Partial Differential Equation

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(x1,……xn) 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 non-linear equations.

Quasi Linear PDE

A PDE is said to be quasi-linear 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 Quasi-linear equation.

Homogeneous PDE

If all the terms of a PDE contains the dependent variable or its partial derivatives then such a PDE is called non-homogeneous partial differential equation or homogeneous otherwise. In the above four examples eqn (4) is non-homogeneous 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 three-types of second-order PDEs in mechanics . They are

  • Elliptic PDE
  • Parabolic PDE
  • Hyperbolic PDE

Consider the example, auxx+buyy+cuyy=0, u=u(x,y). For a given point (x,y), the equation is said to be elliptic if b2-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 b2-ac>0. For parabolic PDEs, it should satisfy the condition b2-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, ux = – sin (at) sin (x) and uxx= – 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.

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Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is?