🧮Physical Sciences Math Tools Unit 11 – PDEs and Separation of Variables

What are PDEs?

• Partial Differential Equations (PDEs) describe systems that change with respect to multiple variables, often space and time
• Contain partial derivatives of an unknown function with respect to two or more independent variables (e.g., $\frac{\partial u}{\partial x}$, $\frac{\partial^2 u}{\partial t^2}$)
• Model a wide range of physical phenomena such as heat transfer, fluid dynamics, and quantum mechanics
• Classified based on their order (highest derivative), linearity, and coefficients
• Require boundary conditions and/or initial conditions to obtain unique solutions
• Analytical solutions often involve advanced mathematical techniques like separation of variables or Fourier series
• Numerical methods (finite difference, finite element) are used when analytical solutions are not feasible

Types of PDEs

• Elliptic PDEs: Laplace's equation ($\nabla^2 u = 0$) and Poisson's equation ($\nabla^2 u = f(x, y, z)$)
  • Describe steady-state or equilibrium problems (electrostatics, gravitational potential)
  • Require boundary conditions on a closed domain
• Parabolic PDEs: Heat equation ($\frac{\partial u}{\partial t} = \alpha \nabla^2 u$) and diffusion equation ($\frac{\partial u}{\partial t} = D \nabla^2 u$)
  • Model time-dependent diffusion processes (heat conduction, mass transfer)
  • Require initial conditions and boundary conditions
• Hyperbolic PDEs: Wave equation ($\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$) and transport equation ($\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u = 0$)
  • Describe propagation phenomena (sound waves, light waves, fluid flow)
  • Require initial conditions and boundary conditions
• Linear PDEs: Coefficients and unknown function appear linearly (e.g., $a\frac{\partial^2 u}{\partial x^2} + b\frac{\partial^2 u}{\partial y^2} = f(x, y)$)
• Nonlinear PDEs: Coefficients or unknown function appear nonlinearly (e.g., $\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0$)

Boundary and Initial Conditions

• Boundary conditions specify the behavior of the solution at the edges of the domain
  • Dirichlet boundary condition: Prescribes the value of the function on the boundary (e.g., $u(0, t) = 0$)
  • Neumann boundary condition: Prescribes the value of the normal derivative on the boundary (e.g., $\frac{\partial u}{\partial x}(L, t) = 0$)
  • Robin (mixed) boundary condition: Linear combination of function value and normal derivative (e.g., $\alpha u + \beta \frac{\partial u}{\partial x} = \gamma$)
• Initial conditions specify the state of the system at the initial time (e.g., $u(x, 0) = f(x)$)
• Well-posed problems require sufficient boundary and initial conditions to ensure existence, uniqueness, and stability of the solution
• Homogeneous boundary conditions: Zero function value or zero normal derivative on the boundary
• Inhomogeneous boundary conditions: Non-zero function value or non-zero normal derivative on the boundary

Separation of Variables Method

• Powerful technique for solving linear, homogeneous PDEs with homogeneous boundary conditions
• Assumes the solution can be written as a product of functions, each depending on a single variable (e.g., $u(x, t) = X(x)T(t)$)
• Substituting the separated solution into the PDE leads to ordinary differential equations (ODEs) for each variable
• ODEs are solved using standard techniques (integration, eigenvalue problems) and combined to obtain the general solution
• Boundary and initial conditions are applied to determine the specific solution and arbitrary constants
• Suitable for various geometries (rectangular, cylindrical, spherical) and coordinate systems (Cartesian, polar, spherical)
• Eigenfunctions obtained from the separation of variables form a complete orthogonal basis for the solution space

Solving PDEs Step-by-Step

1. Identify the type of PDE (elliptic, parabolic, hyperbolic) and its order
2. Determine the domain and boundary conditions
3. If applicable, identify the initial condition(s)
4. Assume a separated solution in the form $u(x_1, x_2, \ldots, t) = X_1(x_1)X_2(x_2)\ldots T(t)$
5. Substitute the separated solution into the PDE and divide by the product of functions
6. Separate the equation into terms, each depending on a single variable
7. Equate each term to a separation constant (eigenvalue) to obtain ODEs
8. Solve the ODEs using appropriate techniques (integration, eigenvalue problems)
9. Apply boundary conditions to the spatial solutions to determine eigenfunctions and eigenvalues
10. Apply initial conditions to determine the time-dependent solution and arbitrary constants
11. Combine the spatial and temporal solutions to obtain the general solution
12. Verify that the solution satisfies the PDE, boundary conditions, and initial conditions

Applications in Physical Sciences

• Heat transfer: Modeling temperature distribution in solids, fluids, and gases
  • Steady-state heat conduction (Laplace's equation)
  • Transient heat conduction (heat equation)
• Fluid dynamics: Describing the motion of fluids and gases
  • Potential flow (Laplace's equation)
  • Navier-Stokes equations (momentum conservation)
• Quantum mechanics: Modeling the behavior of particles at the atomic and subatomic scales
  • Schrödinger equation (wave function, energy states)
  • Dirac equation (relativistic quantum mechanics)
• Electromagnetism: Describing electric and magnetic fields and their interactions
  • Poisson's equation (electrostatic potential)
  • Maxwell's equations (electromagnetic waves)
• Elasticity: Modeling deformation and stress in solid materials
  • Navier-Cauchy equations (displacement, stress)
  • Plate and shell equations (thin structures)

Common Challenges and Tips

• Identifying the appropriate coordinate system and separation of variables based on the geometry of the problem
• Dealing with inhomogeneous boundary conditions or source terms
  • Use superposition principle for linear PDEs
  • Apply change of variables or Green's functions
• Solving eigenvalue problems and determining eigenfunctions
  • Sturm-Liouville theory for self-adjoint operators
  • Orthogonality and completeness of eigenfunctions
• Convergence and stability of numerical methods for PDEs
  • Courant-Friedrichs-Lewy (CFL) condition for explicit schemes
  • Implicit schemes for improved stability
• Verifying the solution and checking for physical plausibility
  • Dimensional analysis and scaling arguments
  • Asymptotic behavior and limiting cases

Practice Problems and Examples

1. Solve the one-dimensional heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0, t) = 0$, $u(L, t) = 0$, and initial condition $u(x, 0) = \sin(\frac{\pi x}{L})$.
2. Find the steady-state temperature distribution in a rectangular plate with dimensions $a \times b$, governed by Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$, and boundary conditions $u(0, y) = 0$, $u(a, y) = 100$, $\frac{\partial u}{\partial y}(x, 0) = 0$, $\frac{\partial u}{\partial y}(x, b) = 0$.
3. Solve the wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ for a vibrating string of length $L$ with boundary conditions $u(0, t) = 0$, $u(L, t) = 0$, and initial conditions $u(x, 0) = \sin(\frac{\pi x}{L})$, $\frac{\partial u}{\partial t}(x, 0) = 0$.
4. Determine the electrostatic potential $V(r, \theta)$ inside a hollow conducting sphere of radius $R$ with a charge distribution $\rho(r, \theta) = \rho_0 \cos(\theta)$, governed by Poisson's equation $\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial V}{\partial r}\right) + \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial V}{\partial \theta}\right) = -\frac{\rho(r, \theta)}{\varepsilon_0}$ and boundary condition $V(R, \theta) = 0$.