Linear Algebra and Linear Operators in Engineering: With Applications in Mathematica 1st. Edition by H. Ted Davis, Kendall T. Thomson | 9780080510248 | 80510248

Linear Algebra

Linear Algebra and Linear Operators in Engineering: With Applications in Mathematica

33 SOLVED PROBLEMS

Question: 10.8.2

Consider the problem of diffusion of carbon into an iron cannonball initially free of carbon. This process is often conducted in order to case-harden (i.e., produce a steel casing on) cannonballs. Suppose the cannonball is a sphere of radius r = b. Initially, the cannonball contains no carbon, ...

Verified Answer:

Assuming that the concentration of carbon in the c...

Question: 10.5.3

Find Green’s function for the problem L u = d²u / dx² + 3 du / dx + 2u = f (x), -1 < x <1, (10.5.66) u (-1) - 2 u' (-1) = 0 u (1) + u' (1) = 0 (10.5.67) and solve the inhomogeneous problem for the cases (a) f (x) = x (10,5.68) and (b) f(x) = exp (-x²). (10.5.69) ...

Verified Answer:

The general solution to Lu = 0 for Eq. (10.5.66) i...

Question: 10.5.2

Find Green’s function for the problem L u = -(1 + x)² d²u / dx² + 4 (1 + x) du / dx + 2u = f (x), 0 < x <1, (10.5.61) B1 u = u (0) = 0, B2 u = u (1) = 1. (10.5.62) ...

Verified Answer:

A solution to

L u_{1} = 0

with [lat...

Question: 4.5.1

How many of the vectors a1 = [1 2 3], a2 = [1 1 1], a3 = [2 3 4], a4 = [3 5 7] (4.5.26) are linearly independent? ...

Verified Answer:

Since the rank r of A,

A = [a_{1}, a_{2}, a...

Question: 9.5.4

Consider in L2 (-∞, ∞) the operator Kv = ∫-∞^∞ exp (-(x²+y²)/2) v (y) dy. (9.5.79) (a) Prove that K is self-adjoint. (b) Prove that K is completely continuous. (c) Find the eigenvectors of K. ...

Verified Answer:

Since

k (x, y) = exp (-(x^{2} + y^{2} / 2) ...

Question: 10.5.4

Use the method just described to find Green’s function g (x, y) for the differential expression Lu = – d² u/dx², (10.5.84) with B1 u ≡ u (0) = 0 B2 u ≡ u (1) = 0 . (10.5.85) ...

Verified Answer:

We have already clearly determined g(x,y) directly...

Question: 10.5.1

Find Green’s function for the problem d²u / dx² + u = x, u (0) = u’ (0) = 0. (10.5.42) ...

Verified Answer:

The formal solution to

\frac{d^{2} u_{y} (x...

Question: 9.5.5

Consider the equation du/dt = iKu, u(t = 0) = u0, (9.5.126) Where i = √-1 and K is a completely continuous, normal operator in H. Assume that the imaginary part of the eigenvalue λn goes as 1/n^1/2 and examine the asymptotic behavior of u(r). ...

Verified Answer:

The formal solution to Eq. (9.5.126) is

u =...

Question: 9.4.2

Let us define p as the capillary pressure needed to force water out of a certain sample of porous rock until it occupies only the fraction s (saturation) of the pore space. How p depends on s is an important question in soil science and in the characterization of aquifers and oil reservoirs. ...

Verified Answer:

where

r_{1}

and

r_{2}[/latex...

Question: 9.4.1

Find the Volterra equation corresponding to the initial value problem d² y/d x² + exp (-x) d y / d x + x² y = x, (9.4.64) Where d y/ d x = 1 and y = 2 at x = 0. (9.4.65) ...

Verified Answer:

Set

\frac{d^{2} y }{dx^{2} } = u.

...