- #1
reterty
- 30
- 2
There is the following linear Volterra equation of the second kind
$$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$
with kernel
$$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$
where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I need a help in the solution (numerical or even analytical) of this equation.
I tried to solve this equation numerically using the trapezoidal method. A surprising result emerged: for $\beta>0.43$, the solution decreases monotonically for all positive values of the argument. If $\beta<0.43$, however, the function decreases but has a finite number of oscillations (maybe a numerical glitch?)
$$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$
with kernel
$$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$
where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I need a help in the solution (numerical or even analytical) of this equation.
I tried to solve this equation numerically using the trapezoidal method. A surprising result emerged: for $\beta>0.43$, the solution decreases monotonically for all positive values of the argument. If $\beta<0.43$, however, the function decreases but has a finite number of oscillations (maybe a numerical glitch?)