ABSTRACT
A partial differential equation is a differential equation that contain unknown multivariable functions and their partial derivatives while ordinary differential equations contains function of a single variables and their derivatives. Therefore, an ordinary differential equation is a special case of partial differential equation but the behaviour of a solution is quite different. It is much more complicated in the case of partial differential equation because it has more than one independent variable. Partial differential equation can be used to describe the various topics in physics such as propagation of sound and wave, heat, electrostatics, Maxwell equation in electromagnetism, elasticity, quantum mechanics, Hamilton-Jacobi theory.
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The independent variables will be denoted by x and y and the dependent variables denoted by z. The partial differentials coefficients are denoted as follows:
∂z/∂x=p ∂z/∂y=q (∂^2 z)/〖∂x〗^2 =r (∂^2 z)/∂x∂y=s (∂^2 z)/〖∂y〗^2
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(∂^2 V)/〖∂x〗^2 =LC(∂^2 V)/〖∂t〗^2
(∂^2 i)/〖∂x〗^2 =LC(∂^2 i)/〖∂t〗^2
These equations are called radio equations.
Where,
V=potential i=current C=capacitance L inductance
IN HEAT FLOW[8]
Heat is the transfer of energy between two bodies. One dimensional heat equation
When the heat flow along the straight lines, then the flow is called one-dimensional.
Let us consider the heat flow in a bar or rod along the x-axis. Consider a rod of homogeneous material of density is ρ (gm⁄〖cm〗^3 ) and having a constant cross-sectional area A (〖cm〗^2 ). Let c be the specific heat and k be thermal conductivity of material. We suppose that the sides of the bar are insulated and the loss of heat from the sides by conduction or radiations is negligible. Take an end of the bar as the origin and the direction of heat flow as the positive x-axis.
Materials of different types will exhibit varied changes in temperature when transferred the same amount of heat. This variation is a result of the difference in properties displayed from one material to another, known as "heat capacity." Every substance has a variable, positive valued heat capacity that represents the amount of heat required to initiate a specific temperature change. (Hechinger, page 1) For ideal gases, there are heat capacities at constant volume and constant pressure given by:
The data which was collected in Procedure A was able to produce a relatively straight line. Even though this did have few straying points, there was a positive correlation. This lab was able to support Newton’s Law of Heating and Cooling.
In section II of this paper, theoretical background relevant to this problem is presented. Section III is a brief summary of the numerical data from Giorgini, Boronat, and Casulleras.
In this equation, Y is the dependent variable, and X is the independent variable. α is the intercept of the regression line, and β is the slope of the regression line. e is the random disturbance term.
The input variables are the ones that I can change in order to affect the experiment and the outcome variables are the ones I will measure to see how the input variable has affected it. Input Variables --------------- Amount of calcium carbonate Amount of hydrochloric acid Surface area of calcium carbonate Concentration of hydrochloric acid Temperature of hydrochloric acid Introduction of a catalyst Outcome variables ----------------- Amount of calcium chloride released Amount of water released Amount of carbon dioxide released Change in weight
good emitter of heat radiation so a lot of heat will be lost to the
Quantum Mechanics developed over many decades beginning as a set of controversial mathematical explanations of experiments that the math of classical mechanics could not explain. It began in the turn of the 20th century, a separate mathematical revolution in physics that describes the motion of things at high speeds. The origins of Quantum Mechanics cannot be credited to any one scientists. Multiple scientists contributed to a foundation of three revolutionary principles that gradually gained acceptance and experiment verification from 1900-1930 (Coolman). Quantum Mechanics is
Part 1. (a) Define each, (b) Explain its significance, (c) where indicated with this symbol * provide an example.
Electromagnetic waves are waves that can propagate even though there is no medium. A magnetic field that changes with time can generate an electric field that also changes with time, and an electric field that changes with time can also produce a magnetic field. If the process is continuous it will produce a magnetic field and electric field continuously. If these magnetic fields and electric fields simultaneously propagate (spread) in space in all directions then this is a symptom of the wave. Such a wave is called an electromagnetic wave because it consists of an electric field and a magnetic field that travels in space.
Independent variables: The temperature of hcl gas will be decreased and increased throughout the experiment.
Cengel, Y. A., & Boles, M. A. (2011). Thermodynamics: An engineering approach (7th ed.). New York, NY: McGraw-Hill.¬¬¬¬
When we place two objects with different temperatures in contact with each other, the heat from the hotter object will immediately and automatically flow to the colder object. This is known as conduction. Some objects make excellent conductors of heat while others make poor conductors of heat or excellent insulators. Silver, copper, and gold make excellent conductors of heat. Foams and plastics make good insulators of heat but make poor conductors. Last night for dinner, I made myself a grilled cheese sandwich and a bowl of tomato soup. I heated the soup faster than I cooked the sandwich so I poured the hot soup into a bowl and finished cooking the sandwich. Once I was done cooking, I gabbed the soup bowl and burned my hand. The heat from the soup made the bowl hot. This is an example of conduction.
As discussed in class, submission of your solutions to this exam will indicate that you have not communicated with others concerning this exam. You may use reference texts and other information at your disposal. Do all problems separately on clean white standard 8.5” X 11” photocopier paper (no notebook paper or scratch paper). Write on only one side of the paper (I don’t do double sided). Staple the entire solution set in the upper left hand corner (no binders or clips). Don’t turn in pages where you have scratched out or erased excessively, re-write the pages cleanly and neatly. All problems are equally weighted. Assume we are working with “normal” pressures and temperatures with ideal gases unless noted otherwise. Make sure you list all assumptions that you use (symmetry, isotropy, binomial expansion, etc.).
Quantum mechanics is a form of physics that is used to study very tiny objects like atoms. Many people have heard of quantum mechanics before whether it was from a book or a television show. Automatically people think “nerd” or “geek” which is pretty much correct, but those people themselves have no idea how quantum mechanics improved their lives or even how it works. It may sound difficult, but it is really not that hard to understand.