MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

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

Equation data
Category:5. Integral Equations
Subcategory:5.6. Nonlinear Equations with Constant Limits of Integration
Equation(s):$\displaystyle y(x)+\int^{\infty}_{-\infty}\bl[\lambda e^{-|x-t|}y(t)+\varphi(x)\Psi(t,y(t))]\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions for $\lambda>-\frac12$:
$$
y_m(x)=Y_f(x)+A_mY_\varphi(x),
$$
where
$$
Y_f(x)&=f(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)f(t)\,dt,
$$
$$
Y_\varphi(x)&=\varphi(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)\varphi(t)\,dt,
$$
and $A_m$ are roots of the
algebraic (transcendental) equation
$$
A+\int^b_a \Psi(t,Y_f(t)+AY_\varphi(t))\,dt=0.
$$
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Tue 03 Jul 2007 17:18
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin