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Diffstat (limited to 'content/know/concept/impulse-response')
| -rw-r--r-- | content/know/concept/impulse-response/index.pdc | 15 |
1 files changed, 13 insertions, 2 deletions
diff --git a/content/know/concept/impulse-response/index.pdc b/content/know/concept/impulse-response/index.pdc index fa921fa..f4c40a8 100644 --- a/content/know/concept/impulse-response/index.pdc +++ b/content/know/concept/impulse-response/index.pdc @@ -28,9 +28,9 @@ This can be used to find the response $u(t)$ of $\hat{L}$ to by simply taking the convolution with $u_p(t)$: $$\begin{aligned} + \hat{L} \{ u(t) \} = f(t) + \quad \implies \quad \boxed{ - \hat{L} \{ u(t) \} = f(t) - \quad \implies \quad u(t) = (f * u_p)(t) } \end{aligned}$$ @@ -68,6 +68,17 @@ $$\begin{aligned} </div> </div> +This is useful for solving initial value problems, +because any initial condition can be satisfied +due to the linearity of $\hat{L}$, +by choosing the initial values of the homogeneous solution $\hat{L}\{ u_h(t) \} = 0$ +such that the total solution $(f * u_p)(t) + u_h(t)$ +has the desired values. + +Meanwhile, for boundary value problems, +the related [fundamental solution](/know/concept/fundamental-solution/) +is preferable. + ## References |