posted 1 years ago

posted 1 years ago

\frac{\partial u}{\partial t}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leq x \leq 1, \quad t \geq 0

subject to the conditions

\frac{\partial u}{\partial x}(0, t)=0, \quad u(1, t)=9 \cos (1) e^{-c^{2} t}, \quad t \geq 0

and

u(x, t)=9 e^{-c^{2} t} \cos x

has the (exact) solution

u(x, 0)=9 \cos x, \quad 0 \leq x \leq 1

posted 1 years ago

posted 1 years ago

\gg \text { rand ('seed', } 5678 \text { ); }

\gg c=1+5^{*} r a n d

This will generate the following output.

posted 1 years ago

posted 1 years ago

а.In the calculation of heat conduction equation, describe the assumption that need to be considered.

b.For one dimensional heat equation in insulated thin rod, derive equations for the first, internal and the last point of the grid.

d.Select thebest method from the TWO (2) numerical methods in Part (2)(c). Justify your answer.

posted 1 years ago

\frac{d x}{d t}=28-2.5 x

Given the initial time, t = 0 minute, the concentration of sodium in the tank is 45| L. Choose a step size in a range of 1 <h< 3 minutes for the interval from 0 to 3h minutes,

а.Calculate the sodium concentration using the TWO (2) numerical methods known as Runge Kutta 4th order method and Euler's method.

b.Describe ONE (1) advantage and ONE (1) disadvantage for the TWO(2) numerical methods in Part (1)(a) respectively.

c.Determine the best method from the TWO (2) numerical methods in Part (1)(b) based on true percent relative error. Justify your answer.

posted 1 years ago

g(x)=g(0) \frac{R^{2}}{(R+x)^{2}}

\frac{d v}{d t}=g-\frac{c}{m} v

where g(x) = gravitational acceleration altitude x (in m) measured upward from the earth's surface (m/s²), g(0)=gravitational acceleration at the earth's surface (9.81 m/s³),and R = the earth’s radius (6.37 x 106 m).

(a)In a fashion similar to the derivation of (Eq. 2) use a force balance to derive a differential equation for velocity as a function of time that utilizes this more complete representation of gravitation. However, for this derivation, assume that upward velocity is positive. (20 points)

(b)For the case where drag is negligible, use the chain rule to express the differential equation as a function of altitude rather than time. Recall that the chain rule is:

\frac{d v}{d t}=\frac{d v}{d x} \frac{d x}{d t} \text { (25 points) }

(c)Use calculus to obtain the closed form solution where v = vo at x = 0.

(d)Use Euler's method to obtain a numerical solution from x = 0 to 100.000 m.using a step of 10,000 m where the initial velocity is 1500 m/s upward. (25

(e)Compare your result with the analytical solution.

posted 1 years ago

v=\frac{g m}{c}\left(1-e^{-\left(\frac{c}{m}\right) t}\right)

where g = 9.8 m/s². For a parachutist with a drag coefficient c = 15 kg/s, compute the mass m so that the velocity is v = 35 m/s at t=9 s. Write MATLAB programs using both the bisection method and the false-position method to determine m to a level of relative error ɛ, <= 0.1%. Compare the two methods and discuss.

posted 1 years ago

a) Graphically.

b) Using the quadratic formula;

c) Write a MATLAB function using the bisection method to determine the highest root. The function should take two inputs of x1, Xu, the max number of iterations, and output the root, estimated error, and true error.

d) Employ initial guesses of xỊ = 5 and xu = 10, call the function you write in part (c) to compute the estimated error and true error at 1s', 5th, 10th, and 30th iteration respectively.