3.1. Linear and Non-linear Partial Differential Equations

A Partial Differential Equation (PDE) is an equation involving partial derivatives of a function of several independent variables. General form of a second-order PDE in two variables 𝑥 and 𝑦:

∂²𝑢 / ∂𝑥² + ∂²𝑢 / ∂𝑥∂𝑦 + ∂²𝑢 / ∂𝑦² + ... = 0

A Partial Differential Equation (PDE) connects the rates of change of a function that depends on several variables. Unlike ordinary differential equations, which involve derivatives with respect to only one variable, PDEs consider multiple independent variables. The general form given shows how second-order derivatives are combined to produce the PDE, indicating relationships between these variables.

Imagine you're watching a video of a wave on a string. The height of the string at any point depends on both time and position along the string, making it a multi-variable situation—hence, a PDE is needed to describe its behavior.

A linear PDE is one in which the dependent variable 𝑢 and its partial derivatives appear to the first power, and are not multiplied together.

General Linear Form:

∂²𝑢 / ∂𝑥² + ∂²𝑢 / ∂𝑥∂𝑦 + ∂²𝑢 / ∂𝑦² + ∂𝑢 / ∂𝑥 + ∂𝑢 / ∂𝑦 + 𝑓(𝑥, 𝑦)𝑢 = 𝑔(𝑥, 𝑦)

Characteristics:

• Coefficients may be functions of independent variables.
• No product or nonlinear functions of 𝑢, ∂𝑢/∂𝑥, etc.

Example: ∂²𝑢 / ∂𝑥² + ∂²𝑢 / ∂𝑦² = 0 (Laplace’s Equation)

A linear PDE maintains a structure where the dependent variable and its derivatives are only raised to the first power and are not multiplied together. This form allows for easier analytical solutions. The characteristics listed highlight that while coefficients can vary, the fundamental linearity must be preserved. An important example is Laplace’s equation, which is used in various fields including physics and engineering to describe steady states.

Think of a recipe where you only add ingredients without mixing them; each ingredient contributes independently without changing one another's property. In linear PDEs, each portion of the equation contributes separately to the outcome, much like how ingredients contribute to a final dish without altering the basic properties of each other.

A non-linear PDE is one in which: • The dependent variable or its derivatives appear with powers other than 1. • Products of the function and/or its derivatives appear. Example: ∂𝑢 / ∂𝑥 + (∂𝑢 / ∂𝑦)² = 1 Non-linear PDEs are typically more difficult to solve and often arise in complex systems such as fluid dynamics and general relativity.

Non-linear PDEs introduce complexity as the dependent variable and its derivatives can be raised to any power or combined in ways such as multiplication. These equations often model real-world phenomena where interactions are non-linear, like turbulence in fluids. The example given demonstrates a scenario where the rate of change in one direction is influenced by its square, illustrating the complications posed by non-linearity.

Imagine trying to predict how a small stone dropped into a pond generates splashes and ripples. The interaction of the water's surface with the stone creates a complex and interconnected pattern that isn't straightforward to predict—this unpredictability reflects what happens in non-linear PDEs.

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