# Vector Calculus

**Topics:**Partial differential equation, Differential equation, Laplace operator

**Pages:**2 (531 words)

**Published:**May 3, 2013

TUTORIAL 1: PARTIAL DIFFERENTIAL EQUATIONS

1. Solve the following equations

a) ∂2u∂x2=24x2(t-2), given that at x=0, u=e2tand ∂u∂x=4t. b) ∂2u∂x∂y=4eycos2x, given that at y=0, ∂u∂x=cosx and at x=π, u=y2. 2. A perfectly elastic string is stretched between two points 10 cm apart. Its centre point is displaced 2 cm from its position of rest at right angles to the original direction of the string and then released with zero velocity. Applying the equation

∂2u∂x2=1c2∙∂2u∂t2

with c2=1, determine the subsequent motion ux,t.

3. One end A of an insulated metal bar AB of length 2 m is kept at 0°C while the other end B is maintained at 50°C until a steady state of temperature along the bar is achieved. At t=0, the end B is suddenly reduced to 0°C and kept at that temperature. Using the heat conduction equation

∂2u∂x2=1c2∙∂u∂t , determine an expression for the temperature at any point in the bar distance x from A at any time t. 4. A square plate is bounded by the lines x=0, y=0, x=2, y=2. Apply the Laplace equation ∂2u∂x2+∂2u∂y2=0

to determine the potential distribution ux,y over the plate, subject to the following boundary conditions.

5. Show that the equation

∂2u∂x2-1c2∙∂2u∂t2=0

is satisfied by u=fx+ct+F(x-ct) where f and F are arbitrary functions. 6. If ∂2u∂x2=1c2∙∂2u∂t2 and c=3, determine the solution u=f(x,t) subject to the boundary conditions u0,t=0 and u2,t=0 for t≥0 ux,0=x(2-x) and ∂u∂tt=0=0 for 0≤x≤2. 7. The centre point of a perfectly elastic string stretched between two points A and B, 4 m apart , is deflected a distance 0.01 m from its position of rest perpendicular to AB and released initially with zero velocity. Apply the wave equation ∂2u∂x2=1c2∙∂2u∂t2 where c=10 to determine the...

Please join StudyMode to read the full document