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 ...
2
votes
1answer
35 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 ...
0
votes
0answers
17 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) = ...
2
votes
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
0answers
35 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
89 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
40 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
46 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
67 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
197 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
66 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
51 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
23 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
42 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 ...
0
votes
1answer
76 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
75 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
38 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
52 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
22 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 ?
1
vote
1answer
44 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
38 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
39 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
34 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
106 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
77 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
86 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
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
66 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
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
25 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
46 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 ...