AMath350.F13.A8.Sol

A 2 b cosx c 1 sin x b2 4ac cos2 x 81 sin2

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = 2, b = cos(x), c = (1 sin (x)) ) b2 4ac = cos2 (x) 8(1 sin2 (x)) = 7(sin2 (x) 1). Equation is hyperbolic if sin2 (x) &gt; 1. Never occurs. Equation is parabolic if sin2 (x) = 1 ) x = 2 + k ⇡ , k Equation is elliptic if sin2 (x) &lt; 1 6. uxx 10uxy + 16uyy (a) a = 1, b = Z Z xux + sin(y )u = 0. 10, c = 16 ) b2 4ac = 36 &gt; 0. PDE is hyperbolic. (b) Characteristic equations: p b + b2 4ac dy = dx 2a =2 Integrating: y = 2 x + k1 dy b = dx =8 y= p b2 2a 8 x + k2 4ac AMATH 350 Page 3 Assignment #8 Solutions - Fall 2013 3. With the transformation ⇠ = p ax, ⌘ = y , we have p @u @u @ u d⇠ ˆ ˆ = =a @x @⇠ dx @⇠ ✓◆ ✓ ◆ ✓◆ p @ @ u d⇠ @ 2u @ @u @ p @u ˆ ˆ @ 2u ˆ and = = a =a = a 2. 2 @x @x @x @x @⇠ @⇠ @⇠ dx @⇠ @u @u ˆ @ 2u @ 2u ˆ = , and = 2 . Hence the given equation uxx + auyy + 2 @y @⌘ @y @⌘ bux + cuy + du = f becomes p au⇠⇠ + au⌘⌘ + abu⇠ + cu⌘ + du = f, ˆ ˆ ˆ ˆ ˆ Meanwhile, that is, b c d f u⇠⇠ + u⌘⌘ + p u⇠ + u⌘ + u = . ˆ ˆ ˆ ˆ ˆ a a a a 4. uxx + 2uxy 15uyy = 1 Notice that b2 4ac = 64 &gt; 0, so the equation is hyperbolic. This means that we can solve it by the method of characteristics. To ﬁnd the characteristics, we need to solve the equations p p dy b ± b2 4ac 2 ± 4 + 60 = = = 3, 5. dx 2a 2 Solving the ﬁrst of these, we ﬁnd dy = dx 3 =) y = 3x + C =) 3x + y = C1 . The second one gives dy = 5 =) y = 5x + C, =) 5x dx y = C2 . Therefore a transformation which will work is ⇠ = 3x + y, ⌘ = 5x y (we could also use the reverse, or use multiples of these functions). Next, we need several applications of the chain rule to rewrite the PDE: ux = u⇠ ⇠x + u⌘ ⌘x = 3ˆ⇠ + 5ˆ⌘ . ˆ ˆ u u u y...
View Full Document

This note was uploaded on 02/08/2014 for the course AMATH 350 taught by Professor Davidhamsworth during the Fall '12 term at University of Waterloo, Waterloo.

Ask a homework question - tutors are online