Assignment
Response 1b
From the derived calculations, it is evident that there is a bigger difference between the Euler’s equation solution and the analytical solution as compared to the margin between the Runge-Kutta’s solution and the analytical solution. This shows that Runge-Kutta’s method is more precise. Actually, the difference between Runge-Kutta and analytical method is negligible.
Numerical methods
The numerical methods used in this question are under numerical differentiation and integration. Euler method main advantage is that, being the most basic numerical method of integrating ordinary differential equations (ODEs) it is efficient and easy to understand. The disadvantage however is that, sometimes it is not consistently stable thus resulting in poor approximations. In such cases, some other techniques are preferable such as the fourth-order Runge-Kutta technique. It uses higher order terms of the Taylor series expansion. It is simple to execute and yields more accurate results. However, it is more expensive than Euler integration.
Engineering entails solving of technical and practical problems using scientific and mathematical tools. Often at times, it is not possible to get the exact analytical solutions to these problems. Therefore, approximate solutions are sufficient, permitting the use of numerical methods in engineering applications. Numerical Methods are categorized as: Linearization, Numerical Integration and Differentiation, estimating roots, solving equations of systems, and Optimization.
Linearization involves solving for the linear estimation to a problem at a certain point. Linear estimation uses the first order of Taylor series expansion. Linearization is a technique is used for finding the stability of an equilibrium point of a dynamical structures. Nonlinear equations are more complicated as compared to linear equations. Taylor’s series offers a suitable method of estimating nonlinear function using a linear equation. Linearization example is in calculating forces and motion in swinging pendulum.
Iteration is applied in generating a chain of numbers that concentrate at the root in root finding. It needs a number of random approximations of initial values of the root, then every iteration of the algorithm leads to a progressively more precise estimation of the root. This method does not give the exact answer. Most techniques calculate the values by finding an auxiliary function for the available values. The bar is therefore a stationary point of the auxiliary function. This function is selected because they have the roots of the initial mathematical statement as stationary stations. Also, for concentrating to these points. Application of these method is in calculating statically indeterminate structural analysis.
Optimization is about evaluating the “best available” values of a mathematical function specified the input. It includes various objective functions and varying domains. In optimization problems, it involves maximizing or minimizing a real function through systematically selecting the inputs from a given range and evaluating the value of the function. Optimization techniques make up a huge section of applied mathematics and engineering. Common applications: minimal error, optimal engineering design, and variation principles.
Numerical differentiation approximates the derivative of a mathematical problem while Numerical integration functions can approximate the value of an integral whether or not the functional expression is known. May also be used when an arbitrary continuous function is known only through discretely sampled data points. Numerical Integration example is a Falling Climber when calculating the kinetic energy of the climber given his the force on the rope. An instance in which numerical differentiation is applied is computing for velocity and acceleration in motion studies.
Estimation of solutions of equations is done using numerical methods when algebraic techniques cannot crack the problem. These methods build up progressive estimations that converge to the precise answer to an equation. Runge-Kutta methods are used to compute initial-value problems for ODEs. However, it is limited to equations that have a single variable. Equations that solve for several variable are such as a Quasi-Newton technique and Finite Difference method.
Response 2b: Laplace transform
The Laplace Transform has lots of applications in science and engineering. Inputs and outputs in time domain which are functions of time are transformed to frequency domain. In this domain, inputs and outputs operate in angular frequency. The advantage of Laplace transform is that it solves higher order differential equations also of more than second degree equations. Its limitation is that it can’t be used in nonlinear systems. This theory plays a huge role in analyzing and designing engineering systems. These notions are applied in Electric circuit analysis, Communication engineering, Control engineering and Nuclear physics.
Laplace transform is used in analysis of electronic circuits. Electrical elements linked together form an electric circuit. The basic electric circuit is made up of three units; a battery, a lamp and connecting wires. It has a number of uses example of in a torch. Electrical circuits are applied in many electrical structures to execute different duties. Laplace Transform is vastly applied by engineers to compute ordinary differential equations in analyzing electronic circuits.
Systems are used to process signals to modify or extract information. They are characterized by their input output relations. System modeling is used when converting signals in time-domain and frequency-domain images. The conversion of signals is essential in signal processing. Processing in either domain is important depending on the intended purpose. For instance, when a sine wave is affected by a noise signal, it appears a messy in time domain. While in frequency-domain, the energy is distributed in the spectrum and the sine wave is not interrupted by the noise. Laplace Transform simplifies calculations in system modeling, when a number of differential equations are involved.
In digital signal processing, Laplace transform is employed in solving problems. For instance, the voltage across an inductor is proportional to the derivative of the current through the device. Engineering has lots of such relations. Consistency of frequency and impulse responses are essential to differential equations of these systems. Their impulse responses are made up of exponentials and sinusoids. The Laplace transform is used to analyze these continuous signals.
Detailed studies of dynamic traits of heat transfer is important in practical engineering. When given density, thickness, thermal conductivity and specific heat capacity, it is possible to compute the heat transfer through a model using Laplace transform. This offers an analytical basis for the optimization of performance of multilayered composite materials. Laplace transform may be applied in real-time digital controls of dynamic heat transfer. It has high accuracy and effectively cuts down the experimental charges associate with measurement of temperature.
Laplace Transform when utilized in process controls assists in examining the variables which when interfered with, give suitable output. Majority of control system analysis are established on linear systems theory. So are design methods. Even though it is possible to come up with these steps using state space models, it is way easier use transfer functions. Transfer functions give room to make algebraic alterations instead of using linear differential equations directly. The creation of transfer functions requires Laplace transform
Response 3b: Fourier series and Fourier transform
Fourier series disintegrates into a periodic signal into trigonometric with different frequencies and amplitudes while Fourier Transform is a mathematical problem that decomposes a signal into frequencies that compose it. The Fourier transform converts the initial signal illustration from time domain to frequency domain illustration as it banks on the frequency. Fourier transform is consist of the frequency domain illustration and transformation process used. Fourier series is for periodic functions while as the Fourier transform is used on aperiodic functions. Both, Fourier series and Fourier transforms are widely applied in engineering.
In communications concept the function is often a voltage. Fourier theory is essential in interpreting signal behavior through filters, communication vessels and sound amplifiers. Frequency contents are also applied in discrete digital communications that utilize zeros and ones to transfer data. Probably, this is the easiest to understand when sending a pulse through a channel. Communication sector has utilizations ranging from network management to transferring single bits through a medium. Mostly, Fourier transform is related with the low level aspects.
Not only is Fourier transform applicable in laboratory examples, but it is also used in real situations. It has extensive implications to the world at large. Example of in astronomy. Often at times, it is challenging to acquire all necessary data from a telescope. Instead, radio waves are used in place of light. These waves are taken like all other ordinary time varying voltage signal, then processed digitally.
The radioactivity of a materials is a factor of a number of components. These include: the mixture making up the sample, and branching disintegration. The basic law of radioactive degeneration is depends on that, the decomposition is wholly a statistical process. The possibility of disintegration is a function of an atomic nucleus. Decay is constant over time. In nuclear physics, when calculating radioactive decomposition, Laplace transform is applied. It eases the research on analytic parts.
Vibrations of machines are examined with assessment of the vibration frequency, displacement, velocity, and acceleration. The latter three are usually in time realm measurements. Thereby, their magnitudes are plotted against time. The vibrational signals of both noise and harmonic information have important content that may be challenging to notice when their amplitudes are drawn in time domain. However, when the values in time domain are expressed in frequency realm, it is possible to detect abnormalities based on differences in magnitude at specified frequencies.
In electromagnetic theory, the intensity of light is proportional to the square of the oscillating electric field which exists at any point in space. The Fourier transform solves this through a mathematical spectrometer which decomposes the rays into constituent parts of the spectrum. Optics apply Fourier transform in the diffraction of light as it moves through holes.
Response 4 d: Applications of Poisons Distribution
In a manufacturing company, it is important to approximate failure rates on a daily basis when measuring the operational performances. Machine failures are caused mechanical wear, adhesive wear and metal fatigue. Mechanical wear takes place due to surfaces abrasively wearing against each other. This abrasive wear causes the most wear. Dirt particles accelerate the abrasion. Adhesive wear results from friction between two machine surfaces. This occurs in sections where there is lubricant scarcity.
In the case that metal is persistently subjected to too much stress, it weakens and ultimately fractures. This metal fatigue may occur on machine parts. For example, too much stress, over a period of time, on a conveyor belt may propagate its wear and tear. Such failures may lead to liability risks related with more on site workers. When a machine loses its functionality, it leads to major financial loss to a business. This is where Poisson distribution comes in handy. It is used in planning out maintenance depending on the frequency and nature of utilization of a machine. It is also applied when laying out a plan of replacing individual parts.
In business operations, Poisson distribution may be applied in estimating the number of anticipated customers by producers. When using this approach, environmental conditions and other factors that may influence frequency of customers are considered as constant. These calculations are prone to either being a success or failing in real life applications. In the “Empirical Law of Change”, is looked into, the stability of these computations increases with increase in n which is infinity. Therefore, the probability of the customer estimation being a total success is negligible. Regardless, it still assists in planning on expected sales when producing.
References
Applications of Fourier series. (2003). Fourier and Laplace Transforms, 113–134. doi: 10.1017/cbo9780511806834.008
Introduction. (n.d.). Numerical Methods In Engineering. doi:10.4324/9780203215753_chapter_2
Poisson Distribution. (2007). Wolfram Demonstrations Project. doi: 10.3840/001281