5.4 Convolution and Duhamel's principle revisited

Convolution and Duhamel's principle are powerful tools for solving non-homogeneous PDEs. They connect Green's functions, which represent system responses to impulses, with source terms to yield solutions. This approach is especially useful for problems with time-dependent sources.

Fourier transforms play a crucial role in this process. By converting PDEs to the frequency domain, complex equations become simpler algebraic ones. This technique, combined with convolution, allows for efficient solutions to a wide range of PDEs in various fields.

Convolution Theorem for PDEs

Convolution Theorem and Green's Functions

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• Convolution theorem states Fourier transform of convolution of two functions equals product of their individual Fourier transforms
• Green's functions represent system's response to point source or impulse in PDEs
• Convolution of Green's function with source term yields solution to non-homogeneous PDE
• Convolution integral for PDEs typically takes form $u(x,t) = \int G(x-y,t)f(y)dy$
  • G represents Green's function
  • f represents source term
• Convolution interpreted as weighted average of Green's function over domain of source term
• Examples:
  • Heat equation: $u(x,t) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{4\pi \alpha t}} e^{-\frac{(x-y)^2}{4\alpha t}} f(y) dy$
  • Wave equation: $u(x,t) = \frac{1}{2} \int_{x-ct}^{x+ct} f(y) dy$

Efficient PDE Solutions Using Convolution Theorem

• Convolution theorem enables efficient PDE solution through:
  • Transforming problem to frequency domain
  • Performing multiplication
  • Inverting back to time domain
• Numerical techniques employed for complex PDE convolutions
  • Fast Fourier Transform (FFT) algorithm
  • Discrete convolution methods
• Examples:
  • Solving diffusion equation: $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} + f(x,t)$
  • Solving wave equation with source term: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} + f(x,t)$

Green's Functions and Fundamental Solutions

Properties of Green's Functions

• Green's functions satisfy equation $L[G] = \delta(x-x')$
  • L represents differential operator
  • δ represents Dirac delta function
• Fundamental solution represents system response to unit impulse at specific point in space and time
• Green's functions derived for various PDE types
  • Elliptic equations (Laplace equation)
  • Parabolic equations (Heat equation)
  • Hyperbolic equations (Wave equation)
• Symmetry and translation properties reflect underlying PDE symmetries and boundary conditions
• Green's functions construct general solutions through superposition principle
• Examples:
  • 1D heat equation Green's function: $G(x,t;x',t') = \frac{1}{\sqrt{4\pi \alpha (t-t')}} e^{-\frac{(x-x')^2}{4\alpha (t-t')}}$
  • 2D Laplace equation Green's function: $G(x,y;x',y') = -\frac{1}{2\pi} \ln \sqrt{(x-x')^2 + (y-y')^2}$

Behavior and Insights from Green's Functions

• Singularity behavior near source point provides insight into local solution behavior
• Decay properties at large distances or times relate to long-term or far-field solution behavior
• Green's functions reveal fundamental properties of PDEs
  • Causality
  • Energy conservation
  • Wave propagation characteristics
• Examples:
  • Wave equation Green's function decay: $G(x,t;x',t') \sim \frac{1}{\sqrt{|x-x'|}}$ as $|x-x'| \to \infty$
  • Heat equation Green's function decay: $G(x,t;x',t') \sim \frac{1}{t^{3/2}}$ as $t \to \infty$

Duhamel's Principle for Non-Homogeneous PDEs

Duhamel's Principle and Time-Dependent Sources

• Duhamel's principle extends Green's functions to PDEs with time-dependent source terms
• Solution to non-homogeneous PDE expressed as integral involving Green's function and time-dependent source term
• Duhamel integral takes form $u(x,t) = \int_0^t G(x,t-\tau)f(x,\tau)d\tau$
  • G represents Green's function
  • f represents time-dependent source term
• Interpreted as continuous superposition of solutions to instantaneous source problems
• Useful for solving initial-boundary value problems with time-varying forcing or source terms
• Examples:
  • Heat equation with time-dependent source: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} + f(x,t)$
  • Wave equation with time-dependent forcing: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} + f(x,t)$

Applications and Properties of Duhamel's Principle

• Combines with other solution techniques
  • Separation of variables
  • Fourier transforms
• Causality of physical systems reflected in Duhamel integral structure
  • Upper integration limit is current time t
• Duhamel's principle applies to various PDE types
  • Parabolic equations (Heat conduction with time-varying heat source)
  • Hyperbolic equations (Vibrating string with time-dependent external force)
• Examples:
  • Diffusion in a medium with time-varying concentration source
  • Acoustic wave propagation with time-dependent sound source

Fourier Transform and Green's Functions for Complex PDEs

Fourier Transform Techniques in PDE Solutions

• Fourier transform converts PDEs from spatial or temporal domain to frequency domain
• PDEs often reduce to simpler algebraic equations in frequency domain
• Green's functions in frequency domain (transfer functions) obtained by applying Fourier transform to spatial Green's function
• Convolution theorem expresses frequency domain solution as product of transfer function and Fourier transform of source term
• Inverse Fourier transforms yield final solution in original spatial or temporal domain
• Examples:
  • Solving heat equation in unbounded domain: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
  • Analyzing wave propagation in dispersive media: $\frac{\partial^2 u}{\partial t^2} = c^2(k) \frac{\partial^2 u}{\partial x^2}$

Advanced Applications and Numerical Implementations

• Combined approach powerful for solving PDEs with complex geometries or boundary conditions
• Method extends to multi-dimensional problems
  • Multi-dimensional Fourier transforms
  • Corresponding Green's functions
• Numerical implementations utilize Fast Fourier Transform (FFT) algorithms for efficient computation
• Applications in various fields
  • Signal processing
  • Image reconstruction
  • Electromagnetic wave propagation
• Examples:
  • Solving 2D Poisson equation: $\nabla^2 u = f(x,y)$
  • Analyzing seismic wave propagation in layered media