91Ó°ÊÓ

Discuss consistency, stability, and convergence for the Implicit Trapezoidal method $$ w_{i+1}=w_{i}+\frac{h}{2}\left(f\left(t_{i+1}, w_{i+1}\right)+f\left(t_{i}, w_{i}\right)\right), \quad \text { for } i=0,1, \ldots, N-1, $$ with \(w_{0}=\alpha\) applied to the differential equation $$ y^{\prime}=f(t, y), \quad a \leq t \leq b, \quad y(a)=\alpha. $$

Use the Taylor method of order two with \(h=0.1\) to approximate the solution to $$ y^{\prime}=1+t \sin (t y), \quad 0 \leq t \leq 2, \quad y(0)=0. $$

Derive the Adams-Bashforth Three-Step method by the following method. Set $$ y\left(t_{i+1}\right)=y\left(t_{i}\right)+\operatorname{ahf}\left(t_{i}, y\left(t_{i}\right)\right)+\operatorname{bhf}\left(t_{i-1}, y\left(t_{i-1}\right)\right)+\operatorname{chf}\left(t_{i-2}, y\left(t_{i-2}\right)\right) $$ Expand \(y\left(t_{i+1}\right), f\left(t_{i-2}, y\left(t_{i-2}\right)\right)\), and \(f\left(t_{i-1}, y\left(t_{i-1}\right)\right)\) in Taylor series about \(\left(t_{i}, y\left(t_{i}\right)\right)\), and equate the coefficients of \(h, h^{2}\) and \(h^{3}\) to obtain \(a, b\), and \(c\).

Derive the Adams-Moulton Two-Step method and its local truncation error by using an appropriate form of an interpolating polynomial.

Derive Milne's method by applying the open Newton-Cotes formula (4.29) to the integral $$ y\left(t_{i+1}\right)-y\left(t_{i-3}\right)=\int_{t_{i-3}}^{t_{i+1}} f(t, y(t)) d t. $$

In a book entitled Looking at History Through Mathematics, Rashevsky [Ra], pp. 103-110, considers a model for a problem involving the production of nonconformists in society. Suppose that a society has a population of \(x(t)\) individuals at time \(t\), in years, and that all nonconformists who mate with other nonconformists have offspring who are also nonconformists, while a fixed proportion \(r\) of all other offspring are also nonconformist. If the birth and death rates for all individuals are assumed to be the constants \(b\) and \(d\), respectively, and if conformists and nonconformists mate at random, the problem can be expressed by the differential equations $$ \frac{d x(t)}{d t}=(b-d) x(t) \quad \text { and } \quad \frac{d x_{n}(t)}{d t}=(b-d) x_{n}(t)+r b\left(x(t)-x_{n}(t)\right) $$ where \(x_{n}(t)\) denotes the number of nonconformists in the population at time \(t\). a. Suppose the variable \(p(t)=x_{n}(t) / x(t)\) is introduced to represent the proportion of nonconformists in the society at time \(t\). Show that these equations can be combined and simplified to the single differential equation $$ \frac{d p(t)}{d t}=r b(1-p(t)) $$ b. Assuming that \(p(0)=0.01, b=0.02, d=0.015\), and \(r=0.1\), approximate the solution \(p(t)\) from \(t=0\) to \(t=50\) when the step size is \(h=1\) year. c. Solve the differential equation for \(p(t)\) exactly, and compare your result in part (b) when \(t=50\) with the exact value at that time.

Show that the Midpoint method and the Modified Euler method give the same approximations to the initial-value problem $$ y^{\prime}=-y+t+1, \quad 0 \leq t \leq 1, \quad y(0)=1 $$ for any choice of \(h\). Why is this true?

Water flows from an inverted conical tank with circular orifice at the rate $$ \frac{d x}{d t}=-0.6 \pi r^{2} \sqrt{2 g} \frac{\sqrt{x}}{A(x)} $$ where \(r\) is the radius of the orifice, \(x\) is the height of the liquid level from the vertex of the cone, and \(A(x)\) is the area of the cross section of the tank \(x\) units above the orifice. Suppose \(r=0.1 \mathrm{ft}\), \(g=32.1 \mathrm{ft} / \mathrm{s}^{2}\), and the tank has an initial water level of \(8 \mathrm{ft}\) and initial volume of \(512(\pi / 3) \mathrm{ft}^{3}\). Use the Runge-Kutta method of order four to find the following. a. The water level after 10 min with \(h=20 \mathrm{~s}\) b. When the tank will be empty, to within \(1 \mathrm{~min}\).

Show that the difference method $$ \begin{aligned} w_{0} &=\alpha \\ w_{i+1} &=w_{i}+a_{1} f\left(t_{i}, w_{i}\right)+a_{2} f\left(t_{i}+\alpha_{2}, w_{1}+\delta_{2} f\left(t_{i}, w_{i}\right)\right) \end{aligned} $$ for each \(i=0,1, \ldots, N-1\), cannot have local truncation error \(O\left(h^{3}\right)\) for any choice of constants \(a_{1}, a_{2}, \alpha_{2}\), and \(\delta_{2}\).