Applications of Partial Differential Equations

Applied Analysis

Applied Linear Algebra

Introduction to Complex Variables

Multivariable and Vector Calculus

Ordinary Differential Equations

Partial Differential Equations

Probability With Physical Applications

MATH | 301 | Applied Analysis | homework | 

Lecture_4.pdf-Lecture 4 Teodor I. Burghelea September

April 2008 Mathematics 307: Applied Linear Algebra Page 2 of 10 pages Marks [15] 1. Let f ( x ) for 0 ≤ x ≤ 2 be the solution to the diﬀerential equation f ′′ ( x ) − 2 f ( x )= − 2 x 2 satisfying the boundary conditions f (0) = 1, f (2) = 5. The ﬁnite diﬀerence approximation to this equation with boundary conditions is a matrix equation of the form ( L + h 2 Q ) f = b (a) [5] Let N = 5 and consider the case where the discretization has N + 1 points including endpoints ( N − 1 interior points). What are L , h , Q and b ? Continued on page 3 April 2008 Mathematics 307: Applied Linear Algebra Page 3 of 10 pages (b) [5] Write down the MATLAB/Octave commands that compute the ﬁnite diﬀerence ap- proximation to f ( x ) using a discretization with N = 50 (i.e., 49 interior points). (c) [5] The exact solution to this equation turns out to be f ( x )= x 2 + 1. Explain how you could use MATLAB/Octave to compare your approximation to the exact solution by plotting them both on the same plot. Continued on page 4 April 2008 Mathematics 307: Applied Linear Algebra Page 4 of 10 pages [15] 2. Let a 1 =      1 1 2 1 − 1      a 2 =      1 0 1 0 1      a 3 =      1 2 3 2 1      be threee vectors in ﬁve dimensional space. (a) [5] How would you check whether a 1 , a 2 and a 3 are linearly dependent? Describe the com- putation or give MATLAB/Octave commands and say what results for your computation would indicate dependence or independence. Continued on page 5 April 2008 Mathematics 307: Applied Linear Algebra Page 5 of 10 pages (b) [5] What 3 × 3 matrix is invertible if and only if the vectors a 1 , a 2 and a 3 are linearly independent? How could you use this matrix to check whether these vectors are almost (but not quite) linearly dependent? (c) [5] Given that the vectors a 1 , a 2 and a 3 are, in fact, linearly independent, how can you compute the projection onto the their span? Continued on page 6 April 2008 Mathematics 307: Applied Linear Algebra Page 6 of 10 pages [15] 3. (a) [5] Explain in what sense the functions e 2 πinx for 0 ≤ x ≤ 1 and n = ..., − 2 , − 1 , 0 , 1 , 2 ,... form an orthonormal set. (b) [5] Let f ( x ) be the function deﬁned on the interval 0 ≤ x ≤ 1 by f ( x )=1 − x . Compute the coeﬃcients c n in the Fourier series expansion f ( x )= ∞  n = −∞ c n e 2 πinx (Hint:  xe iαx dx = e iαx /α 2 − ixe iαx /α + C for α  = 0) Continued on page 7 April 2008 Mathematics 307: Applied Linear Algebra Page 7 of 10 pages (more space for part (b)) (c) [5] What does Parseval’s theorem say in this example? Continued on page 8

MATH | 307 | Applied Linear Algebra | lecture notes | April 2008 Mathematics 307: Ap

Midterm1-2005F-SOLUTIONS.pdf-UBC Mathematics 255 Section 101 Midterm

MATH255Midterm2Notes.docx-MATH 255 Midterm 2 Review Secon

All courses in MATH

Popular Documents