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Equation data
Category:5. Integral Equations
Subcategory:5.4. Linear Equations of the Second Kind with Constant Limits of Integration
Equation(s):$\displaystyle 
y(x)+\lambda\int^\infty_0tJ_\nu(xt)y(t)\,dt=f(x),\qquad \nu>-1$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution for $\lambda\ne\pm 1$: 
$$
y(x)=\frac{f(x)}{1-\lambda^2}-\frac{\lambda}{1-\lambda^2}
\int^\infty_0tJ_\nu(xt)f(t)\,dt.
$$


2. For the characteristic values $\lambda=\pm 1$ and $f(x)\equiv 0$, 
the integral equation has infinitely many linearly independent eigenfunctions.

Eigenfunctions for $\lambda=+1$ have the form
$$
y_+(x)=\varphi(x)-\int^\infty_0tJ_\nu(xt)\varphi(t)\,dt,
$$
where $\varphi=\varphi(x)$ is an arbitrary function.

Eigenfunctions for $\lambda=-1$ have the form
$$
y_-(x)=\varphi(x)+\int^\infty_0tJ_\nu(xt)\varphi(t)\,dt,
$$
where $\varphi=\varphi(x)$ is an arbitrary function.
Remarks:Here, $J_\nu(x)$ is the Bessel function.
Novelty:Material has been partially published elsewhere
References:A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Wed 01 Aug 2007 10:01
Edits by author:0

Edit (Only for author/contributor)


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