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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51 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. ...
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2answers
145 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? ...
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0answers
118 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, ...
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1answer
121 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 ...
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57 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}) ...
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2answers
78 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 ...
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4answers
89 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 ...
3
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2answers
50 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
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1answer
61 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
45 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
53 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
50 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 $$\int_a^b x(t) \frac{dx(t)}{dt} dt$$
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1answer
90 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
78 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 ...
4
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0answers
83 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 ...
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2answers
55 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
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1answer
232 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 _+ ...
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1answer
50 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 ...
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1answer
68 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 ...
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0answers
45 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 ...
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0answers
149 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 ...
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0answers
23 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 ...
2
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1answer
68 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 ...
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1answer
53 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) $ ?
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55 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 ...
3
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2answers
63 views

Finding function $f(x)$

How do we find the function(s) $f(x)$ given that $$f(x)=\int_{0}^{x} te^tf(x-t) \ \mathrm{d}t$$ My Try : I first used the property $\int_{0}^{a}g(x) \ \mathrm{d}x=\int_{0}^{a}g(a-x) \ \mathrm{d}x$ ...
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1answer
34 views

Is it possible to find a solution to this integral equation?

I have an integral equation of the following form: $y(t)=\lambda x(t) + x(t)\int_{-\infty}^{\infty}K(t,s)x(s)ds$ I haven't been able to find any discussion online of integral equations with the ...
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0answers
39 views

Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
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1answer
36 views

How to find the solution of integer equation group

I have the following problem: to find the general item of the following equation: let $a_1=b_1=1$, $$a_{n+1}=6a_n+2b_n, b_{n+1}=3a_n+2b_n$$ for any $n\geq 1$, find $a_n=?, b_n=?$
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0answers
54 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
4
votes
1answer
69 views

How to solve $xy=2\int_1^xy(t)dt+5$?

Could you please give me some hint how to solve this equation: $xy=2\int_1^xy(t)dt+5$. It is not known whether $y(x)$ is continuous or not, so I could not use Fundamental Theorem of Calculus for ...
2
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0answers
76 views

FM signals and non-trivial solutions to a homogeneous Fredholm integral equation of the first kind

I am looking for any non-trivial solution to the following integral equation. That is, find any function $q(\theta)\neq 0$ which satisfies the following equation: $$\int_0^a ...
3
votes
2answers
100 views

Volterra equation for a Bessel type IVP that appears in inverse scattering

I have the following i.v.p. (Colton Kress-Inverse acoustic and electromagnetic scattering theory, Springer) $$y''(r)+(k^2n(r)-\frac{l(l+1)}{r^2})y(r)=0$$ $$y(0)=0, y'(0)=1$$ using the Liouville ...
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0answers
241 views

Integral equation involving magnitude/modulus squared

I wish to solve the following integral equation that has popped up in my studies of focused light. If you notice, it looks almost like a homogenous Fredholm integral equation of the first kind. ...
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1answer
35 views

Define all functions using the main statement

Define all functions that are continious and fullfill the equation $$ f(x) = -1 + \int_0^{x^2} \frac{(f(\sqrt{t})^2 \sin t}{\cos^2t} dt$$ I'm completely lost on this one. I think that you should ...
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2answers
163 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
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1answer
148 views

Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
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66 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
2
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0answers
73 views

Reciprocal Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...
5
votes
1answer
139 views

Existence of integral equation solution

I am trying to prove a differentiable solution in some open interval about the origin for the equation: $$u(x) + u(x)^2 + \int_0^x (1+\cos(x+u(y))) dy = 0$$ I have been trying to prove it as a ...
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2answers
128 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
2
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1answer
35 views

Solution sets/ existence and uniqueness of solutions to $Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x)$

Given $$ Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x) $$ A) For what values of $\lambda$ does there exist a unique solution for all $f\in L^2(0,1)$? B) Find the solution set ...
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86 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
2
votes
2answers
49 views

Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
2
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1answer
54 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
3
votes
0answers
87 views

Contraction mapping principle application

I'm to prove that the following equation has a unique solution: $$f(x) = \int_0^1 e^{-sx} \cos(\alpha f(s)) ds.$$ (Here, $\alpha \in (0,1)$.) The form of the exercise screams to apply the ...
0
votes
1answer
55 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
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1answer
83 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
2
votes
2answers
70 views

Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
0
votes
0answers
63 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?