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probabilty and stas

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UNIT I
1.
When gauss elimination method is not suitable to solve a system of equations? 2.
State the sufficient condition to be satisfied so that an iterative method for solving a system of equations will converge. 3.
Explain direct and iterative methods for solving numerical problems. 4.
Iteration method is a self correction method. Say true or false. 5.
“Iterative method of solving a system of simultaneous linear algebraic equation can be applied to all systems”-say true or false. If false correct the statement 6.
What are the important steps involved in solving the system of linear algebraic equations using Crout’s method? 7.
By Gauss elimination solve x+y=2; 2x+3y=5. 8.
Compare Gauss elimination and Gauss Jordan methods. 9.
Compare Gauss sediel and Jacobi methods. 10.
State the condition for convergence of Gauss seidal. 11.
Gauss elimination and Gauss Jordan are direct methods while _________ and ________ are iterative methods. 12.
What is the basic principle used in Gauss elimination method? 13.
Give two examples for direct method of solving linear equations. 14.
Explain Gauss elimination method 15.
What are the conditions for the convergence of relaxation method? 16.
Gauss seidel method is better than Gauss Jacobi method. Why? 17.
Explain diagonally dominant property in Gauss Jacobi method. 18.
Define residuals in relaxation method. 19.
What are the two types of numerical methods of solving linear equations? 20.
Give two examples for the iterative method of solving a system of linear equations 21.
When Gauss elimination method fails? 22.
What are the principles of Gauss Jordan method? 23.
States true or false, if false correct the statement. “Gauss Jordan method can be applied to any system of simultaneous linear algebraic equations.” 24.
While applying the relaxation method for solving 8x-7y+4z=32, x+5y-3z=18, -2x+2y+7z=19, write the residuals in the initial stage. 25.
As soon as a new value for a variable is found by iteration it is used immediately in the following equations. This method is called ________. 26.
What do you mean by pivoting / partial pivoting? 27.
Explain the diagonally dominant property in iterative methods to solve a system of equations. 28.
If x
(r),
y
(r),
z
(r)
are the r
th
iterates in gauss siedel method, write the scheme for x
(r+1)
, y
(r+1)
, z
(r+1)
. 29.
When do we prefer to apply an iteration method to solve the system of n equations in the same number of variables? 30.
Write down the sufficient conditions for the application of relaxation method. 31.
When does relaxation method succeed? 32.
The rate of convergence in Gauss Siedel method is roughly ______ times than that of Gauss Jacobi method. 33.
Differentiate direct and indirect method. 34.
Can the relaxation method is successful always? If not when it is successful? 35.
Explain ill-conditioned system of equations.
36.
Distinguish between direct and indirect method for solving simultaneous equations. ODE 37.
Using Taylor’s method find y(0.1) given y’=x+y, y(0)=1 correct to two decimal places. 38.
Define pointwise solution. 39.
Explain step-by-step methods for solving ODE. 40.
State the order of error for Taylor series method. 41.
State the order of error for Euler and Improved Euler method. 42.
What are the single step methods of solving a differential equation numerically? Give an example. 43.
Fill in the blank : ____________ is the same as modified Euler method 44.
By Taylors series find y(1.1) given y’=x+y, y(1)=0. 45.
In solving dy/dx=f(x,y), y(x
0
)=y
0
, write down the formula for Eulers improved method. 46.
Compare the Taylor series and Runge Kutta methods. 47.
State any one multi step methods of solving ODE. 48.
State the disadvantages of Taylors series method. 49.
Given y’=1-y and y(0)=0, find y(0.2) by modified Eulers method. 50.
Compute y(0.1) by improved Eulers method given that dy/dx=y+sinx, y(0)=2. 51.
By Eulers method find y(4.1) given y’=(2-y
2
)/5x. 52.
State the RK method of first order for the solution of dy/dx=f(x,y), y(x
0
)=y
0
53.
Write the merits and demerits of Taylor method of solution. 54.
What is the Predictor Corrector method of solving a differential equation? 55.
Write R K method of second order for the solution of dy/dx=f(x,y), y(x
0
)=y
0
56.
Write the improved Euler’s formula. 57.
Write the Milne’s Corrector formula 58.
Write down any 2 demerits of Milne’s method. 59.
Write down the R K formula of fourth order to solve dy/dx=f(x,y) with y(x
0
)=y
0
60.
Define a point wise solution. 61.
State modified Euler’s method to solve y’=f(x,y). y(x
0
)=y
0
at x=x
0
+h. 62.
A particular case of R K method of second order is ________. 63.
What is the use of Corrector formula? 64.
Write down the formula for the predictor used in Milne’s P-C method. 65.
Differentiate Improved Euler and Modified Euler methods. 66.
Explain the distinguishing properties of Runge-Kutta methods. PDE 67.
Classify the following partial differential equation U
xx
+2U
xy
+U
yy
=0 68.
Classify the equation U
xx
+U
xy
+(x
2
+y
2
)U
yy
+x
3
y
2
U
x
+cos(x+y)=0 as elliptic, parabolic and hyperbolic. 69.
Write down the general explicit formula that is used to solve parabolic equations. 70.
Write down the standard five point formula used in solving Laplace equation. 71.
Classify the equation f
xx
+2
xy
+4f
yy
=0. 72.
Write down the diagonal five point formula in solving Laplace equation over a region. 73.
Define a difference quotient. 74.
Define hyperbolic type of partial equation. 75.
Classify the PDE y
2
U
xx
+x
2
U
yy
=0
76.
What is the standard five point formula used in the numerical method solution of
2
∇
U=0? 77.
Classify the equation x
2
U
xx
+(1-y
2
)U
yy
=0, -1<y<1. 78.
Define standard five point formula for laplace equation. 79.
Classify the PDE x
22
/
xU
∂∂
+(y
0)/
22
=∂∂
yU
, x>0,y<0 80.
Classify one dimensional heat equation. 81.
How many conditions are required to solve one dimensional wave equation? 82.
Classify PDE U
xx
+2U
xy
+4U
yy
=0. 83.
Write the standard five point formula for
2
∇
U=f(x,y). 84.
Write down the formula of Crank Nicholson difference scheme to solve the parabolic equation. 85.
Write the difference equation corresponding to the hyperbolic equation. 86.
Classify the equation f
xx
+2f
xy
+4f
yy
=0. 87.
Write down the diagonal five point formula for
2
∇
U=0. 88.
Define hyperbolic equations. 89.
Define parabolic equations. 90.
Classify the PDE : (i) (x+1)U
xx
-2(x+2)U
xy
+(x+3)U
yy
=0. (ii) xf
xx
+yf
yy
=0, x>0,y>0. 91.
Write an explicit formula to solve
.//
22
xUtU
∂∂=∂∂
92.
What is the classification of f
xx
-f
yy
=0? 93.
Write the five point formula to solve u
xx
+u
yy
=0. 94.
Define Liebmanns iteration process. 95.
Classfify the PDE xU
xx
+yU
yy
=0, x>0,y>0. 96.
Define elliptic type of PDE. 97.
Classify the PDE x
2
U
xx
-y
2
U
yy
=0. 98.
Give examples for parabolic equations. 99.
Classify the equation
)./(/1/
22222
tuxu
∂∂=∂∂
α
100.
Name the two methods to solve one dimensional heat equation. 101.
Reduce the Poisson’s equation U
xx
+U
yy
=f(x,y) to a difference scheme at the points of a square mesh, taking x=ih, y=jk(=jh). 102.
Discuss the stability analysis of Bender-Schmidt scheme and Crank-Nicholson scheme. 103.
Discuss the stability analysis of explicit formula for solving wave equations. 104.
Write the finite difference explicit formula for solving U
tt
=a
2
U
xx
105.
Express
2
∇
u=f(x,y) in terms of difference quotients. 106.
In solving the wave equation how will you express the initial condition
.0)0,(
=
xu
t
107.
Classify the PDE : x
2
U
xx
-y
2
U
yy
=0. 108.
Classify the equation x
2
f
xx
-(1-y
2
)f
yy
=0 for all x and y
∈
[-1,1]. 109.
Give examples of elliptical and parabolic equations. 110.
What is the classification of f
xx
=2f
xy
? 111.
Write down the explicit scheme to solve one dimensional wave equation 112.
What is Bender-Schmidt Recurrence equation? For what purpose you use it? 113.Write the purpose of Liebmann’s process. 114. Write down Crank-Nicholson explicit formula. 115. What should be step size in ‘t’ direction to use simplified Crank-Nicholson formula? 116. What should be step size in ‘t’ direction to use simplified Bender-Schmidt formula? 117. List out any two methods to solve parabolic pde. 118. Write the difference scheme for Poisson equation.

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