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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. LaplaceEquation 2. Poisson Equation (1812) 3. The diffusion or heat flow Equation (Joseph Fourier1822) 4. Schrodinger Wave Equation(1925)
- 3. INTRODUCTION: What is Partial DifferentialEquation? 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.
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- 6. POISSON’S EQUATION: Itis 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. NEWTONIANGRAVITY. HYDRODYNAMICS. DIFFUSION.
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- 14. INTRODUCTION OF HEATEQUATION 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 ORHEAT 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 onBrownian motion Particle diffusion Schrodinger equation for a free particle Thermal diffusivity in polymers
- 18. BACKGROUND 1.Particle Aspectof 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 Schrodingerequation 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 energyEigenstates 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
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- 25. A harmonic oscillatorin 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 OFTHE 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 Equationshelp 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