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Mathematics (maths) - Boundary Value Problems In Ordinary And Partial Differential Equations - Important Short Objective Question and Answers: Mathematics (maths) - Boundary Value Problems In Ordinary And Partial Differential Equations

**1.What is the error for solving Laplace and Poisson’s equations by finite difference method?**

__Sol:__

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.**

__Sol:__

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?**

__Sol:__

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?**

__Sol:__

(i)Finite difference method (ii)Shooting method.

**5. What is the classification of one dimensional heat flow equation.**

__Sol:__

One dimensional heat flow equation is

Here A=1,B=0,C=0

*B*2* *−4*AC *= 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) **

__Sol:__

--------- ----------

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*

__Sol:__

The equation is elliptic if (2*B2* ) −4*AC* < 0

(i.e) *B2* −*AC* < 0. It is parabolic if *B2* −*AC* = 0 and hyperbolic if *B2*−4*AC* > 0

9. **Write a note on the stability and convergence of the solution of the difference**

**equation corresponding to the hyperbolic equation** .

__Sol:__

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.**

__Sol:__

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.

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