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Questions and Answers of Linear Algebra

In the bar with fixed ends discussed in Example 11.2. which point experiences the greatest displacement? Which point experiences the greatest stress? The greatest strain?
A bar of length ℓ = 1 has one fixed and one free end and stiffness function c(x) = 1 - x. Find the displacement when subjected to a unit force. Pay careful attention to the boundary condition at
Analyze the periodic boundary value problem for the angular displacement of a circular ring along the same lines as the free case done in Example 11.3. Let 0 ≤ x ≤ 2n denote the angular
What are the physical units (using metric grams, meters and seconds) for measuring the various quantities appearing in the equilibrium equations for a bar?
Analyze the displacement and stress of a homogeneous bar, c = I, of length £ = 1, when subject to a discontinuous external forcewith both ends are fixed at their reference positions: «(0)
(a) Find all possible equilibrium configurations in the free bar of Example 11.3 when subject to the external force(b) Find the corresponding stress. Does the stress depend upon which equilibrium
A bar of length γ = 1 has stiffness function c(x) = I -F x. Find and graph the displacement, stress and strain when the bar is subjected to a unit gravitational force, assuming there is no
Repeat Exercise 11.1.4 in the case when the bar has one fixed and one free end.
A bar of length ℓ = 2 has a constant stiffness function c = 1/3. Find and graph the displacement and stress when the bar is subjected to a force f(x) = 1 - x and both ends are fixed at their
A composite bar is obtained by gluing a uniform bar of length 1 and stiffness c = 1 to a second bar of length I and stiffness c = 2. Using the fact that the stress remains continuous, find and graph
Suppose the composite bar in Exercise 11.1.8 is only fixed at its top end and subject to a uniform gravitational force. Will the free end hang lower if its stiffer half is on top or bottom? Make a
Evaluate the following integrals:a.b. c. d. e. f.
Explain why the Gaussian functionshave the delta function i a i as their limit as n †’. ˆž Hint: Use the known formula
In this exercise, we realize the delta function 5v(a) as a limit of functions on a finite interval [0. i]. Let 0 (a) Prove that the functionswhere gn(x) is given by 111.31). and satisfy (11.29-30).
For each positive integer n, leta Sketch a graph of g"(x).b. c. Evaluateand sketch a graph. Does the sequence fn(x) converge to the step function a (x) as n †’ ˆž?(d) Find the derivative hn(x)
Answer Exercise 11.2.12 for the hat functions
Justify the formula
Let y a.b.
Letdenote the L2 inner product on the interval [0, „“]. Suppose u(x) satisfies u(0) = u(t) = 0. Write out the integration by parts formula (Î´y, u) = (Ïƒy, u) = -
Let δ(k)lx) denote the kth derivative of the delta function
According to (11.41), x δ(x) = 0. On the other hand, by Leibniz' rule, (x δ (x))' = δ(x) + x δ'(x) is apparently not zero. Can you explain this paradox?
If f ∈ C1. should (f δy)' = f δ' or f δ + f δ'?
Simplify the following generalized functions; then write out how they act on a suitable test function u( a):a. ex Î´(a)b. a Î´ (a - 1)c.
(a) Use duality to justify the formula f(x)δ'(x) = f(0)δ'(,x) - f'(0)8(x) when f ∈ C1. (b) Find a similar formula for f(x) δ (n) for the product of a sufficiently smooth function and the nth
Use Exercise 11.2.20 to simplify the following generalized functions; then write out how they act on a suitable test function u(x):(a) φp(x) = (x - 2)δ'(x)(b) ψ(x) = (1 + sin x) [δ (x) +
Prove that if /(a) is a continuous function, andfor every interval [a,b], then /(a) = 0 everywhere.
Explain in detail why there is no continuous function δy(x) such that the inner product identity (11.39) holds for every continuous function u(x).
Explain in detail why the sequence (11.43) converges non-uniformly.
Derive the Green's function for the boundary value problem (11.69) by direct integration. Make sure that you obtain the same answer (11.70).
A uniform bar of unit length t = 4 has constant stiffness c = 2. Find the Green's function for the case when(a) Both ends are fixed(b) One end is fixed and the other is free(c) Why is there no
A bar of length i = 1 and stiffness function c(x) = 1 +x has both ends clamped: n(0) = «(1) = 0. (a) Construct the Green's function for this boundary value problem. (b) Use the Green's function to
An elastic bar has constitutive function(a) Find the displacement and internal force when the bar is subjected to a constant external force, f = 1. and the ends are held fixed: u(0) = u(1)=0.(b) Find
Justify the formula a δ (x) = 0 by using (a) Limits (b) Duality
Consider the boundary' value problem - u" = f(x), u(0) = 0, u(l) = 2u'(l). (a) Find the Green's function. (b) Which of the fundamental properties does your Green's function satisfy? (c) Write down an
When n is a positive integer, set(a) Find the solution un(x) to the boundary value problem -u" = fn(x), u(0) = u(l) = 0, assuming 0 (b) Prove thatconverges to the Green's function (11.59). Why should
Prove directly that formula (11.71) gives the solution to the boundary value problem (11.69)
A system of n masses is connected to a top support by n unit springs of length ∆.x = 1/n, so that the overall length of the mass-spring chain is ℓ = 1. Let K = AT A, where A is the rescaled
Reformulate Exercise 11.2.33 for a mass-spring chain connected to both top and bottom supports. Do the same conclusions hold?
Prove the differentiation formula (11.64).
Define the generalized function φ p(x) = 5(a) - 3δ (x - 1) (a) As a limit of ordinary functions; (b) By using duality.
(a) Justify the formula δ(2a) = ½ δ(a) by (i) Limits (ii) Duality (b) Find a similar formula for δ(ax) when a > 0.
Find and sketch a graph of the derivative (in the context of generalized functions) of the following functions:a.b. c. d.
Find the first and second derivatives of the functionsa.b. c.
Find the first and second derivatives of a e-|x| b 2|x| - |x - 1| c |x2 + x| d a sgn(x2 - 4) e sin |x| f |sin x | g sgn(sin x)
(a) Prove that Ïƒ(Î».x 1 = Ïƒ 0.(b) What about if Î». (c) Use pans (at. (bi to deduce thatfor any Î» ‰ 0.
Let P[u](a) Find the function u,(x) that minimizes P[u] among all C2 functions that satisfy u(0) = (1) = 0. (b) Test your answer by computing P[u,] and then comparing with the value of P[u] when u(x)
Find a function u(x) such thatHow do you reconcile this with the claimed positivity in (11.94)?
Does the inequality (11.94) hold when u(x) ≠ 0 is subject to the Neumann boundary conditions u'(0) = u'(ℓ) = 0?
Let U = C°[0, 1 ]. Find the adjoint I * of the identity operator I: {J †’ U under the weighted inner products
Let c(x) ∈ C°[a, b] be a continuous function. Prove that the linear multiplication operator K[u) = c(x)u(x) is self-adjoint with respect to the L2 inner product. What sort of boundary conditions
Compute the adjoint of the derivative operator v = D[u] = u' under the weighted inner products (11.96) on, respectively, the displacement and strain spaces. Verify that all four types of boundary'
(a) Determine the adjoint of the differential operator u = L[u] = u' + 2xu with respect to the L2 inner products on [0, I] when subject to the fixed boundary conditions u(0) = u(1) = 0.(b) Is the
(a) Show that a differential equation of the form a(x) u"+b(x) u' = /(x) is in self-adjoint form (11.12) if and only if b(x) = a'(x).(b) If b(x) ≠ a'(x) and a(x) 0 everywhere, show that you can
Consider the linear operator
Prove that the complex differential operatoris self-adjoint with respect to the L2 Hermitian inner product on the vector space of continuously differentiable, complex-valued, 2 n periodic functions:
Consider the boundary value problem -u" = x, u(0) = u(1) =0. (i) Find the solution. (ii) Write down a minimization principle that characterizes the solution. (iii) What is the value of the quadratic
In Exercise 7.5.6, you determined the adjoint of the derivative operator D when acting on the space of quadratic polynomials with respect to the L2 inner product
Find the function u.(x) that minimizes the integralsubject to the boundary conditions n(l) = 0, m(2) = 1.
For each of the following functionals and associated boundary conditions, (i) write down a boundary value problem satisfied by the minimizing function, and (ii) find the minimizing function
For each the following boundary value problems,(i) write down a minimization principle, carefully specifying the space of functions, and(ii) find the solution:a. -u" = cosx, u(0) = 1, u(Ï€)
Explain how to solve the inhomogeneous boundary value problem -u" - f(x), u(0) = a, u(1) = β, by using the Green's function (11.59).
A bar 1 meter long has stiffness c(x) = 1 +x at position 0 < x < 1. It is subject to an external force f(x) = 1 -x. The left end of the bar is fixed, while the right end is extended 1 cm.(a)
Prove that the solution to the mixed boundary value problemis the unique C2 function that minimizes the modified energy functional when subject to the inhomogeneous boundary conditions. Hint: Mimic
Find the function u(x) that minimizes the integralsubject to the boundary conditions u(1) = 1, u'(2) = 0. Hint: Use Exercise 11.3.26.
Prove that the functionalsubject to the mixed boundary conditions u(0) = 0, u'(l) = I has no minimizer! Thus, omitting the extra boundary term in (11.106) is a fatal mistake.
SupposeProve that ail solutions to the inhomogeneous Neumann boundary value problem are minimizers of the modified energy functional
Answer Exercise 11.3.2 for the boundary value problemsa.b.c.d.
For each of the following functionals and associated boundary conditions,(i) write down a boundary value problem satisfied by the minimizing function, and(ii) find the minimizing function
Which of the following quadratic functionals possess a unique minimizer among all functions satisfying the indicated boundary conditions? Find the minimizer if it exists.a.b. c. d. e.
Does the quadratic functionalhave a minimum value when u(x) is subject to the homogeneous Neumann boundary value conditions u'(0) = u'(1) = 0? If so, find all functions that minimize p[u]. Hint: What
Let c(x) > 0 for 0achieves a minimum value if and only if f(x) has mean zero.
Show that the internal energy in a bar (the first term in (11.92)) is one half the unweighted L2 inner product between stress and strain: 1/2 ||D[u||2 = l/2(v, w).
Let u.(x) be the equilibrium solution to the Dirich- let boundary value problem for a bar: k[u.] = f, u,(ℓ) = u.(ℓ) = 0. Prove that if f(x) = 0. then the total energy at equilibrium is strictly
When possible, find and graph the displacement u(x) and bending moment w(x) of a beam of unit length l = 1 with constant bending stiffness c(x) = 1 subject to a unit gravitational force when it
Verify the solution formula - 11.126 > by direct substitution into the differential equation and boundary conditions (11.125
(a) List all possible combinations of the listed self- adjoint homogeneous boundary conditions that lead to a unique equilibrium solution to the boundary value problem when subject to any external
A beam has unit length ℓ = 1 and constant bending stiffness clx) = 1. Explain the physical configuration that each of the following inhomogeneous boundary' conditions represents, and then, if
Write down a minimization principle that characterizes the solution to the inhomogeneous boundary value problems in Exercise 11.4.12. Hint: Adapt the argument at the end of Section 11.3; see also
Find and graph the natural cubic spline interpolant for the following data:a.b. c. d.
Repeat Exercise 11.4.14 when the spline has horns clamped boundary conditions.
Find graph the periodic cubic spline that inter-polates following data:a.b. c. d.
(a) Give 30°, n the known values of sinx at x = 0°, 45°, 60°, construct the natural cubicspline interpolant.(b) Compare the accuracy of the spline with the least squares and interpolating
(a) Using the exact values for √x at x = 0, ¼ , 9/16 1 , construct the natural cubic spline interpolant.(b) What is the maximal error of the spline on the interval [0, 1 ]?(c) Compare the error
According to Figure 4.10, the interpolating polynomials for the function 1/(1 + x2) on the interval [ - 3, 3] based on equally spaced mesh points are very inaccurate near the ends of the interval.
Which of the boundary conditions in Exercise 11.4.1 has resulted in the furthest displacement of the beam? Which leads to the most stress?
(a) Draw outlines of the block capital letters I, C, H, and S on a sheet of graph paper. Fix several points on the graphs and measure their x and y coordinates. (b) Use periodic cubic splines x =
Repeat Exercise 11.4.20, using the Lagrange interpolating polynomials instead of splines to parametrize the curves. Compare the two methods and discuss advantages and disadvantages
Let x0 (a) Construct and graph the natural cardinal splines corresponding to the nodes x0 = 0, x1 = 1, x2 = 2, and x3 =3.(b) Prove that the natural spline that interpolatesthe data yo,... ,yn can be
A bell-shaped or B-spline u = 0(x) interpolates the data(a) Find the explicit formula for the natural 5- spline and plot its graph.(b) Show that 0(x) also satisfies the homogeneous clamped boundary
Let 0(x) denote the 5-spline function of Exercise 11.4.23. Assuming n > 4, let P" denote the vector space of periodic cubic splines based on the integer nodes xj = j for j = 0,... , n. (a) Prove
Write down a minimization principle, when possible, for each of the problems in Exercise 11.4.1.
For each of the problems in Exercise 11.4.1,(i) find the Green's function G(x, y), when possible;(ii) graph G (x, f);(iii) write down an integral formula for the displacement of the beam under a
Suppose an elastic bar and an elastic beam have the same length and the same elastic constant c(x) = 1; both have fixed ends, and are both subjected to a constant external force of the same
An elastic beam of unit length has bending stiffness c(x) = I + x2 for 0 < x < 1. Find the strain in the beam under a constant unit force if both ends are simply supported. Would you call this
Consider a beam of unit length ℓ = 1 of constant bending stiffness c(x) = 1 with two free ends.(a) Determine, by direct integration two constraints that must be imposed on the external force f (,v)
Answer Exercise 11.4.7 for a beam with one sliding end and one simply supported end.
Solve the Sturm-Liouville boundary value problem - 4u" + 9u =0, u(0) = 0, u(2) = 1. Is your solution unique?
Find all values of λ > 0 such that the boundary value problem y´´ + 2y´ + (λ + 1)y = 0, y(0) = 0. y(2) = 0. has a nonzero solution y(x).
Let s > 0. (a) Find the solution m(x, e) to the boundary value problem - u" + e2 u = 1, u(0) = u(l) = 0. (b) Show that as ∈ -> 0+, the solution u(x. s) -> u.(x) converges uniformly to the solution
Let ∈ > 0. (a) Find the solution u(x, s) to the boundary value problem -∈2u" + u = 1, u(0) = u(1) = 0. (b) Show that as e -> 0+, the solution u(x, ∈) → u.(x) converges to the solution to u. =

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