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
\(f(x_{1},…x_{n};u,\frac{\partial u}{\partial x_{1}},…\frac{\partial u}{\partial x_{n}};\frac{\partial ^{2}u}{\partial x_{1}\partial x_{1}},…\frac{\partial ^{2}u}{\partial x_{1}\partial x_{n}};… )=0\)
The PDE is said to be linear, if f is a linear function of u and its derivatives.The simple PDE is given by
\(\frac{\partial u}{\partial x}(x,y)=0\)
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.
Examples for PDE
Some of the examples which follows second order PDE is given as
 \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}}=0\) …. (1)
 \(u_{xx}+u_{yy}=0\) …..(2)
 \(ux\frac{\partial ^{u}}{\partial x^{2}}+u^{2}xy\frac{\partial ^{2}u}{\partial x\partial y}+uy\frac{\partial ^{2}u}{\partial y^{2}}+\left ( \frac{\partial u}{\partial x} \right )^{2}+\left ( \frac{\partial u}{\partial y} \right )^{2}+u^{3}=0\) ……(3)
 \(\frac{\partial ^{2}u}{\partial x^{2}}+\left ( \frac{\partial ^{2}u}{\partial x\partial y}\right )^{2}+\frac{\partial ^{2}u}{\partial y^{2}}=x^{2}+y^{2}\) …..(4)
In writing partial differential equation, it is common to use subscript notation. That is,
\(u_{x}=\frac{\partial u}{\partial x}\)
\(u_{xx}=\frac{\partial^{2} u}{\partial x^{2}}\)
\(u_{xy}=\frac{\partial^{2} u}{\partial y\partial x}=\frac{\partial }{\partial y}(\frac{\partial u}{\partial x})\)
Linear PDE
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 equations (1) and (2) are said to be linear equations whereas (3) and (4) are said to be nonlinear equations.
Quasi Linear PDE
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 lower order derivatives of the dependent variables. However, terms with lower order derivatives can occur in any manner. Equation (3) in the above list is a Quasilinear equation.
Homogeneous PDE
If all the terms of a PDE contains 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 eqn (4) is nonhomogeneous where as the first three equations are homogeneous.
Types of PDE
Each type of PDE has certain functionalities that helps 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 contains 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.
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.
Register with BYJU’S learning app to get more information about the maths related articles and formulas. start practice with the problems.
Related Links 
