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MA 3065 Lecture 1: hw1
Exam

Math 5588 Homework 1 (Due Thursday January 19) 1. Read the entire course syllabus: http:math.umn.edu~jwcalder5588S17. Note the midterm and nal exam dates, homework policy, and grading scheme (you do not need to hand in any...

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MA 3065 Lecture 3: hw3
Exam

Math 5588 Homework 3 (Due Thursday February 2) 1. Find and solve the Euler-Lagrange equation for the functional Z log(2) 2 2 I(u) = u(x) + (u (x)) dx 0 subject to boundary conditions u(0) = 0 and u(log(2)) = 1, where log i...

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MA 3065 Lecture 5: hw5
Exam

Math 5588 Homework 5 (Due Thursday February 23) 1. Properties of the convolution. (a) Show that (f g) = f g = f g . (b) Show that f (g h) = (f g) h. 2 2. Compute the convolution of a Gaussian kernel u(x) = e 2 with itself....

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MA 3065 Lecture 4: hw4
Exam

Math 5588 Homework 4 (Due Thursday February 9) 1. Consider the following version of the isoperimetric problem: Z Z 1 1 p max u(x)dx subject to 1 + u (x) dx = L, u:[1,1]R 1 1 u(1)=0=u(1) where L > 2. Show that the optimal c...

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MA 3065 Lecture 2: hw2
Exam

Math 5588 Homework 2 (Due Thursday January 26) In all problems x = (1 ,x2,...,xn) R and u is a real-valued function on R , u : R R. This aim of this homework is to give you practice with multi-variable calculus. p 1. Let x...

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MA 3065 Lecture Notes - Lecture 7: Rayleigh Quotient, Symmetric Matrix, Ellipse
Exam

Math 5588 Homework 7 (Due Thursday March 9) nn T Recall a matrix A = (a ij R is symmetric if aij= a jiso that A = A. We say a symmetric matrix A is positive denite, written A 0, if XX n T n v Av = aij i j 0 for all v R . i...

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MA 3065 Lecture 6: hw6
Exam

Math 5588 Homework 6 (Due Thursday March 2) This homework aims to examine the Fourier transform of generalized functions, or distri- butions, more rigorously. Recall the Fourier transform of a function f(x) is Z 1 f(k) = f...

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MA 3065 Lecture 11: hw11
Exam

Math 5588 Homework 11 (Due Thursday April 20) 1. A weak solution of Burgers equation ut+ uu =x0 for x R, t > 0 has the form ( x t+1 , for x < s(t) u(x,t) = 0, for x > s(t), where s(t) is a shock curve starting at s(0) = 1....

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MA 3065 Lecture Notes - Lecture 9: Leibniz Integral Rule, Unit Disk, Unit Square
Exam

Math 5588 Homework 9 (Due Thursday April 6) 1. Prove the Leibniz integral rule d b(x) b(x)f f(x,t)dt = f(x,b(x))b (x) f(x,a(x))a (x) + (x,t)dt. dx a(x) a(x) x [Hint: For a,b,x R dene b F(a,b,x) = f(x,t)dt, a and apply the ...

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MA 3065 Lecture 12: hw12
Exam

Math 5588 Homework 12 (Due Tuesday May 2) 1. Suppose H(p) is convex and superlinear. Show that L = H is also convex and superlin- ear. [Hint: For convexity, you need to show that L(x+(1)y) L(x)+(1)L(y) for all x,y and 0 < ...

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MA 3065 Lecture Notes - Lecture 8: Maximum Principle, Maxima And Minima, Viscosity Solution
Exam

Math 5588 Homework 8 (Due Thursday March 23) 1. Use an argument based on the maximum principle to show that if maxf 0, u + u u = f(x) for x 1 and u(x) = 0 for x = 1 then u(x) maxf for all x 1. What happens if maxf < 0? 2. ...

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MA 3065 Lecture 10: hw10
Exam

Math 5588 Homework 10 (Due Thursday April 13) 1. Compute the six elemental stiness values = k s for an equilateral triangle. ij ji 2. Suppose your dmain is triangulated by equilateral triangles. Calculate the nonzero value...

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MA 3065 Lecture Notes - Lecture 4: Viscosity Solution, Uniform Convergence, Lipschitz Domain
Exam

Note: Please choose a paper for your term end projectpresentation by Friday Oct 5. Math 8590 Homework 2 (Due Friday Oct 5) Please hand in your solution to 1 problem from those below. Let U R be open. 1. (a) Let u,v USC(U)....

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MA 3065 Lecture 3: hw3 (1)
Exam

find more resources at oneclass.com Math 8590 Homework 3 (Due Friday Oct 19) Please hand in your solution to 1 problem from those below. ...

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MA 3065 Lecture 1: hw5 (1)
Exam

find more resources at oneclass.com Math 8590 Homework 5 (Due Friday Nov 16) Please hand in your solution to 1 problem from those below. 1. Prove the O( h) convergence rate for monotone nite die...

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