Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives. PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, 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.

1.  Group members:
• MUHAMMAD SHAWAIZ KHAN
• SAIF UL ISLAM
• ABDUL QAYYUM
• WASEEM ZAHID
Govt. College University Lahore

2. PARTIAL DIFFERENTIALS
EQUATIONS
1. Laplace Equation
2. Poisson Equation (1812)
3. The diffusion or heat flow Equation (Joseph Fourier1822)
4. Schrodinger Wave Equation(1925)

3. INTRODUCTION:
What is Partial
Differential Equation?
Partial Differential Equation is a differential equation that contains
unknown multivariable functions and their partial derivatives. They
can be used to describe a wide variety of phenomena like Heat,
Sound, Diffusion, Electrostatic, Electrodynamics ,Fluid Dynamics ,
Elasticity and Quantum Mechanics.

5. “PDEs’’
POISSON
EQUATION
SCHROEDING
ER WAVE
EQUATION

6. POISSON’S EQUATION:
 It is partial differential of the form:
ˆ2U= f(x,y,z)
Here, “U” represents the physical quantities in a region containing
mass, electric charge or source of heat or fluid. While f(x,y,z) is
the source density.
For example:
In electrostatics Poisson’s equation is written as:
ˆ2 ɸ = ƿ/€
 It is named after French mathematician ,geometer and
physicist, Simeon Poisson.

7. APPLICATIONS:
 ELECTROSTATICS.
 NEWTONIAN GRAVITY.
 HYDRODYNAMICS.
 DIFFUSION.

12. YYYY

14. INTRODUCTION OF HEAT EQUATION
 In physics and mathematics, the heat equation is a partial
differential equation that describes how the distribution of some
quantity (such as heat) evolves over time in a solid medium, as it
spontaneously flows from places where it is higher towards
places where it is lower.
 It is a special case of the diffusion equation.
 This equation was first developed and solved by Joseph Fourier
in 1822 to describe heat flow
 He used a technique of separating variables t solve heat
equation.
 He used fourier series by using boundary condition.

15. THE DIFFUSION OR HEAT FLOW EQUATION
 ∇2u = (1α2)∂u∂t
 Here u may be the non-steady-state temperature (that is,
temperature varying with time) in a region with no heat
sources; or it may be the concentration of a diffusing
substance (for example, a chemical, or particles such as
neutrons). The quantity α^2 is a constant known as the
diffusivity.

16. APPLICATIONS
 Application on Brownian motion
 Particle diffusion
 Schrodinger equation for a free particle
 Thermal diffusivity in polymers

18. BACKGROUND
 1.Particle Aspect of Radiation
 1.1 Blackbody Radiation
 1.2 Photoelectric Effect
 1.3 Compton Effect
 1.4 Pair Production
 2. Wave Aspect of Particles
 2.1 de Broglie’s Hypothesis: Matter Waves
 2.2 Davisson–Germen Experiment
 2.3 Thomson Experiment

21. IMPORTANCE
 The Schrodinger equation is used to find the allowed
energy levels of quantum mechanical systems.
 The associated wave-function gives the probability of finding
the particle at a certain position.
 The solution to this equation is a wave that describes the
quantum aspects of a system.

22. PROPERTIES
 Real energy Eigenstates
 Quantization
 Consistency with the de Broglie relations
 Non relativistic quantum mechanics

23. THE SCHRÖDINGER PICTURE
 The Schrödinger picture is useful when describing
phenomena with time-independent Hamiltonians
 The Schrödinger picture in which state vectors depend
explicitly on time, but operators do not

24. Applications

25. A harmonic oscillator in
classical mechanics (A–B) and
quantum mechanics (C–H). In
(A–B), a ball, attached to
a spring, oscillates back and
forth. (C–H) are six solutions to
the Schrödinger Equation for
this situation. The horizontal
axis is position, the vertical
axis is the real part (blue) or
imaginary part (red) of
the wave function. Stationary
states, or energy eigenstates,
which are solutions to the time-
independent Schrödinger
equation, are shown in C, D, E,
F, but not G or H.

26. TIME EVOLUTION OF THE SYSTEM’S STATE
1. Time Evolution Operator
2. Stationary States: Time-Independent Potentials
3. Schrödinger Equation and Wave Packets
4. The Conservation of Probability
5. Time Evolution of Expectation Values

27. CONCLUSION
 Laplace Equations help us to understand electromagnetic
theory
 Diffusion equation is like the schrodinger wave equation.
 Wave equation plays vital in order to understand the process of
reflection , transition, interference.
 Maxwell Derive find out the speed of light by using the wave
equation .
 By using the wave equation, laws of optics i.e. law of reflection,
law of incidence , snell’s law, were derived.
 Television
 Telephones