Convolution - differential-equations

Definition of Convolution

Mathematical Expression

Convolution: integral operation combining two functions to produce a third. Defined as:

(f * g)(t) = ∫₀ᵗ f(τ) g(t - τ) dτ

Domain: functions defined on [0, ∞) or ℝ depending on context.

Intuitive Meaning

Measures overlap between f and time-reversed, shifted g. Represents weighted accumulation or memory effect.

Function Spaces

Typically used on integrable functions (L¹), piecewise continuous, or causal functions in engineering.

Properties of Convolution

Commutativity

f * g = g * f. Order of functions interchangeable.

Associativity

(f * g) * h = f * (g * h). Grouping does not affect result.

Distributivity

f * (g + h) = f * g + f * h. Linear over addition.

Scaling

a(f * g) = (af) * g = f * (ag), for scalar a.

Identity Element

Convolution with Dirac delta δ(t) satisfies f * δ = f.

Convolution Theorem in Laplace Transforms

Theorem Statement

Laplace transform of convolution equals product of transforms:

ℒ{f * g}(s) = ℒ{f}(s) · ℒ{g}(s)

Implication for Differential Equations

Transforms integral convolutions into algebraic multiplications, simplifying solutions.

Inverse Laplace

Inverse transform of product: convolution of original functions in time domain.

Applications in Differential Equations

Solving Linear ODEs

Used to solve nonhomogeneous ODEs with forcing functions via integral representation.

Impulse Response and Systems

System output as convolution of input with impulse response function.

Integral Equations

Convolution integral equations arise, solvable by Laplace transform techniques.

Calculation Techniques

Direct Integration

Evaluate convolution integral explicitly when possible.

Laplace Transform Method

Transform both functions, multiply transforms, inverse transform result.

Use of Tables

Refer to convolution tables for common function pairs.

Worked Examples

Example 1: Convolution of Exponentials

Compute (f * g)(t) for f(t) = e^{-at}, g(t) = e^{-bt}, a,b > 0.

(f * g)(t) = ∫₀ᵗ e^{-aτ} e^{-b(t-τ)} dτ= e^{-bt} ∫₀ᵗ e^{(b - a) τ} dτ= e^{-bt} [ (e^{(b - a) t} - 1) / (b - a) ]

Example 2: Using Laplace Transform

Find convolution when ℒ{f}(s) = 1/(s + a), ℒ{g}(s) = 1/(s + b).

ℒ{f * g}(s) = 1/(s + a) · 1/(s + b) = 1/[(s + a)(s + b)]

Inverse transform yields convolution result.

Convolution Integral

Definition

Integral expression defining convolution in continuous time:

(f * g)(t) = ∫₀ᵗ f(τ) g(t - τ) dτ

Limits of Integration

Integral from 0 to t for causal functions; from -∞ to ∞ in general.

Physical Interpretation

Represents system output as accumulation of weighted inputs over time.

Relationship with Laplace Transforms

Transform Domain Simplification

Convolution in time domain corresponds to multiplication in Laplace domain.

Inverse Transform Role

Product of Laplace transforms inverted as convolution integral in time.

Operational Use

Enables solving integral and differential equations efficiently.

Role in Operational Calculus

Operator Representation

Convolution viewed as operator acting on functions, facilitating algebraic manipulation.

Solving Equations

Transforms differential operators into multiplication by Laplace variable s.

Extension to Distributions

Generalized functions like δ(t) used as identity under convolution.

Numerical Approaches

Discretization

Approximate convolution integrals via numerical methods (trapezoidal, Simpson’s rule).

Fast Convolution Algorithms

Use Fast Fourier Transform (FFT) for efficient discrete convolution computation.

Error Analysis

Numerical methods introduce approximation errors; stability depends on kernel smoothness.

Tables and Formulas

Common Convolution Pairs

f(t)	g(t)	(f * g)(t)
1	t	t²/2
e^{at}	e^{bt}	(e^{bt} - e^{at})/(b - a), a ≠ b
t^{n}, n > -1	t^{m}, m > -1	B(n+1, m+1) t^{n+m+1}

Key Formulas

ℒ{f * g}(s) = ℒ{f}(s) · ℒ{g}(s)(f * δ)(t) = f(t)Associativity: (f * g) * h = f * (g * h)

References

Debnath, L., & Bhatta, D. "Integral Transforms and Their Applications." Chapman & Hall/CRC, 2015, pp. 120-145.
Doetsch, G. "Introduction to the Theory and Application of the Laplace Transformation." Springer, 1974, pp. 89-112.
Bracewell, R. N. "The Fourier Transform and Its Applications." McGraw-Hill, 2000, pp. 235-260.
Kaplan, W. "Advanced Calculus." Addison-Wesley, 1979, pp. 400-420.
Widder, D. V. "The Laplace Transform." Princeton University Press, 1946, pp. 60-90.