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Partial Differential Equations

Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations.

Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.

1.	What are Partial Differential Equations?
2.	Partial Differential Equations Formula
3.	Order and Degree of Partial Differential Equations
4.	Partial Differential Equations Types
5.	Partial Differential Equations Classification
6.	Solving Partial Differential Equations
7.	Partial Differential Equations Applications
8.	FAQs on Partial Differential Equations

What are Partial Differential Equations?

Partial differential equations are abbreviated as PDE. These equations are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables.

Partial Differential Equations Definition

Partial differential equations can be defined as a class of differential equations that introduce relations between the various partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables. An equation that can solve a given partial differential equation is known as a partial solution.

Partial Differential Equations Example

An example of a partial differential equation is \(\frac{\partial^2 u}{\partial t^2} = c^{2}\frac{\partial^2 u}{\partial x^2}\). This is a one dimensional wave equation.

Partial Differential Equations Examples

Partial Differential Equations Formula

Partial differential equations can prove to be difficult to solve. Hence, there are certain techniques such as the separation method, change of variables, etc. that can be used to get a solution to these equations. The general formulas for partial differential equations are given below:

First-Order Partial Differential Equations: \(F\left ( x_{1}, x_{2},...,x_{n}, w,\frac{\partial w}{\partial x_{1}},\frac{\partial w}{\partial x_{2}},...,\frac{\partial w}{\partial x_{n}} \right ) = 0\). Here, w = (\(x_{1}\), \(x_{2}\), ...,\(x_{n}\)) is the unknown function and F is the given function.
Second-Order Partial Differential Equations: The general formula of a second-order PDE in two variables is given as \(a_{1}\)(x, y)\(u_{xx}\) + \(a_{2}\)(x, y)\(u_{xy}\) + \(a_{3}\)(x, y)\(u_{yx}\) + \(a_{4}\)(x, y)\(u_{yy}\) + \(a_{5}\)(x, y)\(u_{x}\) + \(a_{6}\)(x, y)\(u_{y}\) + \(a_{7}\)(x, y)u = f(x, y).

In the following section, we will learn more about the types of partial differential equations

Order and Degree of Partial Differential Equations

Order and degree of partial differential equations are used to categorize partial differential equations. The most commonly used partial differential equations are of the first-order and the second-order.

Order of Partial Differential Equations

Order of a partial differential equation can be defined as the order of the highest derivative term that occurs in the PDE. Suppose a partial differential equation is given as \(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = z + xy\). As the order of the highest derivative is 1, hence, this is a first-order partial differential equation.

Degree of Partial Differential Equations

The degree of a partial differential equation is the degree of the highest derivative in the PDE. The partial differential equation \(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = z + xy\) will have the degree 1 as the highest derivative is of the first degree.

Partial Differential Equations Types

Partial differential equations can be broadly divided into 4 types based on the order of the partial derivatives as well as the nature of the equation. These are given below:

First-Order Partial Differential Equations

Partial differential equations where the highest partial derivatives of the unknown function are of the first order are known as first-order partial differential equations. If the equation has n number of variables then we can express a first-order partial differential equation as F (\(x_{1}\), \(x_{2}\), ...,\(x_{n}\), \(k_{x_{1}}\), ...\(k_{x_{n}}\)). First-order PDEs can be both linear and non-linear. A linear partial differential equation is one where the derivatives are neither squared nor multiplied.

Second-Order Partial Differential Equations

Second-order partial differential equations are those where the highest partial derivatives are of the second order. Second-order PDEs can be linear, semi-linear, and non-linear. Linear second-order partial differential equations are easier to solve as compared to the non-linear and semi-linear second-order PDEs. The general formula for a second-order partial differential equation is given as \(au_{xx}+bu_{xy}+cu_{yy}+du_{x}+eu_{y}+fu = g(x,y)\). Here, a, b, c, d, e, f, and g are either real-valued functions of x and/or y or they are real constants.

Quasi Linear Partial Differential Equations

In quasilinear partial differential equations, the highest order of partial derivatives occurs, only as linear terms. First-order quasi-linear partial differential equations are widely used for the formulation of various problems in physics and engineering.

Homogeneous Partial Differential Equations

A partial differential equation can be referred to as homogeneous or non-homogeneous depending on the nature of the variables in terms. The partial differential equation with all terms containing the dependent variable and its partial derivatives is called a non-homogeneous PDE or non-homogeneous otherwise.

Partial Differential Equations Classification

Suppose we have a linear second-order PDE of the form A\(u_{xx}\) + 2B\(u_{xy}\) + C\(u_{yy}\) + other lower-order terms = 0. Then the discriminant of such an equation will be given by B² - AC. Using this discriminant, second-order partial differential equations can be classified as follows:

Parabolic Partial Differential Equations: If B² - AC = 0, it results in a parabolic partial differential equation. An example of a parabolic partial differential equation is the heat conduction equation.
Hyperbolic Partial Differential Equations: Such an equation is obtained when B² - AC > 0. The wave equation is an example of a hyperbolic partial differential equation as wave propagation can be described by such equations.
Elliptic Partial Differential Equations: B² - AC < 0 are elliptic partial differential equations. The Laplace equation is an example of an elliptic partial differential equation.

Classification	Canonical Form	Type	Example
b² - ac > 0	\(\frac{\partial^{2} u}{\partial \xi \partial\eta } + ... = 0\)	Hyperbolic Partial Differential Equation	Wave propagation equation
b² - ac = 0	\(\frac{\partial^{2} u}{ \partial\eta^{2} } + ... = 0\)	Parabolic Partial Differential Equation	Heat conduction equation
b² - ac < 0	\(\frac{\partial^{2} u}{ \partial\alpha ^{2} } +\frac{\partial^{2} u}{ \partial\beta ^{2} }+ ... = 0\)	Elliptic Partial Differential Equation	Laplace equation

Solving Partial Differential Equations

There can be many methods that can be used to solve a partial differential equation. Suppose a partial differential equation has to be obtained by eliminating the arbitrary functions from an equation z = yf(x) + xg(y). The steps to do so are as follows:

Step 1: Differentiate both sides with respect to x and y.

\(\frac{\partial z}{\partial x}\) = yf'(x) + g(y) ---(1)

\(\frac{\partial z}{\partial y}\) = f(x) + xg'(y) ---(2)

Step 2: Now differentiate (1) w.r.t to y and (2) w.r.t x.

\(\frac{\partial^{2} z}{\partial x\partial y}\) = f'(x) + g'(y)

Step 3: Multiply the first equation by x and the second equation by y then add the resultant.

x\(\frac{\partial z}{\partial x}\) + y\(\frac{\partial z}{\partial y}\) = xg(y) + yf(x) + xy(f'(x) + g'(y))

= z + xy(f'(x) + g'(y))

Substituting from step 2 we get,

x\(\frac{\partial z}{\partial x}\) + y\(\frac{\partial z}{\partial y}\) = z + xy(\(\frac{\partial^{2} z}{\partial x\partial y}\))

The general, particular or singular solution can be determined for this equation by using various methods such as change of variables, substitution, etc.

Partial Differential Equations Applications

Partial differential equations are widely used in scientific fields such as physics and engineering. Some applications of partial differential equations are given below:

Partial differential equations are used to model equations to describe heat propagation. The equation is given by \(u_{xx}\) = \(u_{t}\)
Propagation of light and sound is given by the wave equation. This equation is a second-order partial differential equation and is given by \(u_{xx}\) - \(u_{yy}\) = 0.
The Black-Scholes equation is another important second-order partial differential equation that is used to construct financial models.

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Important Notes on Partial Differential Equations

A partial differential equation is an equation consisting of an unknown multivariable function along with its partial derivatives.
There a broadly 4 types of partial differential equations. These are first-order, second-order, quasi-linear partial differential equations, and homogeneous partial differential equations
Second-order partial differential equations can be classified into three types - parabolic, hyperbolic, and elliptic.

Examples on Partial Differential Equations

Example 1: Find the partial differential equation of u = f(x² - y²).
Solution: Differentiate w.r.t x and y.
\(\frac{\partial u}{\partial x}\) = 2x . f'(x² - y²)
\(\frac{\partial u}{\partial y}\) = -2y . f'(x² - y²)
\(\frac{\partial u}{\partial x}\) / \(\frac{\partial u}{\partial y}\) = -x / y
Thus, the differential equation is given as y.\(\frac{\partial u}{\partial x}\) + x.\(\frac{\partial u}{\partial y}\) = 0.
Answer: y.\(\frac{\partial u}{\partial x}\) + x.\(\frac{\partial u}{\partial y}\) = 0.
Example 2: Find the partial differential equation of all spheres that have their centers in the x-y plane and have a fixed radius. [Hint: use the equation (x - a)² + (y - b)² + z² = r²]
Solution: Differentiate w.r.t x and y.
2z \(\frac{\partial z}{\partial x}\) = -2(x - a)
2z \(\frac{\partial z}{\partial y}\) = -2(y - a)
(x - a) = -z\(\frac{\partial z}{\partial x}\), (y - a) = -z\(\frac{\partial z}{\partial y}\)
Substituting these values in the given equation we get the partial differential equation as,
z² = \(\frac{r^{2}}{(\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y})^{2} + 1}\)
Answer: z² = \(\frac{r^{2}}{(\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y})^{2} + 1}\)
Example 3: Given p(x, t) = sin(bt)cosx, prove \(\frac{\partial^2 p}{\partial t^2} = b^{2}\frac{\partial^2 p}{\partial x^2}\)

Solution: \(\frac{\partial p}{\partial t} = bcos(bt)cos(x)\)

\(\frac{\partial^2 p}{\partial t^2} = -b^{2}sin(bt)cos(x)\)

\(\frac{\partial p}{\partial x} = -sin(bt)sin(x)\)

\(\frac{\partial^2 p}{\partial x^2} = -sin(bt)cos(x)\)

\(b^{2}\frac{\partial^2 p}{\partial x^2} = -b^{2}sin(bt)cos(x)\) = \(\frac{\partial^2 p}{\partial t^2}\)

Hence, proved.

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Practice Questions on Partial Differential Equations

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FAQs on Partial Differential Equations

What are Partial Differential Equations?

Partial differential equations can be defined as differential equations that consist of an unknown function, with several dependent and independent variables as well as their partial derivatives.

What is the Formula for Partial Differential Equations?

The general form of a first-order partial differential equation is given as F (\(x_{1}\), \(x_{2}\), ...,\(x_{n}\), \(k_{x_{1}}\), ...\(k_{x_{n}}\)) while that of a second order PDE is given by \(au_{xx}+bu_{xy}+cu_{yy}+du_{x}+eu_{y}+yu = g(x,y)\).

How to Solve Partial Differential Equations?

There are many methods available to solve partial differential equations such as separation method, substitution method, and change of variables. Depending upon the question these methods can be employed to get the answer.

Are Partial Differential Equations Linear?

All partial differential equations may not be linear. There can be semi-linear and non-linear partial differential equations also.

What are the Types of Partial Differential Equations?

The types of partial differential equations are listed below:

First-order partial differential equations
Second-order partial differential equations
Quasi-linear partial differential equations
Homogeneous partial differential equations

What are Ordinary Differential Equations and Partial Differential Equations?

Ordinary differential equations (ODE) are equations that contain differentials with respect to one variable only. Partial differential equations have partial derivatives with respect to several independent variables. ODE are a subclass of PDE.

What are the Applications of Partial Differential Equations?

Partial differential equations are widely used in engineering and physics to model natural phenomena such as heat transfer, wave propagation, diffusion, and electrostatics.

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