Five Steps to the Regular Form of Partial Differential Equations: Simplifying Complex Equations to a Neat Form

# Overview of Regular Forms of Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that describe the relationship between an unknown function and its partial derivatives with respect to multiple independent variables. The regular form is a special form of PDEs that can simplify the process of solving equations.

The definition of the regular form is as follows: a PDE can be written in the following form:

F(x, y, z, u, p, q) = 0

Where:

* `x`, `y`, `z` are independent variables
* `u` is the unknown function
* `p`, `q` are the first partial derivatives of the unknown function

# Theoretical Foundations of Regular Forms of Partial Differential Equations

### 2.1 Types and Characteristics of Partial Differential Equations

Partial Differential Equations (PDEs) are equations that include multiple independent variables, an unknown function, and partial derivatives of the unknown function. Depending on the highest order partial derivative of the unknown function, PDEs can be divided into:

- First-order partial differential equations: the highest order partial derivative is first-order.
- Second-order partial differential equations: the highest order partial derivative is second-order.
- Higher-order partial differential equations: the highest order partial derivative is greater than second-order.

The characteristics of PDEs include:

- **Linear/Non-linear:** If the coefficients of the unknown function and its partial derivatives in the equation are constants or only related to the independent variables, then it is a linear PDE; otherwise, it is a non-linear PDE.
- **Homogeneous/Non-homogeneous:** If the right-hand side of the equation is zero, then it is a homogeneous PDE; otherwise, it is a non-homogeneous PDE.
- **Elliptic/Parabolic/Hyperbolic:** Depending on the sign of the second-order partial derivatives of the unknown function in the equation, PDEs can be divided into elliptic, parabolic, or hyperbolic types.

### 2.2 Definition and Significance of Regular Forms

The regular form of partial differential equations refers to converting PDEs into a standard form with specific structure and characteristics. The definition of the regular form is as follows:

∂u/∂t + a(x,y)∂u/∂x + b(x,y)∂u/∂y = f(x,y)

Where:

- u(x,y,t) is the unknown function.
- t is the time variable.
- x and y are space variables.
- a(x,y) and b(x,y) are coefficient functions.
- f(x,y) is a known function.

The significance of the regular form includes:

- **Simplifying Solutions:** The regular form converts PDEs into a form that is easier to analyze and solve.
- **Physical Meaning:** The regular form can reveal the essence of the physical processes described by the PDE.
- **Uniformity:** The regular form allows for a unified solution method for different types of PDEs.

# 3.1 Regular Form of First-Order Partial Differential Equations

The general form of **first-order partial differential equations** is:

P(x, y, u) dx + Q(x, y, u) dy = 0

Where, P(x, y, u) and Q(x, y, u) are functions of x, y, and u.

The **regular form** refers to the equation being simplified into the following form:

du = M(x, y) dx + N(x, y) dy

Where, M(x, y) and N(x, y) are functions of x and y.

**Derivation Steps:**

1. **Integrability Condition:**

If the equation is integrable, there exists a function f(x, y, u) such that:

    P = f_x,   Q = f_y

2. **Integrating Factor:**

If the equation is not integrable, introduce an integrating factor μ(x, y) such that:

    μP = f_x,   μQ = f_y

3. **Regular Form:**

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