FAMILY NAME: . . . . . . . . . . . . . . . . . . . . . . . . . . . . OTHER NAME(S):. . . . . . . . . . . . . . . . . . . . . . . . . . STUDENT NUMBER: . . . . . . . . . . . . . . . . . . . . . . SIGNATURE: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS June 2011 MATH2089 Numerical Methods and Statistics (1) TIME ALLOWED – 3 Hours (2) TOTAL NUMBER OF QUESTIONS – 6 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) THIS PAPER MAY NOT BE RETAINED BY THE CANDIDATE (6) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER MAY BE USED (7) STATISTICAL FORMULAE ARE ATTACHED AT END OF PAPER STATISTICAL TABLES ARE ATTACHED AT END OF …show more content…
ii) Find an initial value problem (IVP) satisfied by y. Please see over . . . June 2011 MATH2089 Page 4 c) When using N equal intervals of width h, the error in estimating the integral τ f (t)dt 0 by the Trapezoidal rule is Etrap = O(h2 ) and by Simpson’s rule is ESimp = O(h4 ). i) For each of these numerical integration rules, what conditions are required on the integrand f so these error estimates are valid? ii) Suppose that the error using h = 5 × 10−3 is E0 = 1.19 × 10−4 when using either the Trapezoidal rule or Simpson’s rule. For both rules, ˆ estimate the error if an interval width of h = 1 × 10−3 is used. iii) The Matlab command [z, w] = gauleg(N); calculates the N Gauss-Legendre nodes z and weights w for the interval [−1, 1]. Show how z and w can be used to numerically calculate τ f (t)dt. 0 Please see over . . . June 2011 3. MATH2089 Page 5 Answer in a separate book marked Question 3 Fick’s second law states how the concentration of a chemical changes with time because of diffusion. Let c(x, y) denote the concentration at position (x, y) ∈ Ω. The steady state version of Fick’s second law (without interior sources of the chemical) is Laplace’s equation ∂2c ∂2c + 2 = 0. ∂x2 ∂y Consider a problem with domain Ω = {(x, y) : 0 ≤ x ≤ 1, xi = ih, i = 0, . . . , n 0 ≤ y ≤ 1.5} . j = 0, . . . , (3.1) Ω is discretised using h = 1/n (where n is even) and 3n . 2 The domain is illustrated in Figure 3.1 for n = 6. The
Weight 10 dry post-82 pennies which get 77.12g, using 30ml initial volume measuring the volume of 10 pennies, record the data 9.1ml. Using equation Density= Mass/Volume, get the density of the pre-82 pennies is 8.47g/ml. Then calculate the error%=0.04%, and the deviation%=7.13%.
Small groups are the proper environment to develop and grow disciples of Jesus. The purpose of a small group is to develop sacrificial, relational, transformed people who can continue the cycle of disciple development. Small groups must be intentional, individual and missional. There are five primary passages that can be used to form a small group ministry philosophy. Each of these passages have accompanying principles that we can apply to our small group ministries.
Results for 81×81grid with Re=1000 are shown in Figure 1.It contains contours and graphs at Re =1000, with convergence criteria =1e-6, No of iterations = 15000, and No of time steps =50000.
Use the trapezium rule with five equally spaced ordinates to estimate the area of the region
initial conditions, or on the non-negative integers t 2 N. In this case, the relation above starts from
(Mrs Bolding I can’t really talk about accuracy and precision because I don’t now how to create a graph from these results…I tried to dicuss about error and unexpected results in the discussion… that’s about it, so
As to derive the functions used in this paper, I had to gather data for use in the analysis. This is
As we know the zeros of Y2, it will be very easy to determine the complex zeros of Y1. 2 will equal to the real part of the complex zeros of Y1, and subsequently, ±3 will equal to the imaginary part of the complex roots. Although, as we know that y-axis is the imaginary axis, the imaginary parts of the complex roots obtained, of the Y1, will be plot according to the y-axis.
Using a statistical model they created (See Appendix), Entine and Small ended up with the following results:
3 This equation has an associated uncertainty of (∂T1 gδT1 ) + (∂T2 gδT2 ) + (∂L gδL ) + (∂l1 gδl1 ) (II.22) or δg = (
(a) Number of Iterations – The program performs successive iterations with check on convergence. The analysis is terminated if e < 0.0001 or upon reaching the specified number of iterations. This is introduced to circumvent the possibility of divergence.
Now we demonstrate the use of the method of separation of variables by applying it to the one-dimensional transient heat conduction problem given in Eqs. (1). First,
intervals are not true. In this paper, we will use one instrument which is the RGDP of
The major topics explored in Calculus C are largely defined by derivatives of vector-valued and parametrically defined functions, integration by partial fractions, improper integrals, series convergence (Taylor and Maclaurin), L’Hopitals Rule, and numerous applications. All of the following topics require a solid foundation in not only Calculus A but also Calculus B.
In [28], it is shown that the proposed method is not correct. Sharifzadeh and colleagues [29] have proposed another version of the method that eliminates the flaws of the previous version, but it is computationally costly.