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Maths Formulas

PDE

Ordinary and partial differential equations are very important in calculus. These occur in many applications. An ordinary differential equation is the special case of a partial differential equation. But it is much more complicated with partial differential equations because the functions for which we are looking at are functions of more than one independent variable. And in this article, we will discuss the partial differential equation in the short PDE. Let us start learning.

PDE

What is PDE?

A partial differential equation is an equation involving two or more independent variables. Also with an unknown function and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved.

A particular solution to a partial differential equation is a function that solves the equation. Also, it is turned into an identity when substituted into the equation.  Its solution is called general if it contains all particular solutions of the equation concerned.

The exact solution term is often used for second- and higher-order nonlinear PDEs to denote a particular solution. Thus aid the solution of physical and other problems involving the functions of many variables. Some application areas are the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

Explanation

A Partial Differential Equation contains partial derivatives of one or more dependent variable with more than one independent variable. We can show a PDE for a function

\(u(x_1,x_2,x_3,……,x_n)\)

in the form:

\(f(x_1,x_2,…;u\), \(\frac{\partial u}{\partial x_1}\), \(\frac{\partial u}{\partial x_2}\),…..\(\frac{\partial ^2u}{\partial x_1}\), \(\frac{\partial ^2u}{\partial x_2}\),…..) = 0

And the PDE will be linear if f is a linear function of u and its derivatives. We can write the simple PDE as,

\(\frac{\partial u}{\partial x}\) (x,y)= 0

The above relation implies that the function u(x,y) is independent of x and it is the reduced form of above given PDE Formula. The order of PDE is the order of the highest derivative term of the equation.

We denote the partial derivatives using subscripts, such as;

\(u_x\) = \(\frac{\partial u}{\partial x}\)

uxx = \(\frac{\partial^2 u}{\partial x^2}\)

uxy = \(\frac{\partial ^2 u}{\partial y \partial x}\) = \(\frac{\partial }{\partial y}\) \(\frac{\partial  u}{\partial  x}\)

In some cases, for example in Physics when we learn about wave equation or sound equation, partial derivative, ∂ which we can also represent by ∇ i.e. del.

Classification of Partial Differential Equation (PDEs):

Each type of PDE has certain functionalities that help to determine whether a particular finite element approach is appropriate to the problem being described by the PDE. The solution depends on the equation and several variables contain partial derivatives with respect to the variables.

First-Order Partial Differential Equation:

The first-order partial differential equation has only the first derivative of its unknown function having ‘m’ variables. We can represent in the following form of;

\(F(x_1,…,x_m, u,u_x1,….,u_xm)=0\)

So some of the examples of second-order PDE are:

(1) \(\frac{\partial ^2 u}{\partial x^2}\) + \(\frac{\partial ^2 u}{\partial y^2}\)

(2) \((u_xx + u_yy)\) = 0

(3) \(\frac{\partial ^2 u}{\partial x^2}\) + \(\frac{\partial ^2 u}{\partial y^2}\) + (\(\frac{\partial ^2 u}{\partial x\partial y})^2\) = \(x^2\) + \(y^2\)

Some of the examples of second-order PDE are:

PDE

(Source: MathsisFun.com)

Linear Partial Differential Equation

If the dependent variable and all its partial derivatives occur linearly in any PDE then such an equation is linear PDE otherwise a nonlinear partial differential equation. In the above example (1) and (2) are linear equations whereas example (3) and (4) are non-linear equations.

Solved Examples

Q: What is Quasilinear PDE?

Ans: The partial differential equation is quasilinear if we can express it in the form below:

a(x,y,u(x,y)) \(\frac{\partial u(x,y)}{\partial x}\) + b(x,y,u(x,y)) \(\frac{\partial u(x,y)}{\partial y}\) = c(x,y,u(x,y))

It is a kind of first-order partial differential equation.

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