Partial Differential Equations

Partial differential equations in mathematics are differential equations that contain partial derivatives with multi variables. The partial differential equation is abbreviated for common use as PDE, which consists of partial derivatives of dependent or independent variables. Partial differential equations have three types of second-order partial differential equations.
We will discuss partial differential equations with a few examples to give you an overview of this important mathematics concept in this article. The concept of differential equations can be a mixture of surprises and fun with several difficulties.
We will break down this concept of partial differential equations into smaller pieces to help you understand better.

What is the partial differential equation definition?

Partial differential equations can be defined as a mathematical equation consisting of two or more independent variables and an unknown function dependent on those variables and partial derivatives of that function concerning the independent variables present in the equation.

When we find the solution to a partial differential equation, it refers to the function that solves that equation and converts it into an identity when the function is substituted. This is a general solution if it is correct for all particular equation solutions.

Partial differential equations are a mathematical method to formulate physical and several other problems involving various variables’ functions. These physical problems include heat propagation of sound electrostatics, elasticity, electrodynamics, and fluid flow.

Partial differential equations can be difficult to solve, but several techniques are used to solve the partial differential equations more simply. These techniques include the separation method, change of variables method, and many others.

Order and degree

The important concept of holder and degree of partial differential equations is crucial to categorise and understand the concept of partial differential equations. However, the most common partial differential equations are the first and second-order partial differential equations.

The order of a partial differential equation refers to the highest derivative that is present in the partial differential equation.

Suppose the order of the highest derivative present in a partial differential equation is 1, then this partial differential equation is referred to as a first-order partial differential.

Whereas the degree of a partial differential equation is defined as the degree of the highest derivative present in the partial differential equation.

Suppose the highest degree present in the partial differential equation is 1 then it is said that the partial differential equation is of first degree.

Types of partial differential equations

There are four types of ordinary and partial differential equations namely

• First-order partial differential equations
• Linear partial differential equations
• Quasi-linear partial differential equations
• Homogeneous partial differential equations

Let’s understand and discuss these four types of partial differential equations in detail

First order partial differential equations

First order partial differential equation are expressed in the form of

F(x1,…,xm, u,ux1,….,uxm)=0

Linear partial differential equations

a(x,y)∂²w∂x²+2b(x,y)∂²w∂x∂y+c(x,y)∂²w∂y² α(x,y)∂w∂x+β(x,y)∂w∂y+γ(x,y)w+δ(x,y)

Quasi linear partial differential equations

f(x,y,w)∂w/∂x+g(x,y,w)∂w/∂y=h(x,y,w)

Homogeneous partial differential equations

dy/dx = ( x + y )/( x – y)

dy/dx = x ( x – y )/ y²

dy/dx = ( x² + y² )/xy

dy/dx = ( 3x + y )/ ( x – y )

Partial differential equation examples

Now that you have become familiar with the concept of partial differential equations and the different types of partial differential equations, let’s understand the concept of partial differential equations better with a few examples of each type.

First order partial differential equation

F(y (n) (x), y (n-1) (x), . . . , y 0 (x), y(x), x) = 0

Linear partial differential equation

a(x,y)∂²w∂x²+2b(x,y)∂²w∂x∂y+c(x,y)∂²w∂y²= α(x,y)∂w∂x+β(x,y)∂w∂y+γ(x,y)w+δ(x,y)
Quasi-linear partial differential equation

x(∂u)/(∂x))+y(∂u/∂y)=u

Homogeneous partial differential equation

dy/dx = ( x³ + y³)/ ( xy² + yx²)

Conclusion

Partial differential equations are defined as the equations that have a function with various unknown variables and their respective partial derivatives. A partial differential equation offers help in relating a function with several variables to their partial derivatives. There are several uses of partial differential equations that include the study of various phenomena present in nature like fluid flow, waves, heat, and sound. Partial differential equations represent problems consisting of several unknown functions with variables both independent and dependent, with the partial derivatives of this function concerning the independent variables present in the equation.