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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1answer
318 views

Volterra integral equation with variable boundaries

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy$$ I'm also Trying to solve this integral equation like she does Solving an integral equation with a separable kernel. and I also have some doubts ...
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2answers
381 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?
7
votes
2answers
3k views

Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow ...
7
votes
1answer
246 views

Fredholm Equations

I have the following problem to solve $$\phi (x)=\lambda \int_{o}^{\pi }\cos(2x+y)\phi (y) dy+ \sin x$$ following the instructions from the following link early to conclude that: $$\phi (x)=\lambda ...
4
votes
2answers
1k views

Volterra integral equation of secong type solve using resolvent kernel

Solve the integral equation $$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I ...
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votes
2answers
345 views

Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
14
votes
2answers
437 views

if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant, show that $f(x)$ is constant; or ...
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4answers
319 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
13
votes
4answers
440 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
7
votes
2answers
520 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: ...
3
votes
2answers
235 views

Solve integral equation of second kind using Fredholm method

I need to solve this integral equation $$\phi (x)=(x^2-x^4)+ \lambda \int_{-1}^{1}(x^4+5x^3y)\phi (y)dy$$ Using the Fredholm theory of the intergalactic equations of second kind. I really don't ...
3
votes
1answer
615 views

Eigenvalues of an operator

I think this question isn't that hard, but I am a bit confused: Define $$(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$$ Then $A$ is an operator on functions. Find the eigenvalues and the ...
1
vote
1answer
2k views

How to find eigenfunctions of a linear operator (follow-up question)

This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ...
32
votes
2answers
2k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
6
votes
2answers
107 views

Does this condition on the sum of a function and its integral imply that the function goes to 0?

Consider a bounded real-valued function $S:\mathbf{R}\to\mathbf{R}$ so that $$\lim_{x\to\infty} \left( S(x) + \int_1^x \frac{S(t)}{t}dt\right)$$ exists and is finite. Can one say that ...
3
votes
1answer
560 views

How to find eigenfunctions of a linear operator

I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral. As a special case suppose I am interested in this function: ...
6
votes
2answers
232 views

Can we bound from above sub-solutions of Volterra integral equations?

Gronwall's lemma says the following. Assume that $v\in C^0([t_0, T])$ is a nonnegative function. If $u \in C^0([t_0, T])$ satisfies the integral inequality $$u(t) \le c + \int_{t_0}^t u(s)v(s)\, ...
3
votes
4answers
254 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
3
votes
1answer
310 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
3
votes
2answers
436 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 ...
3
votes
1answer
331 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
2
votes
0answers
206 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 ...
1
vote
1answer
55 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 ...
1
vote
1answer
55 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
1
vote
1answer
123 views

Differential equations, integral equations

Is there an analytical way of proving that if $\phi$ is a solution to \begin{equation} y(t)=e^{it}+a\int_{t}^{\infty}\sin (t-s)y(s)s^{-2}ds, \end{equation} then $\phi$ would be a solution to the ...
1
vote
1answer
893 views

Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
0
votes
1answer
164 views

Solving integral equation $a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$

I've got this nasty-looking integral equation involving taking two minimums: $$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$ where $\delta(\cdot)$ is the ...