numerical methods

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NUMERICAL METHODS
QUESTION BANK
UNIT-5
PART-A
1. Define the local truncation error.
2. Write down the standard five point formula used in solving laplace equation U xx + U yy =
0 at the point ( i x, j y ).
3. Derive Crank-Niclson scheme.
4. State Bender Schmidt’s explicit formula for solving heat flow equations
5. Classify x 2 f xx + (1-y 2 ) f yy = 0
6. What is the truncation error of the central difference approximation of
y ' (x)?
7. What is the error for solving Laplace and Poissson’s equation by finite difference method.
8. Obtain the finite difference scheme fore the difference equations 2
2
2
dx
d y
+ y = 5.
9. Write dowm the implicit formula to solve the one dimensional heat equation.
10. Define the diagonal five point formula .
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1. Solve the equqtion Ut = U xx subject to condition u(x,0) = sin x ; 0 x 1,u(0,t) =
u(1,t) =0 using Crank- Nicholson method taking h = 1/3 k = 1/36(do on time step)
2. Solve U xx + U yy = 0 for the following square mesh with boundary values
1 2
1 4
2 5
4 5
3. Solve U xx = Utt with boundary condition u(0,t) = u(4,t) and the initial condition
ut (x,0) = 0 , u(x,0)=x(4-x) taking h =1, k = ½ (solve one period)
4. Solve xy II + y = 0 , y(1) =1,y(2) = 2, h = 0.25 by finite difference method.
5. Solve the boundary value problem xy II -2y + x = 0, subject to y(2) = 0 =y(3).Find
y(2.25),y(2.5),y(2.75).
6 . Solve the vibration problem
2
2
4
x
y
t
y
subject to the boundary conditions
y(0,t)=0,y(8,0)=0 and y(x,0)=
2
1
x(8-x).Find y at x=0,2,4,6.Choosing x = 2, t =
2
1
up
compute to 4 time steps.
7. Solve 2 u = -4(x + y) in the region given 0 x 4, 0 y 4. With all boundaries kept
at 0 0 and choosing x = y = 1.Start with zero vector and do 4 Gauss- Seidal iteration.
0 0 0 0 0 0 0 0 0 0
u1 u 2
u 3 u 4
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0 0
0 0
0 0
0 0 0 0 0 0 0 0 0 0
8. Solve u xx + u yy = 0 over the square mesh of sid e 4 units, satisfying the following
conditions .
u(x,0) =3x for 0 x 4
u(x, 4) = x 2 for 0 x 4
u(0,y) = 0, for 0 y 4
u(4,y) = 12+y for 0 y 4
9. Solve
t
u
x
u
2 2
2
= 0, given that u(0,t)=0,u(4.t)=0.u(x,0)=x(4-x).Assume h=1.Find
the values of u upto t =5.
10. Solve y tt = 4y xx subject to the condition y(0,t) =0, y(2,t)=o, y(x,o) = x(2-x),
(x,0) 0
t
y
. Do 4steps and find the values upto 2 decimal accuracy.