Maths partial differential equation Poster

1. Partial Differential Equation [PDE]
Department Of Electronics and Telecommunication | 2nd Yr. 3rd Sem. (C) | Subject- Maths. | Submitted By:- Ashish Pandey (30) | Mandar Muley (33)
Introduction
Partial differential equations (PDEs) are equations that involve rates of change with respect
to continuous variables. The position of a rigid body is specified by six numbers, but the
configuration of a fluid is given by the continuous distribution of several parameters, such as
the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-
dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional
configuration space. This distinction usually makes PDEs much harder to solve than ordinary
differential equations (ODEs), but here again there will be simple solutions for linear
problems. Classic domains where PDEs are used include acoustics, fluid
flow, electrodynamics, and heat transfer.
A partial differential equation (PDE) for the function is an equation of the form
Application PDE in Wave Equations
The simplest situation to give rise to the one-dimensional wave equation is the motion of
a stretched string - specifically the transverse vibrations of a string such as the string of a
musical instrument. Assume that a string is placed along the x−axis, is stretched and then
fixed at ends x = 0 and x = L; it is then deflected and at some instant, which we call t = 0, is
released and allowed to vibrate. The quantity of interest is the deflection u of the string at
any point x, 0 ≤ x ≤ L, and at any time t > 0. We write u = u(x, t). The diagram shows a
possible displacement of the string at a fixed time t.
Application of PDE in Line Equations
In a long electrical cable or a telephone wire both the current and voltage depend upon
position along the wire as well as the time. It is possible to show, using basic laws of
electrical circuit theory, that the electrical current i(x, t) satisfies the PDE.
PDEs can be used to describe a wide variety of phenomena such
as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.
These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just
as ordinary differential equations often model one-dimensional dynamical systems, partial
differential equations often model multidimensional systems. PDEs find their generalisation
in stochastic partial differential equations.