5.2 Solving PDEs using Fourier transforms

Fourier transforms are powerful tools for solving partial differential equations. They convert complex spatial derivatives into simple algebraic terms, making PDEs easier to solve. This method is especially useful for linear PDEs with constant coefficients.

Once transformed, PDEs often become ordinary differential equations or even algebraic equations. Solving these simplified equations and applying the inverse Fourier transform gives us the original PDE solution. This technique provides insights into solution behavior and stability.

Solving PDEs with Fourier Transforms

Fourier Transform Fundamentals

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• Fourier transform converts function of time or space into function of frequency
• Applied to linear PDEs with constant coefficients to convert spatial derivatives into algebraic expressions
• Transform of partial derivative with respect to x given by $F\{\frac{\partial u}{\partial x}\} = i\omega F\{u\}$
  • ω represents frequency variable
  • i denotes imaginary unit
• Higher-order derivatives transformed as $F\{\frac{\partial^n u}{\partial x^n}\} = (i\omega)^n F\{u\}$
  • Converts complex differential operators into simple algebraic terms
• Initial and boundary conditions require appropriate transformation for frequency domain consistency
• Transformed PDE in frequency domain typically becomes ordinary differential equation (ODE)
  • ODE expressed in terms of remaining untransformed variables and frequency variable

Applications and Examples

• Heat equation in one dimension: $\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$
  • Transformed equation: $\frac{\partial F\{u\}}{\partial t} = -k\omega^2 F\{u\}$
• Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}$
  • Transformed equation: $\frac{\partial^2 F\{u\}}{\partial t^2} = -c^2\omega^2 F\{u\}$
• Laplace's equation in two dimensions: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • Transformed equation (in x): $-\omega^2 F\{u\} + \frac{\partial^2 F\{u\}}{\partial y^2} = 0$

Algebraic Equations in Frequency Domain

Solving Transformed Equations

• Transformed PDE in frequency domain often becomes algebraic equation or simpler ODE
• For purely algebraic equations, isolate transformed function F{u}
  • Express in terms of frequency variable and transformed initial/boundary conditions
• Apply standard ODE solution methods for transformed ODEs
  • Characteristic equations (for linear ODEs with constant coefficients)
  • Variation of parameters (for non-homogeneous ODEs)
  • Separation of variables (for PDEs separable in frequency and remaining variables)
• Pay attention to frequency variable ω dependence
  • Crucial for inverse transformation process
• Ensure solution satisfies transformed boundary or initial conditions in frequency domain
• For PDE systems, transformed equations may form algebraic equation system
  • Solve using matrix methods (Gaussian elimination)
  • Apply elimination techniques (substitution method)

Example Solutions

• Heat equation solution in frequency domain: $F\{u\}(\omega,t) = F\{u_0\}(\omega)e^{-k\omega^2t}$
  • $u_0$ represents initial temperature distribution
• Wave equation solution: $F\{u\}(\omega,t) = A(\omega)\cos(c\omega t) + B(\omega)\sin(c\omega t)$
  • $A(\omega)$ and $B(\omega)$ determined by initial conditions
• Laplace's equation (transformed in x): $F\{u\}(\omega,y) = C(\omega)e^{-|\omega|y} + D(\omega)e^{|\omega|y}$
  • $C(\omega)$ and $D(\omega)$ determined by boundary conditions

Solution Interpretation and Conversion

Frequency Domain Interpretation

• Frequency domain solution represents spectral decomposition of original function
  • Shows behavior across different frequencies
• Analyze amplitude spectrum |F{u}(ω)| to understand dominant frequencies in solution
• Examine phase spectrum arg(F{u}(ω)) to determine phase shifts between different frequency components
• High-frequency components often correspond to rapid spatial variations or discontinuities
• Low-frequency components typically represent overall trends or large-scale behavior

Inverse Fourier Transform Application

• Recover spatial domain solution by applying inverse Fourier transform to frequency domain solution
• Inverse Fourier transform given by $u(x,t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F\{u\}(\omega,t) e^{i\omega x} d\omega$
  • F{u} represents solution in frequency domain
• Evaluate inverse transform integral
  • May involve contour integration techniques for complex-valued integrands
• Utilize common inverse Fourier transform pairs to simplify inversion process
  • Gaussian function: $F\{e^{-ax^2}\} = \sqrt{\frac{\pi}{a}}e^{-\omega^2/(4a)}$
  • Exponential decay: $F\{e^{-a|x|}\} = \frac{2a}{a^2 + \omega^2}$
• Verify obtained spatial domain solution satisfies original PDE and initial/boundary conditions

Example Conversions

• Heat equation solution in spatial domain: $u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} u_0(\xi) e^{-(x-\xi)^2/(4kt)} d\xi$
• Wave equation solution: $u(x,t) = \frac{1}{2}[f(x+ct) + f(x-ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} g(\xi) d\xi$
  • f and g represent initial displacement and velocity respectively

Existence and Uniqueness of Solutions

Solution Analysis in Frequency Domain

• Fourier transform provides insights into existence and uniqueness of solutions
  • Examine behavior of transformed equation in frequency domain
• Well-posed problem requires unique solution for all relevant frequencies
• Analyze denominator of frequency domain solution for potential singularities
  • Singularities may indicate ill-posedness or non-uniqueness
• Apply Paley-Wiener theorem to relate Fourier transform decay rate to original function analyticity
  • Provides information about solution regularity
• Examine frequency domain solution growth rate as |ω| → ∞
  • Determines smoothness and decay properties of spatial domain solution

Stability and Perturbation Analysis

• Investigate solution stability using Fourier transform techniques
  • Analyze how small perturbations in initial or boundary conditions affect frequency domain representation
• Examine amplification factor in frequency domain
  • Stable solutions have bounded amplification factors for all frequencies
• Analyze high-frequency behavior to detect potential instabilities
  • Ill-posed problems often exhibit exponential growth in high-frequency components
• Apply Fourier analysis to study numerical stability of discretization schemes
  • von Neumann stability analysis for finite difference methods
• Examples of stability analysis:
  • Heat equation (stable): amplification factor $e^{-k\omega^2t}$ decays for all ω
  • Backward heat equation (unstable): amplification factor $e^{k\omega^2t}$ grows unboundedly for large ω