Elton Cheng SID: 460390873Assignment 1 AMME2000AbstractThe report demonstrates the operations of different analysis techniques. In the report, both analytical solutions and numerical solutions are tested for accuracy and validity in their applications. Section 1 compares the results obtained from differentiation and various finite difference stencils. Section 2 models the diffusion of gas in chamber using the Fourier’s series.Table of ContentsIntroduction…………………………………………………………………1Section 1…………………………………………………………………1Section 2…………………………………………………………………4Conclusion…………………………………………………………………5Appendix…………………………………………………………………6IntroductionEngineers attempt to implement theories from science to arrive at a solution that defines the physical world and apply it to make life more convenient. During an engineering analysis, obtaining an exact solution is not alwayss plausible, the solution may be difficult or impossible to derive due to irregular and unpredictable nature of the physical world. Fortunately, engineers came up with estimations to approximate the exact solutions. Section1 of the report examines the accuracy of these approximation through a simple comparison between the values solved by second order differentiation and two different finite different stencils. Section 2 uses a Fourier’s Series to derive an equation that imitates the diffusion of gas in a chamber. Section 1 The second differentiation of the function f(x) = cosh(2x) was analysised by using the analytical method and numerical method between the interval of 0 ≤ x ≤ 1. The exact solutionfor the second differentiation is f(x) = 4cosh(2x) and the numerical solution is calculated using the two stencils:Figure 1.1: Two finite difference stencils The interval is divided into 4 points and the second differentiation solution of each point is determined through both analytical and numerical method and is plotted into the graph as shown below.The deviance of the 1storder plot and 2ndorder plot from the exact plot in Figure 2 demonstrates that estimated value is from 1storder is less accurate from 2ndorder. 2nd order plot appears to have overlapped the exact plot which signifies its accuracy. Further analysis ismade through the derivation of leading order error of the two stencils.