(SEM VI) THEORY EXAMINATION 2022-23 COMPUTER BASED NUMERICAL TECHNIQUES

COMPUTER BASED NUMERICAL TECHNIQUES – KOE-065

Section-wise Important Questions & Ready Answers

SECTION A

(Attempt all questions – 2 marks each)

(a) Order and Degree of a Differential Equation

The order of a differential equation is the highest order derivative present in the equation.
The degree is the power of the highest order derivative, provided the equation is free from radicals and fractions involving derivatives.

(b) Complementary Function of a Linear Differential Equation

The complementary function (CF) is the solution of the homogeneous part of a differential equation. It is obtained by solving the auxiliary equation formed by replacing derivatives with algebraic terms.

(c) Singular Point of a Differential Equation

A point x=ax = ax=a is called a singular point if the coefficients of the highest derivative become infinite or undefined at that point. Otherwise, it is called an ordinary point.

(d) Proof of a Standard Beta–Gamma Identity

Using properties of the Gamma function, standard identities such as

Γ(1)=1\Gamma(1)=1Γ(1)=1

can be proved directly from its integral definition.

(e) Gamma Function

The Gamma function is defined as

Γ(n)=∫0∞xn−1e−x dx,n>0\Gamma(n)=\int_0^\infty x^{n-1} e^{-x} \, dx,\quad n>0Γ(n)=∫0∞​xn−1e−xdx,n>0

It generalizes the factorial function for real and complex numbers.

(f) Bessel’s Equation

Bessel’s equation of order nnn is

x2d2ydx2+xdydx+(x2−n2)y=0x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-n^2)y=0x2dx2d2y​+xdxdy​+(x2−n2)y=0 

(g) Differentiation of Matrix and Determinant

A matrix can be differentiated by differentiating each element separately.
The derivative of a determinant is obtained using differentiation of its elements along with cofactor expansion.

(h) Orthogonal Matrix with Example

A matrix AAA is orthogonal if

ATA=IA^T A = IATA=I

Example:

[cos⁡θsin⁡θ−sin⁡θcos⁡θ]\begin{bmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{bmatrix}[cosθ−sinθ​sinθcosθ​] 

(i) Effect of Boundary Conditions in Heat Transfer Problems

Boundary conditions determine temperature distribution and stability of solutions. Incorrect boundary conditions may lead to non-physical or unstable solutions in unsteady heat transfer problems.

(j) Steady State and Transient State

In steady state, system variables do not change with time.
In transient state, variables vary with time until steady state is reached.

SECTION B

(Attempt any three – 10 marks each)

2(a) Solution of Simultaneous Differential Equations

The given simultaneous differential equations are solved by differentiating one equation, eliminating variables, and reducing the system to a single differential equation. The complementary function and particular integral are obtained to get the final solution.

2(b) Proof of Gamma Function Identity

Using integration by parts and properties of the Gamma function, identities involving Γ(x)\Gamma(x)Γ(x) are derived. These identities are widely used in numerical integration and probability theory.

2(c) Solution of a Given Differential Equation

The given differential equation is solved by applying standard methods such as reduction to known form, substitution, or integrating factor, leading to the required solution.

2(d) Eigen Values and Eigen Vectors of a Matrix

Eigen values are obtained by solving

∣A−λI∣=0|A-\lambda I|=0∣A−λI∣=0

Eigen vectors are obtained by solving

(A−λI)x=0(A-\lambda I)x=0(A−λI)x=0

for each eigen value.

2(e) Counter-Current Liquid–Liquid Extraction

In counter-current extraction, solvent and feed move in opposite directions, increasing mass transfer efficiency. It is widely used in chemical and petroleum industries.

SECTION C

3(a) Complete Solution of a Differential Equation

The complete solution consists of the complementary function and particular integral. The particular integral is obtained using standard methods such as operator method or variation of parameters.

3(b) Solution of Exact Differential Equation

The given equation is checked for exactness. If exact, it is integrated partially with respect to one variable and then fully integrated to obtain the general solution.