0
votes
0answers
37 views

How to solve integral equations like this?

sorry for such a non-specific question and lack of research effort, but I'm new to integral equations and don't know where to start. How does one go about solving equations of the form ...
2
votes
0answers
53 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
1
vote
0answers
35 views

solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
0
votes
0answers
33 views

An integral equation of the first kind with variable limits

How should one solve the following integral equation to find $F(\tau)$: $$ \int_{\tau -T}^{\tau}F(t)X(\tau -t)dt=e^{-(\frac{\tau}{T})^2} $$
0
votes
0answers
47 views

Integral equation problem

I has worked on this problem for a while and still stucked on it. Hopefully someone give me a hint. Consider the following integral equation, find the function $g(r)$: $$\int_0^\infty {K(s,r)g(r)dr} ...
1
vote
1answer
43 views

Combination of Integration and Derivative

How to solve this equation: $$ \int_0^{\frac{\pi}{2}}\left(a\sin y-b\left(\frac{dy}{dx}\right)^2\right)~dy=c, $$ where $a$, $b$, and $c$ are constant. Thank you for your help.
1
vote
1answer
50 views

Integral equation question

If f(x) and f(t) both have the same domain and range, is there a general way to find $\int_{0}^{x^2} f(t) dt = f(x)$ given t? The actual problem tells that t = 9 and f(x) = $5 e \exp{x cos ...
0
votes
1answer
60 views

How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
0
votes
2answers
56 views

Question about integral is equal to zero

Suppose we have the following equation $$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$ where the function $g(x)$ does not depend on $r$, while $f(x,r)$ is function of ...
3
votes
1answer
54 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
2
votes
1answer
101 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
4
votes
1answer
113 views

Integral equation $u(t)=f(t)+a\int_0^t u(s)ds\quad t\geq 0$

Let $a\in\mathbb R$ and $f\colon [0,1]\to\mathbb R$ a continuous function. Solve the integral equation $$u(t)=f(t)+a\int_0^t u(s)\mathrm ds,\quad t\geq 0$$ and find an explicit formula for the ...
4
votes
2answers
79 views

Integration solving problem

A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$ The author said that, equation (2) can be written from equation (1) by ...
3
votes
2answers
357 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 ...
1
vote
1answer
104 views

How to solve integral equation $x(t)-\int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1.$

I was thinking about the problem that was as follows: The integral equation $x(t)-\displaystyle \int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1,$ has (a)no solution, (b)a ...
1
vote
1answer
168 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2) $$ -y''(x)+q(x)y(x)=0 $$ (3) $$ y(0)=0=y(\infty) $$ $ q(x) $ here is ...
3
votes
0answers
100 views

integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related

given two functios $ f(x) $ and $ g(x) $ related by $$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$ what ...
1
vote
2answers
72 views

homogeneous linear differential equation question

I was wondering if there is an analytical solution to the following homogeneous linear differential equation $$\dfrac {dM} {dt}=\dfrac {M} {\alpha \left( t\right) }e^{\beta\left( t\right) t}$$ which ...
1
vote
1answer
121 views

Comparison between solutions of ODE

Could anyone help on the following problem? Let R(t) be the solution to the integral equation: $R(t)=1+\int_{0}^{t}\frac{1}{R(s)}ds$, namely $R(t)=\sqrt{2t+1}$. Assume that X is continuous and ...