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The forced oscillator with $k \geq 0$ : unique global solution and the convolution (Green's function) formula

Statement

Let $m > 0$ , $c \geq 0$ , $k \geq 0$ , let $I \subseteq R$ be an interval with nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) and let $f : I \to R$ be continuous. Fix $t_{0} \in I$ and data $(x_{0}, v_{0}) \in R^{2}$ . Let $g : R \to R$ be the (unique) solution of the associated homogeneous model with time domain $R$ and zero forcing, with $g (0) = 0$ and $g^{'} (0) = 1 / m$ .

Then:

There is exactly one solution $x : I \to R$ of the forced model $m x^{''} + c x^{'} + k x = f$ with $x (t_{0}) = x_{0}$ and $x^{'} (t_{0}) = v_{0}$ . It is twice differentiable on $I$ , with derivatives at included endpoints taken relative to $I$ , and $x^{''}$ is continuous on $I$ in its relative topology.
If $x_{h} : R \to R$ is any homogeneous solution with $x_{h} (t_{0}) = x_{0}$ and $x_{h}^{'} (t_{0}) = v_{0}$ (it exists and is unique by The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data and The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0), then $x (t) = x_{h} (t) + \int_{t_{0}}^{t} g (t - s) f (s) d s (t \in I) .$

Remarks

Explicit kernel. For $k > 0$ and $c = 0$ one has $g (t) = \frac{\sin (ω_{0} t)}{m ω_{0}}$ with $ω_{0} = \sqrt{k / m}$ ; for $k > 0$ , $c > 0$ and $4 m k > c^{2}$ one has $g (t) = \frac{e^{- γ t} \sin (ω t)}{m ω}$ with $γ = c / (2 m)$ and $ω = \sqrt{k / m - γ^{2}}$ ; for $k = 0$ , $c > 0$ one has $g (t) = \frac{1 - e^{- c t / m}}{c}$ ; and for $k = c = 0$ one has $g (t) = t / m$ . These are the solutions of the homogeneous initial value problem displayed in The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data clauses 1-3 and The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0 clauses 1-2, read off at the data $(0, 1 / m)$ ; the overdamped case $Δ > 0$ has the two-exponential form $g (t) = \frac{e^{λ_{+} t} - e^{λ_{-} t}}{m (λ_{+} - λ_{-})}$ .
Scope. This is a statement about the linear model of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data only: $f$ is an arbitrary continuous real function, and no physical interpretation of $m, c, k$ or of the formula as a superposition of physical impulses is asserted. The name "Green's function" here denotes the kernel $g$ of the convolution formula, not a distribution-theoretic object.

Facts & Assumptions

Given: The parameters $m > 0$ , $c \geq 0$ , $k \geq 0$ , the interval $I$ with nonempty interior, the continuous $f : I \to R$ , the point $t_{0} \in I$ and the data $(x_{0}, v_{0})$ . The homogeneous kernel $g$ and member $x_{h}$ solve the associated all- $R$ homogeneous model; the forced model uses the given time domain $I$ . No extension of $f$ outside $I$ is used.

[F1]

The model and its phase-plane system are those of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data; $x$ solves the forced model on $I$ exactly when $t \mapsto (x (t), x^{'} (t))$ solves $z^{'} = G_{f} (t, z)$ on $I$ . [given]

[F2]

For $k > 0$ the homogeneous model has exactly one global solution for every initial datum, and its three explicit solution families are those of The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data clauses 1-3; in particular the solution $g$ with data $(0, 1 / m)$ exists, is unique and is defined on all of $R$ . [given]

[F3]

For $k = 0$ the homogeneous model has exactly the solution families of The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0 clauses 1-2, and every interval solution extends to a global member; in particular $g$ with data $(0, 1 / m)$ exists, is unique and is defined on all of $R$ . [given]

[F4]

Convolution differentiation (Differentiation of a variable-limit convolution integral with a C-one matrix kernel): if $I$ is an interval, $t_{0} \in I$ , $U$ is open with $I - I \subseteq U$ , $G : U \to M_{n} (R)$ is $C^{1}$ and $b : I \to R^{n}$ is continuous, then $t \mapsto \int_{t_{0}}^{t} G (t - s) b (s) d s$ is differentiable with derivative $G (0) b (t) + \int_{t_{0}}^{t} G^{'} (t - s) b (s) d s$ . Here $n = 1$ , $U = R$ , $G = g$ ; when $g$ is $C^{2}$ , the same application with $G = g^{'}$ is legitimate. [given]

[F5]

Homogeneous uniqueness from Variation of constants for $z^{'} = A z + b (t)$ via the matrix exponential: for a fixed $A \in M_{2} (R)$ and the globally continuous forcing $b_{0} (t) \equiv 0$ , the global solution of $z^{'} = A z + b_{0}$ with zero datum is $z \equiv 0$ , and every differentiable solution on an interval containing its initial time with that datum is its restriction. [given]

Proof

technique · constructive; build the solution as homogeneous solution plus convolution with the kernel $g$, verify the equation by two applications of the convolution lemma, and invoke the variation-of-constants uniqueness

1.1F2F3

The kernel $g$ exists, is unique and is defined on all of $R$ by [F2] (case $k > 0$ ) and [F3] (case $k = 0$ ), with $g (0) = 0$ and $g^{'} (0) = 1 / m$ . Since $g$ solves $m g^{''} + c g^{'} + k g = 0$ , we have $g^{''} = - (c g^{'} + k g) / m$ ; the right-hand side is continuous because $g$ and $g^{'}$ are, so $g^{''}$ is continuous and $g$ is of class $C^{2}$ on $R$ .

2.1step 1.1F4

Put $F (t) : = \int_{t_{0}}^{t} g (t - s) f (s) d s$ for $t \in I$ . Applying [F4] with kernel $g$ and $b : = f$ gives, at every interior point, $F^{'} (t) = g (0) f (t) + \int_{t_{0}}^{t} g^{'} (t - s) f (s) d s = \int_{t_{0}}^{t} g^{'} (t - s) f (s) d s,$ and applying [F4] once more with the $C^{1}$ kernel $g^{'}$ (legitimate by step 1.1) gives $F^{''} (t) = g^{'} (0) f (t) + \int_{t_{0}}^{t} g^{''} (t - s) f (s) d s = \frac{f (t)}{m} + \int_{t_{0}}^{t} g^{''} (t - s) f (s) d s .$ If $I$ includes an endpoint, the same split-difference-quotient argument as in the proof of Differentiation of a variable-limit convolution integral with a C-one matrix kernel applies with increments $h$ restricted to the side lying in $I$ ; it gives these formulas for the domain-relative endpoint derivatives, first for $F$ with kernel $g$ and then for $F^{'}$ with kernel $g^{'}$ . The displayed right-hand sides are continuous on $I$ , since $f, g, g^{'}, g^{''}$ are continuous and the integrals vary continuously on compact subintervals. In particular $F (t_{0}) = 0$ and $F^{'} (t_{0}) = 0$ .

3.1step 2.1F1F2F3construct

Let $x_{h}$ be a homogeneous solution with the data $(x_{0}, v_{0})$ (existing by [F2], [F3]) and set $x : = x_{h} + F$ on $I$ . [construct] Then $x$ is twice differentiable on $I$ with continuous second derivative, using domain-relative derivatives at included endpoints, and, by step 2.1 and the linearity of the differential expression, $m x^{''} + c x^{'} + k x = (m x_{h}^{''} + c x_{h}^{'} + k x_{h}) + m F^{''} + c F^{'} + k F = f (t) + \int_{t_{0}}^{t} (m g^{''} + c g^{'} + k g) (t - s) f (s) d s = f (t),$ because the bracket vanishes identically by step 1.1. Moreover $x (t_{0}) = x_{h} (t_{0}) + F (t_{0}) = x_{0}$ and $x^{'} (t_{0}) = x_{h}^{'} (t_{0}) + F^{'} (t_{0}) = v_{0}$ .

4.1step 3.1F1F5discharge-construct: the constructed convolution is the unique solution∎

Uniqueness: let $\tilde{x} : I \to R$ be any other solution with the prescribed data and put $u : = \tilde{x} - (x_{h} + F)$ , where $x_{h} + F$ is the constructed solution of step 3.1. By linearity, $u$ solves the homogeneous equation on $I$ with $u (t_{0}) = u^{'} (t_{0}) = 0$ . Its phase curve $U (t) : = (u (t), u^{'} (t))$ therefore solves $U^{'} = A U$ on $I$ with $U (t_{0}) = 0$ , where $A = (\begin{array}{cc} 0 & 1 \\ - k / m & - c / m \end{array})$ . Apply [F5] with the globally continuous forcing $b_{0} \equiv 0$ : the unique global solution from zero data is zero, and clause 3 of the matrix variation-of-constants result makes every interval solution with the same data its restriction. Thus $U = 0$ and $\tilde{x} = x_{h} + F$ . This proves uniqueness using only a forcing defined on its actual domain $I$ for the constructed solution.

The forced oscillator with $k \geq 0$ : unique global solution and the convolution (Green's function) formula

Statement

Remarks

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The forced oscillator with k≥0k≥0: unique global solution and the convolution (Green's function) formula

Statement

Remarks

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The forced oscillator with $k \geq 0$ : unique global solution and the convolution (Green's function) formula