Laplace transform method for convolution equations

Builds on Volterra equation of convolution type Laplace transform Convolution theorem for the Laplace transform Inverse Laplace transform

Definition

Let $k$ and $f$ be locally integrable functions on $[0, \infty)$ whose Laplace transforms $\hat k(p) = \int_0^\infty e^{-pt} k(t) \, dt$ and $\hat f(p)$ converge absolutely for $\operatorname{Re} p > c$ , for some real $c$ . Consider the Volterra equation of convolution type $x(t) = f(t) + \int_0^t k(t - s) \, x(s) \, ds, \quad t \ge 0.$ The Laplace transform method for this equation consists of the following steps.

Assume that the solution $x$ has a Laplace transform $\hat x$ , and apply the convolution theorem to get $\hat x(p) = \hat f(p) + \hat k(p) \, \hat x(p)$ .
Solve: $\hat x(p) = \dfrac{\hat f(p)}{1 - \hat k(p)}$ for $\operatorname{Re} p$ large enough that $\hat k(p) \ne 1$ .
Recover $x$ as the inverse Laplace transform of $\hat x$ , and check that it solves the equation.

For an equation of the first kind $\int_0^t k(t - s) \, x(s) \, ds = f(t)$ , step 2 becomes $\hat x = \hat f / \hat k$ .

Remarks

Example: $x(t) = t - \int_0^t (t - s) \, x(s) \, ds$ , with $k(u) = -u$ and $f(t) = t$ . Here $\hat k(p) = -1/p^2$ and $\hat f(p) = 1/p^2$ , so $\hat x(p) = \frac{1/p^2}{1 + 1/p^2} = \frac{1}{p^2 + 1}$ and $x = \sin$ . Check: $\int_0^t (t - s) \sin s \, ds = t - \sin t$ , so the right side is $\sin t$ . Non-example: the method does not apply directly to $x(t) = 1 + \int_0^t ts \, x(s) \, ds$ , since the kernel $ts$ is not a function of $t - s$ , and the transform of the integral term is not a product. The step 1 assumption is harmless when $k$ and $f$ are as above: the unique locally integrable solution then has a Laplace transform for $\operatorname{Re} p$ large. The function $r$ with $\hat r = \hat k / (1 - \hat k)$ is the resolvent kernel, and $x = f + r * f$ .

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Function concept
Inverse Laplace transform concept
Laplace transform concept
Volterra equation of convolution type concept
Laplace transform definition
Convolution theorem for the Laplace transform statement
Laplace transform method for convolution equations definition

Tag	`XQTHXW4F1R`
URI	`m2://XQTHXW4F1R`
Kind	definition
Writing	W0 stub
MSC	45D05, 45E10, 44A10
Defines	Laplace transform method for convolution equations
Uses	Volterra equation of convolution type, Laplace transform, Convolution theorem for the Laplace transform, Inverse Laplace transform

Laplace transform method for convolution equations

Definition

Remarks

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