Differential equation has its application in different area of knowledge of mankind. A few such examples are: the motion of a projectile, rocket, planet or satellite, the charge or current in an electric circuit, the reactions of chemicals, the rate of growth of a population, spring mass systems, bending of beams, the conduction of heat in a rod or in a slab etc. The mathematical formulations of all of the problems give rise to differential equations. Basically, most of the differential equations involving physical phenomena are nonlinear. It is simple to solve the differential equations which are linear but the solution of nonlinear equations are laborious and in some cases it is impossible to solve them analytically. In such circumstances …show more content…
But sometimes, such a linearization may lead to real errors not only of a quantitative but also of a qualitative nature. And when the linearization is not possible, the original nonlinear equation itself must be treated. With the discovery of numerous phenomena of self-excitation of circuits containing nonlinear capacitor of electricity, like electron tubes, gaseous discharge etc. and in many cases of nonlinear mechanical vibrations of special types, the method of small oscillations becomes inadequate for their analytical treatment[*]. To this end, a number of methods are developed, such as straight forward expansion method, Lindtsteadt-Poincare (LP) method, Modified LP method, Struble technique, Harmonic balance method, the Krylov-Bogoliubov-Mitropolskii (KBM) perturbation method …show more content…
At present the method is used to obtain oscillatory as well as damped oscillatory, critically damped, over damped, near critically damped, more critically damped solutions of second, third, fourth etc. order nonlinear differential equations by imposing some precise conditions to make the solutions uniformly valid. The method of Krylov and Bogoliubov is called an asymptotic method in the sense An asymptotic series itself may not be convergent but for a fixed number of terms, the approximate solution tends to the exact solution as is very near to zero. Perturbation methods have recently been received much attention for investigating solutions of dynamic, stochastic and economic equilibrium models, both single-agent or rational-expectations models and multi-agent or game-theoretic models. A perturbation method is based on the attribute that the equations to be solved are sufficiently “smooth” or sufficiently differentiable in the required regions of variables and parameters. First of all, Van der Pol concentrated to the self-excited oscillations and indicated that their existence is natural and the systems are governed by nonlinear differential equations. This nonlinearity appears, thus as the very essence of these phenomena and by linearizing the differential equation in the sense of the method of small oscillations, one simply eliminates
In a simple regression model, we are trying to determine if a variable Y is linearly dependent on variable X. That is, whenever X changes, Y also changes linearly. A linear relationship is a straight line relationship. In the form of an equation, this relationship can be expressed as
“Hell-raiser, razor-feathered risers, windhover over Peshawar,”(Majmudar Lines 1-3) That was the first stanza of the poem “Ode to a Drone” by Amit Majmudar. This poem is written by a Muslim American who grew up in Cleveland, Ohio. The first stanza is the beginning to an ode to a drone, but immediately, you know he is not praising the drone for being powerful, he is explaining how it is unnecessarily destructive. A drone is an unmanned flying vehicle that shoots missiles or drops bombs on targets in modern warfare. Modern warfare today is taking place in the Middle East which is a transcontinental region centered on Western Asia and Egypt. Many terrorist groups such as ISIS, which is the largest threat right now.
It teaches us to expect the unexpected. A famous example of chaos theory, referred to as the "butterfly effect, “postulates that the beat of a butterfly's wing could trigger a breath of breeze
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
Kate wanted a relationship for a long time - and finally, she had met Aaron and they started dating. They went on a couple of dates and had a great time together. Aaron recently graduated from university, so he wanted to take a few months off, before applying for a steady job. This change also gave Aaron an opportunity to shake off some of his romantic dust and find new ways for Kate and him to spend time together. At first, Kate was delighted with Aaron’s sense of romance and creativity, but as time went by, she barely found time for herself anymore. Kate didn 't want to hurt Aaron so she “played along”, and only a few weeks later (after she had tried to find *ANY* way to keep him busy…) Kate finally decided to bring up the subject. Aaron
Typical analysis such as analytical analysis (PDE, ODE etc), numerical analysis (finite element analysis) might not be so attractive in the settings of my research problem above. One of the big disadvantages for typical analysis in my research problem is that typical analysis is computationally intractable due to the high dimensions in the above problems. Secondly, typical analysis is usually not as flexible and adaptable as Monte Carlo simulations. Some analytical analysis such as PDE does not even fit the setting of the above research problems, as it is empirical by its natural.
It is interesting to note that the ongoing controversy concerning the so-called conflict between Wilhelm Gottfried Leibniz and Isaac Newton is one that does not bare much merit. Whether one came up with the concepts of calculus are insignificant since the outcome was that future generations benefited. However, the logic of their clash does bear merit.
Born August 13, 1883 in Gibbon, Nebraska, Edwin H. Sutherland grew up and studied in Ottawa, Kansas, and Grand Island, Nebraska. After receiving his B.A degree from Grand Island College in 1904, he taught Latin, Greek, History, and shorthand for two years at Sioux Falls College in South Dakota. In 1906 he left Sioux Falls College and entered graduate school at the University of Chicago from which he received his doctorate. (Gaylord, 1988:7-12) While attending the University of Chicago he changed his major from history to sociology. Much of his study was influenced by the Chicago approach to the study of crime that emphasized human behavior as determined by social and physical environmental factors, rather than genetic or personal characteristics. (Gaylord, 1988:7-12)
Second, the terza rima scheme helps the narrator to express his thoughts. In A Defense of Poetry, Shelley states that there exists harmony between the language that poets employ and the sounds that are contained in each word because both sounds and thoughts are intertwined to convey the message that they attempt to represent (763). In other words, there exists a close proximity between the sense of words and their sound; it is the enchainment of both ideas and sounds that creates an effect of harmony. Thus, language and sound work in conjunction to create a stylized and harmonious message which comes to life each time the poem is either read or recited out loud. We mentioned earlier that the Ode to the West Wind is an ode that addresses, that
Newton-Raphson method is of use when it comes to approximating the root or roots of an equation.
Stoichiometry is very used in our daily life, for example cooking. Just imagine that you really want cookies, but you are almost out of sugar, this is where stoichiometry pops in, you have to figure out how much of the other ingredients you will need compared to the amount of sugar that you have.
Differential analysis is useful in many situations faced by the management and it has to choice between different alternatives for each situation to make the necessary decision. Some of these situations are as follows:
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
In this assessment of the projectile motion of an object, I found that it can be applied to many useful situations in our daily lives. There are many different equations and theorems to apply to an object in motion to either find the path of motion, the displacement, velocity, acceleration, and time of the object in the air.