integral equation

An integral equation involves an unknown function under the integral sign. Most common of them is a linear integral equation

$\displaystyle \alpha(t)\,y(t)+\!\int_a^bk(t,\,x)\,y(x)\,dx = f(t),$ (1)

where $ \alpha,\,k,\,f$ are given functions. The function $ t \mapsto y(t)$ is to be solved.

Any linear integral equation is equivalent to a linear differential equation; e.g. the equation $ \displaystyle y(t)\!+\!\int_0^t(2t-2x-3)\,y(x)\,dx = 1+t-4\sin{t}$ to the equation $ y''(t)-3y'(t)+2y(t) = 4\sin{t}$ with the initial conditions $ y(0) = 1$ and $ y'(0) = 0$ .

The equation (1) is of

If both limits of integration in (1) are constant, (1) is a Fredholm equation, if one limit is variable, one has a Volterra equation. In the case that $ f(t) \equiv 0$ , the linear integral equation is homogeneous.

Example. Solve the Volterra equation $ \displaystyle y(t)\!+\!\int_0^t(t\!-\!x)\,y(x)\,dx = 1$ by using Laplace transform.

Using the convolution, the equation may be written $ y(t)+t*y(t) = 1$ . Applying to this the Laplace transform, one obtains $ \displaystyle Y(s)+\frac{1}{s^2}Y(s) = \frac{1}{s}$ , whence $ \displaystyle Y(s) = \frac{s}{s^2+1}$ . This corresponds the function $ y(t) = \cos{t}$ , which is the solution.

Solutions on some integral equations in EqWorld.



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