Classification of Partial Differential Equation
There is a linear second-order partial differential equation of second degree given as Auxx + 2Buxy + Cuyy + constant = 0. Its discriminant is B2 – AC. On the basis of different values of such discriminant, the partial differential equations can be classified as follows,
- Parabolic PDEs
- Hyperbolic PDEs
- Elliptic PDEs
Below are the classification of Partial Differential Equation.
Parabolic PDE
Such partial equations whose discriminant is zero, i.e., B2 – AC = 0, are called parabolic partial differential equations. These types of PDEs are used to express mathematical, scientific as well as economic, and financial topics such as derivative investments, particle diffusion, heat induction, etc.
Hyperbolic PDE
Such partial equations whose discriminant exceeds zero, i.e., B2 – AC > 0, are called hyperbolic partial differential equations. These types of PDEs are used to express wave progressions and other such concepts and fundamentals which pertain to waves.
Elliptic PDE
Such partial equations whose discriminant is less than zero, i.e., B2 – AC < 0, are called elliptic partial differential equations. The most common example of an elliptic PDE is the Laplace equation.
Partial Differential Equation
Partial Differential Equation contains an unknown function of two or more variables and its partial derivatives with respect to these variables. The order of partial differential equations is that of the highest-order derivatives. Such equations aid in the relationship of a function with several variables to their partial derivatives. They are extremely important in analyzing natural phenomena such as sound, temperature, flow properties, and waves.
They are used to express issues that include an unknown function with numerous dependent and independent variables, as well as the second derivative of this function with respect to the independent variables.
In this article, we will learn the definition of Partial Differential Equations, their representation, their order, the types of partial differential equations, how to solve PDE, and many more details.
Table of Content
- What is a Partial Differential Equation?
- Partial Differential Equation Definition
- Order of Partial Differential Equation
- Examples of Partial Differential Equations
- Degree of Partial Differential Equation
- General Form of Partial Differential Equation
- Representing Partial Differential Equation
- Types of Partial Differential Equations
- First-Order Partial Differential Equation
- Second-Order Partial Differential Equation
- Quasi-Linear Partial Differential Equation
- Homogeneous Partial Differential Equation
- Classification of Partial Differential Equation
- Applications of Partial Differential Equations
- How to Solve Partial Differential Equations
- Steps for Solving Partial Differential Equations
- Partial Differential Equation Class 12
- Partial Differential Equations Examples
- Practice Questions on Partial Differential Equations
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