A Partial Differential Equation commonly denoted as PDE is a differential equation containing partial derivatives of the dependent variable (one or more) with more than one independent variable. A PDE for a function u(x_{1},……x_{n}) is an equation of the form
The PDE is said to be linear if f is a linear function of u and its derivatives. The simple PDE is given by;
∂u/∂x (x,y) = 0
The above relation implies that the function u(x,y) is independent of x which is the reduced form of partial differential equation formula stated above. The order of PDE is the order of the highest derivative term of the equation.
Representation of Partial Differential Equation
In PDEs, we denote the partial derivatives using subscripts, such as;
In some cases, like in Physics when we learn about wave equations or sound equation, partial derivative, ∂ is also represented by ∇(del or nabla).
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. There are threetypes of secondorder PDEs in mechanics. They are
 Elliptic PDE
 Parabolic PDE
 Hyperbolic PDE
Consider the example, au_{xx}+bu_{yy}+cu_{yy}=0, u=u(x,y). For a given point (x,y), the equation is said to be Elliptic if b^{2}ac<0 which are used to describe the equations of elasticity without inertial terms. Hyperbolic PDEs describe the phenomena of wave propagation if it satisfies the condition b^{2}ac>0. For parabolic PDEs, it should satisfy the condition b^{2}ac=0. The heat conduction equation is an example of a parabolic PDE.
FirstOrder Partial Differential Equation
In Maths, when we speak about the firstorder partial differential equation, then the equation has only the first derivative of the unknown function having ‘m’ variables. It is expressed in the form of;
F(x_{1},…,x_{m}, u,u_{x1},….,u_{xm})=0
Partial Differential Examples
Some of the examples which follow secondorder PDE is given as
Linear Partial Differential Equation
If the dependent variable and all its partial derivatives occur linearly in any PDE then such an equation is called linear PDE otherwise a nonlinear PDE. In the above example (1) and (2) are said to be linear equations whereas example (3) and (4) are said to be nonlinear equations.
QuasiLinear Partial Differential Equation
A PDE is said to be quasilinear if all the terms with the highest order derivatives of dependent variables occur linearly, that is the coefficient of those terms are functions of only lowerorder derivatives of the dependent variables. However, terms with lowerorder derivatives can occur in any manner. Example (3) in the above list is a Quasilinear equation.
Homogeneous Partial Differential Equation
If all the terms of a PDE contain the dependent variable or its partial derivatives then such a PDE is called nonhomogeneous partial differential equation or homogeneous otherwise. In the above four examples, Example (4) is nonhomogeneous whereas the first three equations are homogeneous.
Partial Differential Equation Solved Question
Show that if a is a constant ,then u(x,t)=sin(at)cos(x) is a solution to \(\frac{\partial ^{2}u}{\partial t^{2}}=a^{2}\frac{\partial ^{2}u}{\partial x^{2}}\).
Solution
Since a is a constant, the partials with respect to t are
\(\frac{\partial u}{\partial t}=a\cos (at)\cos (x)\) ; \(\frac{\partial^{2} u}{\partial t^{2}}=a^{2}\sin (at)\sin (x)\)Moreover, u_{x} = – sin (at) sin (x) and u_{xx}= – sin (at)cos(x), so that
\(a^{2}\frac{\partial^{2}u }{\partial x^{2}}=a^{2}\sin (at)\cos (x)\)Therefore, u(x,t)=sin(at)cos(x) is a solution to \(\frac{\partial ^{2}u}{\partial t^{2}}=a^{2}\frac{\partial ^{2}u}{\partial x^{2}}\).
Hence proved.
Related Links 

Homogeneous Differential Equation  Linear Equations 
Second Order Differential Equation Solver  First Order Differential Equation 
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