pdes - problems (3)

8/3/2019 PDEs - Problems (3)

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MATH20401(PDEs) Tony Shardlow Problems Part III

1. Explain in your own words why is it important for a PDE to be well posed.

2. Prove that there is at most one solution to

u

(x) + u(x) = f(x), u

(0) = u

(1) = 0.

3. Show that u = 0 is the unique solution of

u(x) + u3(x) = 0, u(0) = u(1) = 0.

4. Consider the linear inhomogeneous second order two point BVP

a(x)u(x) + b(x)u(x) = f(x), 0 x 1

for some functions a,b,f.

(a) If a(x) > 0 and f(x) > 0 for x [0, 1], show that u(x) attains its maximum valueat one of the two end points x = 0, 1.

(b) By substituting v(x) = u(x) + ex , show if a(x) > 0 and f(x) 0 that u(x) attainsits maximum value at one of the two end points x = 0, 1.

(c) Let u satisfy

a(x)u + b(x)u =0,

u(0) = , u(1) = .

Show that |u(x)| max{||, ||} .

5. Let u satisfy

a(x)u (x) + b(x)u

(x) =f(x),

u(0) = + , u(1) = + .

Suppose that a(x) > 0 for x [0, 1]. Show that |e(x)| = |u0(x) u(x)| || for0 x 1.

6. Consider the wave equation

utt =uxx

u(0, t) =g(t), u(1, t) = h(t), t > 0

u(x, 0) =u0(x), ux(x, 0) = u1(x), x [0, 1]

for given functions g(t), h(t) and u0(x), u1(x).

(a) Let

E(u, t) =

1

0

ut(x, t)2 + ux(x, t)

2 dx

and show thatd

dtE(u, t) = 2

1

0

x

ux(x, t)ut(x, t)

dx.

(b) Show that the solution to the above wave equation is unique.

MATH20401(PDEs): Problem Sheet III: Page 1


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7. Let u(x) = ex . Compute the finite difference approximations (u(x + h) u(x h))/2hand (u(x + h) u(x))/h to u(0) at x = 0 for h = 0.1, 0.01. Which is the most accurate?

We write e = O(hk) if there exists a constant C such that |e| C|h|k as h 0 . B yusing Taylors Theorem, show that

u

(x) =

u(x + h) u(x h)

2h + O(h2

)

and

u(x) =u(x + h) u(x)

h+ O(h).

8. For a spacing h , let

2(x) = u(x + h) + u(x h) 2u(x), D2(x) =u(x + 5h) + 5u(x h) 6u(x)

15.

Consider approximating u(x) by 2(x)/h2 and by D2(x)/h2 .

(a) Compute 2(0)/h2 and D2(0)/h2 for u(x) = ex and h = 0.1. Which is most accu-rate?

(b) Show that 2(x)/h2 is a second order accurate approximation to u(x). That is,prove 2(x)/h2 = u(x) + O(h2).

(c) Show that D2(x)/h2 is a first order accurate approximation to u(x). That is, proveD2(x)/h2 = u(x) + O(h).

9. Forw > 0, let u be the solution of

d2u

dx2+ w

du

dx= 0, with u(0) = 1, u(1) = 0. (1)

(a) Derive a centered difference approximation to u on grid xj = jh for j = 0, . . . , nand h = 1/n. Define the truncation error Tj and show that the truncation error

|Tj | = O(h2

).(b) Write down the problem as a linear system of equations Ax = f, where the matrix

A and vectors f and x should be specified.

10. Consider

d2u

dx2= f(x), 0 < x 1, with u(0) = 0,

du

dx(1) = 0

Derive the centred finite difference approximation and formulate as linear system of equa-tions Au = f, where u = [u1, . . . , un]

T . HINT: add an extra grid point at xn+1 andapproximate u(1) with a centered finite difference.

In the case n = 3 and f = 1, solve the resulting linear system of equations to findapproximation at the grid points.

11. Let u be the solution of

d2u

dx2= f(x), with u(0) = 1, u(1) = 0. (2)

For a uniformly spaced grid xj = jh for j = 0, . . . , n and h = 1/n, the centred finitedifference approximation uj to u(xj) is the solution to

1

h22uj = fj , j = 1, . . . , n 1

where 2uj = 2uj + uj1 + uj+1 . Prove that the solution of the linear system for uj isunique.



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12. Consider the radially symmetric diffusion problem: find u(r) satisfying

d2u

dr2

1

r

du

dr= 4, r (0, 1),

with boundary conditions u(0) = 1 and u(1) = 0.

(a) For a spacing h = 1/n, let rj = jh for j = 0, . . . , n and derive the centred finitedifference approximation uj for u(rj).

(b) Write down a linear system of equations Au = f for u = [u1, . . . , un1]T , where the

matrix A and vector f should be defined.

(c) Let h = 1/3 and find the approximation uj to u(rj) for j = 1, 2.

(d) Solve the differential equation to find the exact solution and compare with the nu-merical solution.

13. For the PDE in question 8., write down the upwind finite difference approximation.

(a) Show that the truncation error Tj satisfies |Tj | = O(h).(b) By using the discrete maximum principle, show that the upwind finite difference

method is stable

Prove that the global error |u(x) uj | = O(h).


pdes - problems (3)

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