Tagged Questions

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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 $h$ ...
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Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
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Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$|f(t,x)|\leq \phi(t)|x|.$$ Suppose ...
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Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0$$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
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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 ...
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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 ...
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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 ...
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 ...
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 ...