Course 73110: Computer Lab 14

In this lab you are presented with some problems that require numerical software for their solution. Your task is to find an appropriate function or routine that can be used to solve the mathematical problem. You don't need to solve the problem!

Consider all the following computing alternatives:

1. Matlab 
2. some other interactive tool 
3. Fortran or C
4. some other compiled language

Any commercial software that you select should be available to you, that is, it should be installed

• in the math PC lab or some other PC lab (say where), or
• in the IAC or CSC scientific computing centres (which have the NAG and IMSL fortran libraries, among others), or
• on some other computer to which you can have access (say where)

Some starting points for your search are

• Guide to Available Mathematical Software (GAMS)
• Michael Heath's Mathematical Software Source List

Problem 1

The deflection of a beam-column is described by the equation

E*I*y'' / (1+(y')^2)^(3/2) = P*y - w*x^2/2
y(0)=0, y'(L) = 0

where E, I, b, h, w, P, L are given positive constants. Find y(x) for 0<=x<=L.

Problem 2

The flow rates in a pipeline network satisfy the system of equations

q01= q12 + q13
q12= q24 + q23
q34= q23 + q13
q45= q24 + q34

where

qij=bij*sqrt(pi-pj)

and

b01=0.3, b12=b45=b23=0.2, b13=b24=b34=0.1,

and

p0=500, p5=0.

Find p1, p2, p3, p4.

Problem 3

Find and plot a smooth surface that interpolates or approximates z=f(x,y) for the following x,y,z data:

(172,451,164), (81,671,87, (163,762,81), (350,800,74), (480,711,89), (608,573,121), (668,423,235), (650,160,58), (350,100,63), (97,276,82), (330,360,256), (458,513,260), (518,363,246), (34,100,5), (12,445,50), (26,800,32), (695,800,1), (735,504,89), (740,384,241), (700,102,23)

Problem 4

Compute the surface area of an ellipsoid with semiaxes (a , b, c) , given by the formula

Problem 5

Find values of the special function known as the Sine Integral.

Problem 6

Find values of a, b, c, d so that the function

y(x)=a*exp(-b*x)+c*exp(-d*x)

fits the following (x,y) data:

(0,5.8955), (0.1,3.5639), (0.2,2.5173), (0.3,1.9790), (0.4,1.8990), (0.5,1.3938), (0.6,1.1359), (0.7,1.0096), (0.8,1.0343), (0.9,0.8435), (1.0,0.6856), (1.1,0.6100), (1.2,0.5392), (1.3,0.3946), (1.4,0.3903), (1.5,0.5474), (1.6,0.3459), (1.7,0.1370), (1.8,0.2211), (1.9,0.1704), (2.0,0.2636)

Problem 7

Solve (as efficiently as possible) the system of equations that arise in cubic splines interpolation.

Problem 8

Find the electrical potential u(x,y) between two circular metallic conductors,  given as the solution of the partial differential equation