SRM University 2007 B.Tech Food Process Engineering NUMERICAL SOLUATIONS FOR BOUNDARY VALUE PROBLEMS - Question Paper
Wednesday, 30 January 2013 05:00Web
UNIT – V
NUMERICAL SOLUATIONS FOR BOUNDARY VALUE issues
PART - A
1. Define a difference quotient
2. Define elliptic kind of partial differential equations.
3. Define parabolic kind of partial differential equations.
4. Define Hyperbolic kind of partial differential equations.
5. Classify the formula Uxx+2Uxy + 4Uyy = 0.
6. Write down standard 5 point formula in solving laplace formula over a region.
7. Write down diagonal 5 point formula is solving laplace formula over a region.
8. What is the purpose of liebmann’s process?
9. What is Bender-schmidt recurrence equation?
10. For what purpose Bender-schmidt recurrence formula is used?
11. Write down Crank-Nicholson difference method.
12. What is the purpose of crank-Nicholson difference method?
13. Name two methods that you use to solve 1 dimensional heat formula.
14. Express a2Uxx = Utt in terms of difference quotients
15. Write down the explicit scheme to solve 1 dimensional wave formula.
16. In solving the wave formula how will the critical condition Ut(x,0)=0 be expressed?
17. . In the explicit formula got in connection with solving wave equation, what is the simple form to which it reduces?
18. What is the relaxation operator that we have in solving ?2u = 0 by relaxation method?
19. Express a2uxx = utt in terms of difference quotients.
20. If u is harmonic, will satisfy ?2u = 0?
21. The boundary conditions in solving Uxx + Uyy = 0 are
(1) U(0,y) = 0 for 0? y ? 4
(2) U(4,y) = 12+y for 0? y ? 4
(3) U(x,0) = 3x for 0? x ? 4
(4) U(x,4) = x+2 for 0? x ?4
Plot the values on U on the boundary at the grid points taking h=k=1
.
22. Write down the explicit scheme to solve 1 dimensional wave formula.
23. Name 2 methods that you use to solve 1 dimensional heat formula.
PART-B
1. Given the value of u(x, y) on the boundary of the square region shown in the figure. Evaluare u(x,y) satisfying Laplace formula =0 at the pivotal points.
1000 1000 1000 1000
2000
u1
u2
500
2000
u3
u4
0
1000 500 0 0
2. Solve over Uxx + Uyy = 0 over the square mesh of side four units : satisfying the
subsequent the boundary conditions.
(i) U(0,y) = 0 for ? y ? 4
(ii) U(4,y) = 12+y, for 0? y ? 4
(iii) U(x,0) = 3x for 0? x ? 4
(iv) U(x,4) = x2 for 0? x ? 4
3. Solve = -10(x2+y2+10) over the square mesh with sides x=0, y=0, x=3, y=3 with u=0 on the boundary and mesh length one unit.
4. Solve Uxx + Uyy = 0 for the subsequent square mesh with provided boundary conditions:
0 500 1000 500 0
1000
u1
u2
u3
1000
2000
u4
u5
u6
2000
1000
u7
u8
u9
1000
0 500 1000 500 0
5. Solve provided U(0,t)=0, U(4,t)=0, U(x,0)= x(4-x) assuming h=k=1. obtain the values of U upto t=5.
6. Solve Ut = Uxx subject to U(0,t) = 0, U(1, t) = 0 and U(x, 0) = sin x, 0< x <1.
7. Solve by Crank Nicholson method the formula Uxx=Ut subject to U(x, 0)=0, U(0,t)=0 and U(l, t)=t, for 2 time steps.
8. Using Crank-Nicholson’s Scheme, Solve
Uxx = 16Ut, 0
U(x,0)=0, U(0,t)=0, U(1,t)=100t.
9. Using Crank-Nicholson method solve subject to U(x,0)=0, U(0,t)=0 and U(1,t)=t, (i) taking h=0.5 and k= 1/8 (ii) h=1/4 and k=1/8.
10. Solve numerically 4Uxx = Utt with the boundary conditions U(0,t)=0, U(4,t)=0 and the initial conditions Ut(x,0)=0, and U(x,0)=x(4-x) taking h=1 (for four time steps)
11. Solve 25 Uxx-Utt=0 for U at the pivotal points, provided U(0,t)=U(5,t)=0, Ut(x,0)=0 and U(x,0)=2x for 0?x ?2.5.
= 10-2x for 2.5 ? x ? 5.
12. Evaluate the pivotal values of the subsequent formula h =1 and upto 1 half of the period of the oscillation 16 Uxx = Utt provide U(0,t)=U(5,t) = 0, U(x,0) = x2(5-x) and Ut¬ (x,0)=0.Solve , O
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