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The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:5. Integral Equations
Subcategory:5.1. Linear Equations of the First Kind with Variable Limit of Integration
Equation(s):$\displaystyle \int_0^x(x-t)^\nu J_\nu(\lambda(x-t))y(t)\,dt=f(x),\qquad {\rm Re}\,\nu>-\frac12$.
Solution(s),
Transformation(s),
Integral(s)
:
\eqnitem 1.
If $\nu=1$ and $f(0)=f'(0)=0$ then
$$
  y(x)=\frac{f'(x)}\lambda+\frac1\lambda\int_0^xJ_0(\lambda(x-t))\left(\frac{d^2}{dt^2}+\lambda^2\right)f(t)\,dt.
$$

\eqnitem 2.
If $\nu=n$ (number $n\ge 0$ is an integer) and $f(0)=f'(0)=\ldots=f^{(2n+1)}(0)=0$ then
$$
  y(x)=\frac{2^nn!}{(2n)!\lambda^n}\int_0^xJ_0(\lambda(x-t))\left(\frac{d^2}{dt^2}+\lambda^2\right)^{n+1}f(t)\,dt.
$$

\eqnitem 3.
If $\nu=n-1/2$ (number $n$ is a positive integer) and $f(0)=f'(0)=\ldots=f^{(2n-1)}(0)=0$ then
$$
  y(x)=\frac{\sqrt\pi}{(2\lambda)^{n-1/2}(n-1)!}\left(\frac{d^2}{dx^2}+\lambda^2\right)^nf(x).
$$

\eqnitem 4.
If $[{\rm Re}\,\nu+1/2]+1=m>1$ ($[{\rm Re}\,\nu+1/2]$ is the integer part of the number $\Re\nu+1/2$)
and $f(0)=f'(0)=\ldots=f^{(2m-1)}(0)=0$ then
$$
  y(x)=\frac{\pi(2\lambda)^{1-m}}{\Gamma(\nu+1/2)\Gamma(m-\nu-1/2)}\int_0^x(x-t)^{m-\nu-1}J_{m-\nu}(\lambda(x-t))
  \left(\frac{d^2}{dt^2}+\lambda^2\right)^mf(t)\,dt.
$$
Novelty:Material has been fully published elsewhere
References:Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 5, Inverse Laplace Transforms, Gordon & Breach, New York, 1992 (p. 471).
Author/Contributor's Details
Last name:Manzhirov
First name:Alexander
Country:Russia
City:Moscow
Statistic information
Submission date:Sat 14 Jul 2007 13:57
Edits by author:0

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