Differential Equations & Fourier Analysis Cheat Sheet
This cheat sheet covers essential methods for solving ordinary and partial differential equations, including power series, Fourier series, and Bessel functions, along with their applications in heat and wave problems.
Core Principles
- Separation of Variables: A technique to simplify PDEs into multiple ODEs.
- Power Series Solutions: Representing solutions as infinite series for ODEs, especially around ordinary or singular points.
- Fourier Series: Decomposing periodic functions into a sum of sines and cosines.
- Bessel Functions: Solutions to Bessel's equation, crucial for problems with cylindrical symmetry.
- Sturm-Liouville Problems: Eigenvalue problems for second-order ODEs with boundary conditions.
- Heat and Wave Equations: Fundamental PDEs describing diffusion and wave propagation.
- Laplace's Equation: Governing steady-state phenomena in various geometries.
Action Steps
- Identify the type of differential equation (ODE, PDE, linear, homogeneous, etc.).
- Choose the appropriate solution method (e.g., separation of variables, power series, transform).
- Apply the method systematically, solving intermediate equations.
- Incorporate boundary and initial conditions to find specific solutions.
- Verify the solution by substituting it back into the original equation and conditions.
Formulas
- Separable DE: dy/g(y) = f(x)dx
- Homogeneous 2nd Order Linear DE: ay"+by'+cy=0
- Cauchy-Euler Equation: x²y"+Axy'+By=0
- Power Series Radius of Convergence: r = 1 / lim |am+1 / am|
- Fourier Series: f(x) = a0/2 + Σ[an cos(nπx/L) + bn sin(nπx/L)]
- Bessel's Equation: x²y"(x) + xy'(x) + (x² - ν²)y(x) = 0
- Legendre's Equation: (1 - x²)y" - 2xy' + n(n + 1)y = 0
- Heat Equation (1D): ut = c²uxx
- Wave Equation (1D): utt = c²uxx
- Laplace's Equation (2D): Uxx + Uyy = 0
- Fourier Transform: F{f(x)} = ∫ f(x)e^(-iλx) dx
- Heat Kernel: K(x,t) = 1 / (2√πt) * e^(-x² / 4t)
Key Terms
- Ordinary Point: A point x₀ where the coefficients of a differential equation are analytic.
- Singular Point: A point x₀ where the coefficients of a differential equation are not analytic.
- Regular Singular Point: A singular point x₀ where (x-x₀)p(x) and (x-x₀)²q(x) are analytic.
- Orthogonal Functions: Functions y₁, y₂ such that ∫ p(x)y₁(x)y₂(x)dx = 0 with respect to a weight function p(x).
- Eigenvalue: A value λ for which a Sturm-Liouville problem has a non-trivial solution.
- Eigenfunction: A non-trivial solution corresponding to an eigenvalue in a Sturm-Liouville problem.
- Piecewise Continuous: A function that is continuous except at a finite number of points where it has jump discontinuities.
- Fourier Series Coefficients: Constants (a₀, an, bn) that determine the sine and cosine components of a Fourier series.
- Bessel Function: Solutions to Bessel's differential equation, often denoted by Jν(x) and Yν(x).
- Legendre Polynomial: Polynomial solutions to Legendre's differential equation, denoted by Pn(x).
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