Math 2.0

Integral transforms, operational calculus Definition

Operational calculus

Builds on Laplace transformInverse Laplace transformConvolution on the half-lineLaplace transform of a derivativeConvolution theorem for the Laplace transformLinear differential equation of order n

Definition

Let EE be the space of locally integrable functions f ⁣:[0,∞)→ℂf \colon [0, \infty) \to \C of exponential order, that is, with ∣f(t)∣≤Meat\abs{f(t)} \le M e^{at} for some constants M,aM, a and almost all tt. The operational calculus based on the Laplace transform is the correspondence f↦F=ℒff \mapsto F = \mathcal{L} f, F(s)=∫0∞e−stf(t) dtF(s) = \int_0^\infty e^{-st} f(t) \, dt, used with the following rules.

  1. ℒ\mathcal{L} is linear and injective up to equality almost everywhere, so ff is recovered from FF by the inverse Laplace transform.
  2. If ff is continuous, f′f' exists except at finitely many points on each bounded interval, and f′∈Ef' \in E, then ℒ(f′)(s)=sF(s)−f(0)\mathcal{L}(f')(s) = s F(s) - f(0).
  3. The convolution on the half-line becomes a product: ℒ(f∗g)=ℒf⋅ℒg\mathcal{L}(f * g) = \mathcal{L} f \cdot \mathcal{L} g.

A linear problem for ff is solved by transforming it with these rules into an algebraic equation for FF, solving for FF, and recovering ff.

Remarks

Rule 2 is the Laplace transform of a derivative and rule 3 is the convolution theorem. With them, a linear differential equation with constant coefficients and given initial values becomes a linear equation for FF. Its coefficients are polynomials in ss. The name operational calculus also covers other frameworks with the same structure. One is the formal calculus of Heaviside; another is the Mikusiński calculus, where ss is an element of a field and no integral transform is used.

Example: y′′+y=0y'' + y = 0, y(0)=0y(0) = 0, y′(0)=1y'(0) = 1. Rule 2 twice gives s2Y−1+Y=0s^2 Y - 1 + Y = 0, so Y(s)=1/(s2+1)Y(s) = 1/(s^2 + 1), which is the transform of sin⁡t\sin t. Check: sin⁡′′+sin⁡=0\sin'' + \sin = 0, sin⁡0=0\sin 0 = 0, sin⁡′0=1\sin' 0 = 1.

Non-example: y′=t et2y' = t\, e^{t^2} fits the scheme only formally. Its solutions are et2/2+Ce^{t^2}/2 + C, which are not of exponential order, so they are outside EE and have no Laplace transform.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Convolution on the line, concept, no evidence levelConvolution on the li…conceptFunction, concept, no evidence levelFunctionconceptInverse Laplace transform, concept, no evidence levelInverse Laplace trans…conceptLaplace transform, concept, no evidence levelLaplace transformconceptLebesgue integral, concept, no evidence levelLebesgue integralconceptLinear differential equation of order n, concept, no evidence levelLinear differential e…conceptLaplace transform, definition, no evidence levelLaplace transformdefinitionConvolution on the half-line, definition, no evidence levelConvolution on the ha…definitionConvolution theorem for the Laplace transform, statement, no evidence levelConvolution theorem f…statementLaplace transform of a derivative, statement, no evidence levelLaplace transform of …statementOperational calculus, definition, no evidence levelOperational calculusdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Convolution on the line concept
  2. Function concept
  3. Inverse Laplace transform concept
  4. Laplace transform concept
  5. Lebesgue integral concept
  6. Linear differential equation of order n concept
  7. Laplace transform definition
  8. Convolution on the half-line definition
  9. Convolution theorem for the Laplace transform statement
  10. Laplace transform of a derivative statement
  11. Operational calculus definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Operational_calculus
About this page
Tag2C008K3V28
URIm2://2C008K3V28
Kinddefinition
WritingW0 stub
MSC44A45, 44A40, 44A10
DefinesOperational calculus
UsesLaplace transform, Inverse Laplace transform, Convolution on the half-line, Laplace transform of a derivative, Convolution theorem for the Laplace transform, Linear differential equation of order n

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