Four Methods of Integral Representation in Partial Differential Equations: From Green's Function to Integral Equations

# 1. Overview of Integral Representation of Partial Differential Equations

Partial Differential Equations (PDEs) are extensively applied in the fields of physics, engineering, and mathematics. Integral representation is a significant method for solving PDEs, transforming them into equivalent integral equations, thereby simplifying the solution process.

The advantage of integral representation lies in its ability to express the solution of PDEs in integral form, revealing the properties and structure of the solution. Moreover, integral representation offers numerical methods for solving PDEs, such as the Boundary Element Method and the Finite Element Method.

# 2. Green's Function Method

### 2.1 Definition and Properties of Green's Function

Green's function, also known as the Green's function operator, is a special solution to a differential equation that plays a crucial role in the integral representation of partial differential equations.

**Definition:**

For a given partial differential equation:

Lu(x) = f(x)

where L is a linear differential operator and f(x) is a given source function.

The Green's function G(x, y) satisfies the following equation:

LG(x, y) = δ(x - y)

where δ(x - y) is the Dirac delta function.

**Properties:**

***Linearity:** The Green's function G(x, y) is linear with respect to the source function f(x).
***Symmetry:** For any x and y, G(x, y) = G(y, x).
***Integral Representation:** The solution to the partial differential equation u(x) can be represented as:

u(x) = ∫ G(x, y) f(y) dy

### 2.2 Construction Methods of Green's Function

There are various methods for constructing Green's functions, with common methods including:

**Direct Solution:** For simple partial differential equations, Green's function can be directly solved. For example, for the Laplace equation:

∇²G(x, y) = δ(x - y)

The Green's function is:

G(x, y) = (1/4π) * (1/|x - y|)

**Using Integral Representation:** For some complex partial differential equations, Green's function can be constructed using integral representation. For instance, for the heat equation:

∂u/∂t = ∇²u

The Green's function is:

G(x, y, t) = (1/(4πt)^(3/2)) * exp(-|x - y|²/(4t))

**Using Variational Method:** Variational methods can also be used to construct Green's functions. The specific approach is to transform the partial differential equation into a variational problem, then solve for the extremum of the variational problem.

### 2.3 Derivation of Green's Function Integral Representation

**Theorem:** The solution to the partial differential equation Lu(x) = f(x) can be represented as:

u(x) = ∫ G(x, y) f(y) dy

**Proof:**

Let v(x) = u(x) - ∫ G(x, y) f(y) dy. Then:

Lv(x) = Lu(x) - L∫ G(x, y) f(y) dy

= f(x) - ∫ L[G(x, y)] f(y) dy

= f(x) - ∫ δ(x - y) f(y) dy

= 0

Therefore, v(x) is a solution to the partial differential equation Lv(x) = 0. According to the uniqueness theorem, v(x) = 0. Thus, u(x) = ∫ G(x, y) f