## Partial Differential Equations Representation

In Partial Differential Equations, we denote the partial derivatives using subscripts, such as:

## Partial Differential Equations Types

The different types of PDEs are:-

1.) First-order Partial Differential Equation

2.) Linear Partial Differential Equation

3.) Quasi-Linear Partial Differential Equation

4.) Homogeneous Partial Differential Equation

Let’s get familiar with these PDEs one by one.

### First-Order Partial Differential Equation

In First-Order Partial Differential Equation, the equation has only the first derivative of the unknown function having ‘m’ variables.

It is represented in the form:

### Linear Partial Differential Equation

In any PDE, if the dependant variable and all of its partial derivatives occur linearly, the equation is referred to as a linear PDE; otherwise, it is referred to as a nonlinear PDE.

### Quasi-Linear Partial Differential Equation

Suppose all the terms with the highest order derivatives of dependent variables occur linearly, the coefficients of those terms being functions of only lower-order derivatives of the dependent variables. In that case, the PDE is said to be quasi-linear. On the other hand, lower-order derivative terms can appear in any way.

### Homogeneous Partial Differential Equation

If the dependent variable or its partial derivatives appear in all of the terms of a PDE, it is referred to as a non-homogeneous partial differential equation; otherwise, it is referred to as a homogeneous partial differential equation.

## Partial Differential Equations Order and Degree

Partial differential equations are divided according to their order and degree. The first-order and second-order partial differential equations are the most often used.

### Order of Partial Differential Equation

The order of a partial differential equation (PDE) can be defined as the order of the PDE's highest derivative term.

Let us say a PDE is given as:

Since the order of the highest derivative is 1, this is first-order PDE.

### Degree of Partial Differential Equation

The degree of a Partial Differential Equation is defined as the degree of PDE's highest derivative.

The below given partial differential equation will have degree 1 as the highest derivative is of the first degree.

## Partial Differential Equations Classification

Each form of PDE has specific functions that help determine whether a given finite element approach is appropriate for the problem that the PDE is describing. The equation determines the answer, and numerous variables have partial derivatives with regard to each other.

There are three types of second-order PDEs:

1.) Parabolic PDE: If , it results in a Parabolic partial differential equation.

2.) Hyperbolic PDE: If , it results in a Hyperbolic partial differential equation.

3.) Elliptic PDE: If , it results in an Elliptical partial differential equation.

## Solved Example

**Example:** Show that if ‘a’ is a constant, then u(x,t) = sin(at)cos(x) is a solution to

**Solution: **Since ‘a’ is a constant, the partials with respect to ‘t’ are:

Moreover,

Therefore,

Hence Proved.

## FAQs

**1. Define Partial Differential Equation.**

A partial differential equation is a type of differential equation that comprises equations with unknown multi variables with partial derivatives.

**2. What is the Order of Partial Differential Equation?**

The order of a partial differential equation (PDE) can be defined as the order of the PDE's highest derivative term.

**3. Mention Types of 2nd order PDE.**

There are three types of second-order PDEs: Parabolic, Hyperbolic, and Elliptic PDE.

## Key Takeaways

In this article, we have extensively discussed the Partial Differential Equation, its definition, representation, and types. If you want to learn more, check out our articles on the Determinants, and Doolittle Algorithm.

Do upvote our blog to help other ninjas grow.

Happy Coding!