Question

Find the resolvent kernel to solve the Volterra integral equation of the second kind:

U(x) = f(x) + λ ∫₀ˣ eˣ⁻ᵗ u(t) dt

Answer 1

The resolvent kernel for solving the Volterra integral equation of the second kind is given by
`K(x, t) = e^(x - t).`

Step-by-step explanation:

The resolvent kernel plays a crucial role in solving Volterra integral equations of the second kind. In this context, the resolvent kernel K(x, t) is the solution to the homogeneous equation associated with the given Volterra integral equation. For the provided equation:

`\[ U(x) = f(x) + λ \int_0^x e^(x - t) U(t) dt \]`

the resolvent kernel is K(x, t) = e^(x - t). This is obtained by considering the homogeneous part of the equation, setting f(x) to zero, and solving for U(x) with a kernel function dependent on both x and t.

In the integral equation, the resolvent kernel e^(x - t) represents the weight or influence of the past values of the solution on the current value at x. The exponential term reflects a decay with time, emphasizing the significance of more recent values. The resolvent kernel encapsulates the underlying dynamics of the system and facilitates the solution process by transforming the integral equation into a more manageable form. In practical terms, it allows for the expression of the solution in terms of the given function f(x) and the resolvent kernel, simplifying the solution procedure for Volterra integral equations of the second kind.