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Table of contents
1.
Introduction
2.
Partial Differential Equations Definition
3.
Partial Differential Equations Representation
4.
Partial Differential Equations Types
4.1.
First-Order Partial Differential Equation
4.2.
Linear Partial Differential Equation
4.3.
Quasi-Linear Partial Differential Equation
4.4.
Homogeneous Partial Differential Equation
5.
Partial Differential Equations Order and Degree
5.1.
Order of Partial Differential Equation
5.2.
Degree of Partial Differential Equation
6.
Partial Differential Equations Classification
7.
Solved Example
8.
FAQs
9.
Key Takeaways
Last Updated: Mar 27, 2024

Partial Differential Equations

Author Rajat Agrawal
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Introduction

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

In other words, partial differential equations help calculate partial derivatives for functions having several variables. These equations are classified as differential equations.

Partial differential equations are very helpful in studying various phenomena that occur in nature, such as sound, fluid flow, heat, and waves.

Let’s learn about partial differential equations in-depth.

Partial Differential Equations Definition

Partial differential equations are commonly known as PDE. These equations are used to describe problems involving an unknown function with numerous dependent and independent variables and the partial derivatives of that function with respect to the independent variables. A PDE for a function u(x1,...xn) is an equation of the form:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mi>n</mi></msub><mo>;</mo><mo>&#xA0;</mo><mi>u</mi><mo>,</mo><mfrac><mrow><mo>&#x2202;</mo><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mfrac><mrow><mo>&#x2202;</mo><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></mfrac><mo>;</mo><mo>&#xA0;</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&#x2202;</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&#x2202;</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></mfrac><mo>;</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn><mspace linebreak="newline"/><mspace linebreak="newline"/></math>
The PDE is linear if and its derivatives are a linear function of u. The normal PDE is given by: 

∂u/∂x (x,y) = 0 

The above relation implies that the function u(x,y) is independent of x.

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Partial Differential Equations Representation

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

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><mrow><mo>&#x2202;</mo><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mspace linebreak="newline"/><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"/><msub><mi>u</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><mo>&#x2202;</mo><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mfenced><mfrac><mrow><mo>&#x2202;</mo><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></mfenced></math>

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:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi><mi>m</mi></msub><mo>,</mo><mo>&#xA0;</mo><mi>u</mi><mo>,</mo><msub><mi>u</mi><mrow><mi>x</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>u</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub></mrow></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn></math>

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:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>z</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>&#x2202;</mo><mi>z</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>z</mi><mo>+</mo><mi>x</mi><mi>y</mi></math>

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.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>z</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>&#x2202;</mo><mi>z</mi></mrow><mrow><mo>&#x2202;</mo><mi>y</mi></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>z</mi><mo>+</mo><mi>x</mi><mi>y</mi></math>

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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><mi>C</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn></math>, it results in a Parabolic partial differential equation. 

2.) Hyperbolic PDE: If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><mi>C</mi><mo>&gt;</mo><mn>0</mn></math>, it results in a Hyperbolic partial differential equation.

3.) Elliptic PDE: If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><mi>C</mi><mo>&lt;</mo><mn>0</mn></math>, 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

 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msup><mi>a</mi><mn>2</mn></msup><mo>.</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>

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

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><mi>t</mi></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mo>&#xB7;</mo><mi>cos</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced><mspace linebreak="newline"/><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mo>&#xB7;</mo><mi>sin</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced><mspace linebreak="newline"/></math>

Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mi>sin</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mi>sin</mi><mfenced><mi>x</mi></mfenced><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mi>sin</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mi>cos</mi><mfenced><mi>x</mi></mfenced><mo>,</mo><mo>&#xA0;</mo><mi>s</mi><mi>o</mi><mo>&#xA0;</mo><mi>t</mi><mi>h</mi><mi>a</mi><mi>t</mi></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mo>&#xB7;</mo><mi>sin</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced></math>

Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>sin</mi><mfenced><mrow><mi>a</mi><mi>t</mi></mrow></mfenced><mo>&#xB7;</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced><mo>&#xA0;</mo><mi>i</mi><mi>s</mi><mo>&#xA0;</mo><mi>a</mi><mo>&#xA0;</mo><mi>s</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>&#xA0;</mo><mi>t</mi><mi>o</mi><mo>&#xA0;</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msup><mi>a</mi><mn>2</mn></msup><mo>&#xB7;</mo><mfrac><mrow><msup><mo>&#x2202;</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>&#x2202;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>

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.

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