Articles of integral equations

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y’)^2 – y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ < $\pi$ weak minimum if ${\alpha}$ > $\pi$ strong minimum if ${\alpha}$ > $\pi$ Comparing the given functional to the standard form $J = \int_{a}^{b} F(x, y, y^{‘})$, we […]

Solve this Recursive Integral

How to solve this recursive integral? I do not even begin to understand how to solve this recursive integral. This doesn’t even seem possible? $$H{(x, y)} = \int_{t=0}^{t=2\pi} H(\frac{xt}{2},\frac{xt}{2})*2xt dt$$ This is intriguing me. What approaches exist for solving this? EDIT:$$H{(x)} = \int_{t=0}^{t=2\pi} H\big(\frac{xt}{2}\big)*2xt \, dt$$

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 $\lambda \neq \frac{4}{\pi -2}$; no solution for $\lambda \neq \frac{4}{\pi +2}$, but the corresponding homogeneous equation has a non-trivial solution; or no solution for $\lambda \neq \frac{4}{\pi -2}$, but the corresponding homogeneous equation […]

Volterra integral equation of second type

Solve the Volterra integral equation of second kind : $$ y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds $$ I know two methods for such integral equations: Picard’s method The method of finding the resolvent kernel and the Neumann series I tried using both of these methods but I couldn’t solve it. Which of the […]

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 $\lim_{x\to\infty} S(x)=0$?

Solution of the integral equation $y(x)+\int_{0}^{x}(x-s)y(s)ds=x^3/6$

So the question is to solve the integral equation to find out $y(x)$. $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\dfrac{x^3}{6}$$ So I find out the Resolvent kernel for $(x-s)$ and got $\sin{(x-s)}$, so I conclude that the solution is $$y(x)=\dfrac{1}{6}\int_{0}^{x}s^3\sin{(x-s)}ds$$ But the answer says $$y(x)=\int_{0}^{x}s\sin{(x-s)}ds$$ What did I miss?

Useful reformulation of Goldbach's conjecture?

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$’th prime. Substituting $ y=\exp(\lambda x)$ as a trail solution and factorizing. The differential equation must be simplified and factorized to: $$ \prod_{j=1}^\infty (\lambda-p_n^s) = 0$$ $$ \implies \prod_{j=1}^\infty (1-\frac{p_n^s}{\lambda}) = 0$$ […]

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such functions $f$. So far, I’ve managed to show that $\int_0^1\!{f(x)^t\, \mathrm{dx}}$ is constant for $t \geq 2$. Help would be appreciated.

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 $$f(x)=Ae^{bx}+B$$ where $A\ge 0,|B|\le A$ and $A,B$ are constant, and the positive number $b$ is such $\dfrac{b}{e^b-1}=a$ My try: let $$f(x)-a\int_x^{x+1}f(t)dt=C$$ then we have $$f'(x)-af(x+1)+af(x)=0,\forall x\in R$$ other idea: let […]

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?