1.What is the error for solving Laplace and Poisson’s equations by finite difference method?
The error in replacing by the difference expression is of the order . Since h=k, the error in replaing by the difference expression is of the order .
2. Define a difference quotient.
A difference quotient is the quotient obtained by dividing the difference between two values of a function by the difference between two corresponding values of the independent variable.
3. Why is Crank Nicholson’s scheme called an implicit scheme?
The Schematic representation of crank Nicholson method is shown below.
The solution value at any point (i,j+1) on the (j +1)th level is dependent on the solution values at the neighboring points on the same level and on three values on the j th level. Hence it is an implicit method.
4. What are the methods to solve second order boundary-value problems?
(i)Finite difference method (ii)Shooting method.
5. What is the classification of one dimensional heat flow equation.
One dimensional heat flow equation is
B2 −4AC = 0
Hence the one dimensional heat flow equation is parabolic.
6. 6. State Schmidt’s explicit formula for solving heat flow equation
Sol: ---- ----
7. Write an explicit formula to solve numerically the heat equation (parabolic equation)
x and k is the space in the time direction).
The above formula is a relation between the function values at the two levels j+1 and j and is called a two level formula. The solution value at any point (i,j+1) on the (j+1)th level is expressed in terms of the solution values at the points (i-1,j),(i,j) and (i+1,j) on the j th level.Such a method is called explicit formula. the formula is geometrically represented below.
8. State the condition for the equation to be
(i) elliptic,(ii)parabolic(iii)hyperbolic when A,B,C are functions of x and y
The equation is elliptic if (2B2 ) −4AC < 0
(i.e) B2 −AC < 0. It is parabolic if B2 −AC = 0 and hyperbolic if B2−4AC > 0
9. Write a note on the stability and convergence of the solution of the difference
equation corresponding to the hyperbolic equation .
For ,λ= the solution of the difference equation is stable and coincides with the solution of the differential equation. For λ> ,the solution is unstable.
For λ< ,the solution is stable but not convergent.
10. State the explicit scheme formula for the solution of the wave equation.
The formula to solve numerically the wave equation =0 is
The schematic representation is shown below.
The solution value at any point (i,j+1) on the ( j +1)th level is expressed in terms of solution values on the previous j and (j-1) levels (and not interms of values on the same level).Hence this is an explicit difference formula.