BCA DCA2101 COPUTER ORIENTED NUMERICAL METHOD

Description

 

SESSION	Nov-dec2023
PROGRAM	BCA
SEMESTER	III
course CODE & NAME	DCA2101 & Computer Oriented Numerical Methods

 

 

 

 

 

 

SET-I

 

 

 

 

 

1. Show that

(a)  

(b)  

 

 

Ans 1.

To prove the given identities, let’s start by defining the operators:

= Laplacian operator (also known as the second-order spatial derivative operator)

= Gradient operator (vector of first-order spatial derivatives)

Divergence operator (divergence of a vector field)

Scalar

 

1. b) To prove :

Let’s start with the left-hand side (LHS):

Using the definitions of the Laplacian  and gradient  operators, we have:

Expanding the terms, we can

Its Half solved only

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2. Solve the system of equation by Gauss Elimination’s method

                

                

.

 

 

 

Ans 2.

 

Consider the given system of equations,

2x+y+4z=124

x+11y−z=338

x−3y+2z=20

 

Convert the system to matrix form,

 

The augmented matrix for the above matrix form,

 

 

Q3. Find the equation of the best fitting straight line for the data:

 

X	1	3	4	6	8	9	11	14
Y	1	2	4	4	5	7	8	9

 

 

Ans 3.

To find the equation of the best-fitting straight line for the given data points, you can use linear regression. The equation of a straight line is typically represented as:

Y = mx + b

Where:

• Y is the dependent variable (in this case, the Y values).
• X is the independent variable (in this case, the X values).
• m is the
•  

Set-II

 

 

 

 

Q4. Evaluate f(15), given the following table of values:

 

x	10	20	30	40	50
y = f(x)	46	66	81	93	101

 

Ans 4.

To evaluate f(15) using the given table of values, you can use interpolation. Since the table provides values of y = f(x) for specific values of x, you can interpolate to find the value of f(15) which falls between x = 10 and x = 20.

We can use linear interpolation for this purpose. Linear interpolation assumes that the function f(x) varies linearly between two data points. Here’s how you can calculate f(15):

First, identify the two data

 

 

 

 

Q5. Use Taylor’s series method to solve the initial value problem:

 for   given that .

 

 

Ans 5.

 

The Differential Equation

With initial condition

 

6. Apply Runge-Kutta fourth order method to find an approximate value of y when x = 0.1 given that , .

 

Ans 6.

To apply the Runge-Kutta fourth-order method to solve the given initial value problem, we need to follow these steps:

1. Define the differential equation:
2. Specify the initial