Partial Differential Equations - Examples & Applications

This section provides practical examples of numerical methods for solving partial differential equations (PDEs), focusing heavily on the Finite Difference Method applied to Elliptic (Laplace), Parabolic (Heat Conduction), and Hyperbolic (Wave) equations, as well as an overview of the Finite Element Method.

Classification of PDEs

Case Study: Classifying a PDE

Classify the following partial differential equation representing steady-state heat conduction with internal heat generation in 2D:
k∂2T∂x2+k∂2T∂y2+q=0k \frac{\partial^2 T}{\partial x^2} + k \frac{\partial^2 T}{\partial y^2} + q = 0

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Case Study: Classifying the Wave Equation

Classify the 1D wave equation:
∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

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Elliptic PDEs: Laplace Equation

The Laplace equation, ∇2T=0\nabla^2 T = 0, dictates steady-state distributions where the value at any point is the average of its neighbors.

Intermediate: Finite Difference Method for a Heated Plate

A square steel plate has dimensions 1.2 m×1.2 m1.2 \text{ m} \times 1.2 \text{ m}. The boundary temperatures are fixed:
Top edge: 100∘C100^\circ\text{C} Bottom edge: 0∘C0^\circ\text{C} Left edge: 75∘C75^\circ\text{C} Right edge: 50∘C50^\circ\text{C}
Set up the finite difference equations using a grid spacing of Δx=Δy=0.4 m\Delta x = \Delta y = 0.4 \text{ m} to find the steady-state temperature of the interior nodes.

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Parabolic PDEs: Heat Conduction

The 1D heat equation ∂T∂t=α∂2T∂x2\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} describes the transient distribution of temperature in a medium over time.

Advanced: Explicit Finite Difference Method

An aluminum rod 10 cm10 \text{ cm} long has an initial uniform temperature of 0∘C0^\circ\text{C}. At t=0t = 0, both boundaries are suddenly raised to 100∘C100^\circ\text{C}. The thermal diffusivity of aluminum is α=0.835 cm2/s\alpha = 0.835 \text{ cm}^2/\text{s}.
Use the explicit finite difference method to estimate the temperature distribution at t=0.1 st = 0.1 \text{ s}. Use a spatial step Δx=2 cm\Delta x = 2 \text{ cm} and a time step Δt=0.1 s\Delta t = 0.1 \text{ s}.

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Hyperbolic PDEs: Wave Equation

Hyperbolic PDEs, like the wave equation, dictate the propagation of disturbances. They require two initial conditions (initial position and initial velocity).

Advanced: Finite Difference for Wave Equation

A taut string of length L=10 mL = 10 \text{ m} is plucked. The wave equation is ∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} with c=2 m/sc = 2 \text{ m/s}. The initial displacement is u(x,0)=f(x)u(x, 0) = f(x) and initial velocity is zero ∂u∂t(x,0)=0\frac{\partial u}{\partial t}(x, 0) = 0. Setup the explicit finite difference equation for an interior node at the first time step t=Δtt = \Delta t.

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Finite Element Method (FEM)

While Finite Difference discretizes the differential equation directly on a grid, the Finite Element Method divides the domain into elements, assuming a simple basis function (shape function) within each element, and minimizes the global error. It is particularly powerful for complex geometries.

Case Study: FEM vs FDM in Structural Analysis

An aerospace engineer needs to determine the stress distribution in an aircraft wing bracket with a curved, irregular boundary and a central hole. Compare the application of FDM and FEM to this problem.

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