Analytic Fourier Decomposition The force from a machine pres

Analytic Fourier Decomposition. The force from a machine press has an exact, periodic \"saw-toothed\" shape, f(t), repeating every 4 seconds as shown at right, Analytically determine the Fourier series representation of the force (i.e. calculate the values of B_0 A_k and B_k for all k greaterthanorequalto 1). Use MATLAB to reconstruct an approximation of f(t) over one period using only the Fourier terms up to k = 20. Use a finely-spaced time vector of 1,000 points between 0 and 4 seconds (i.e. t=linspace (0, 4, 1000)). Make a plot (called HW8_1.pdf) comparing this approximation off(t) (in red) with the exact saw-toothed f(t) (in black) over one period. Due on-paper: part (a). Due on-line: plot HW8_1.pdf (no script).

Solution

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The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

Physics

Classical mechanics

So long as the force acting on a particle is known, Newton\'s second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton\'s second law to obtain an ordinary differential equation, which is called the equation of motion.

Electrodynamics

Maxwell\'s equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell\'s equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.