Approximating Solutions for Differential Equations
A differential equation is defined as an equation which relates an unknown function to one or more derivatives. When solved and transformed into its original equation in the form f(x), an exact value can be found at any given point. While some differential equations can be solved, it is important to realize that very few differential equations that come from "real world" problems can be solved explicitly, and often it is necessary to resort to numerical integration for their solutions. For the exploration I will be using an example in which a differential equation is used in the real world, specifically involving Newton's Law of Cooling. To approximate values at various points of the original equation (Which will be able to be found analytically for means of having the exact values to compare to the approximations. For purposes of the exploration, however, we will assume that the differential equation cannot be solved and we must thus resort to numerical methods), Euler's method will be used and compared with other methods to evaluate how accurate each one is when compared to the true value that is being found. Euler's method, being the earliest discovered approach to approximating solutions for differential equations, is an easy, yet rather inaccurate method when compared to more newly discovered methods that differ in their solving processes. I aim to start with Euler's method, and go on to using other methods in order of increasing accuracy for the same example.
Having taken a calculus class two years ago, I was introduced to topics which I either enjoyed or developed a deep hatred towards. From integration by parts to related rates, the one topic that I caught on to the be...
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..., an almost perfect approximation, but with the cost of requiring a great deal of calculations. I can only see this method being used if one has technology on hand, or desires high accuracy and is willing to put in the extra work to get it. Heun's method seems to be the most practical method to use in real-life situations, as it presents a happy medium between accuracy and procedural proccess. Euler's method is really only useful for means of introducing the concept of approximation methods in a classroom context, which is how I had learned about them. However, this method must be given credit, as it was the first one invented, providing a basis for the improved approximation methods. This can be seen by the way that both Heun's method and the Runge-Kutta method incorporate Euler's method in their procedure, and expand on it further to provide a more accurate result.
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.
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Euler’s method is, “pretty old,” and it was, “developed by Leonhard Euler (pronounced oy-ler), a prolific Swiss mathematician who lived 1707-1783,” it was a great solution to predict the force of gravity would have on the soon to be spaceship ascending into space (Meyers, 2017). Since this proposed a solution to the given issue of safety winning the world wide space race, “Katherine Johnson and other women of NASA [use Euler’s method] to send astronauts [to space],” the boldness that these courageous women use to overcome their task is simply astounding. On top of their perseverance for the mathematical aspect of the space race the scientist, “Johnson and engineer Ted Skopinski” use physics and math/science methods to determine the exact place the astronaut would be if he fell to the ocean so that the navy could find and hopefully rescue him (Meyers, 2017). Skopinski and Johnson also use their like mentality to come to the development that the earth is, “not a perfect sphere, as assumed in idealized orbital mechanics calculations, but bulges slightly in the middle, like a squashed ball, which causes the capsule's orbit to shift slightly over time,” (Meyers,
To investigate the relationship between the air pressure in a ball and the bounce height of that ball where the drop height (gravitational potential energy), temperature and location are kept constant.
Zervakis, Peter, 2006. “Differentiated Integration”: An Alternative Path to Classical Integration?, in: The Federalist (Paria), 48 (2006) 3, 205-213.
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:
Some of the fluid flow, heat flow and particle flow problems are the most complex problems. It is difficult to find the analytic solution of these problems due to their extra-ordinary complexity. Scientists and engineers are facing this problem from decades and then they turned to the numerical methods. By developing increasingly better numerical schemes, researchers have been able to solve increasingly more complex flow problems [11]. The heat diffusion equations are used to describe the flow of heat. In chapter 2 we use Nodal Integral Method to solve the heat diffusi...
The argument in this paper that even though the onus of the discovery of calculus lies with Isaac Newton, the credit goes to Leibniz for the simple fact that he was the one who published his works first. Appending to this is the fact that the calculus wars that ensue was merely and egotistic battle between humans succumbing to their bare primal instincts. To commence, a brief historical explanation must be given about both individuals prior to stating their cases.
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.
Integration and differentiation are systems that stand for general dialectical processes of constant and change. An integrated system is which an individual part is that cover the system true form successfully interconnected and reinforced. To integrate an item means to organize and incorporate different parts creatively breaking the rules to somehow make something that originally separate work great together. The differentiate system has a unique function that cannot be changed or molded. To differentiate is to be bias toward different parts that are unique to themselves.
According to Newton’s Second Law of Motion, the vector sum of the total forces in a system is equal to the product of the mass (m) and acceleration ( a ) of the system.
In my previous studies, I have covered all the four branches of mathematics syllabus and this has made me to develop a strong interest in pure mathematics and most importantly, a very strong interest in calculus.
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.
This evaluation has not only allowed me explore calculus more in depth, but also physics, and the way the world works. This has personally allowed me to explore the connections between math and real-world situations, which is hard to find in textbooks.
The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world. A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest richness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof.