# Syllabus - 3rd Semester -Engineering Mathematics III - Subject Code - 06MAT31

## Syllabus - 3rd Semester -Engineering Mathematics III - Subject Code - 06MAT31

Engineering Mathematics III

PART – A

UNIT 1:

Periodic functions, Fourier expansions, Half range expansions, Complex form of Fourier series, Practical harmonic analysis.

7 Hours

UNIT 2:

Finite and Infinite Fourier transforms, Fourier sine and consine transforms, properties. Inverse transforms.

6 Hours

UNIT 3:

Partial Differential Equations (P.D.E)

Formation of P.D.E Solution of non homogeneous P.D.E by direct integration, Solution of homogeneous P.D.E involving derivative with respect to one independent variable only (Both types with given set of conditions) Method of separation of variables. (First and second order equations) Solution of Lagrange’s linear P.D.E. of the type P p + Q q = R.

6 Hours

UNIT 4:

Derivation of one dimensional wave and heat equations. Various possible solutions of these by the method of separation of variables. D’Alembert’s solution of wave equation. Two dimensional Laplace’s equation – various possible solutions. Solution of all these equations with specified boundary conditions. (Boundary value problems).

7 Hours

PART – B

UNIT 5:

Introduction, Numerical solutions of algebraic and transcendental equations:- Newton-Raphson and Regula-Falsi methods. Solution of linear simultaneous equations : - Gauss elimination and Gauss Jordon methods. Gauss - Seidel iterative method. Definition of eigen values and eigen vectors of a square matrix. Computation of largest eigen value and the corresponding eigen vector by Rayleigh’s power method.

6 Hours

UNIT 6:

Finite differences (Forward and Backward differences) Interpolation, Newton’s forward and backward interpolation formulae. Divided differences – Newton’s divided difference formula. Lagrange’s interpolation and inverse interpolation formulae. Numerical differentiation using Newton’s forward and backward interpolation formulae. Numerical Integration – Simpson’s one third and three eighth’s value, Weddle’s rule.

(All formulae / rules without proof)

7 Hours

UNIT 7:

Variation of a function and a functional Extremal of a functional, Variational problems, Euler’s equation, Standard variational problems including geodesics, minimal surface of revolution, hanging chain and Brachistochrone problems.

6 Hours

UNIT 8:

Difference equations – Basic definitions. Z-transforms – Definition, Standard Z-transforms, Linearity property, Damping rule, Shifting rule, Initial value theorem, Final value theorem, Inverse Z-transforms. Application of Z-transforms to solve difference equations.

0. Text Book: Higher Engineering Mathematics by Dr. B.S. Grewal (36th Edition – Khanna Publishers

1. Higher Engineering Mathematics by B.V. Ramana (Tata-Macgraw Hill).

2. Advanced Modern Engineering Mathematics by Glyn James – Pearson Education.

1. One question is to be set from each unit.

2. To answer Five questions choosing atleast Two questions from each part.

PART – A

UNIT 1:

**Fourier Series**Periodic functions, Fourier expansions, Half range expansions, Complex form of Fourier series, Practical harmonic analysis.

7 Hours

UNIT 2:

**Fourier Transforms**Finite and Infinite Fourier transforms, Fourier sine and consine transforms, properties. Inverse transforms.

6 Hours

UNIT 3:

Partial Differential Equations (P.D.E)

Formation of P.D.E Solution of non homogeneous P.D.E by direct integration, Solution of homogeneous P.D.E involving derivative with respect to one independent variable only (Both types with given set of conditions) Method of separation of variables. (First and second order equations) Solution of Lagrange’s linear P.D.E. of the type P p + Q q = R.

6 Hours

UNIT 4:

**Applications of P.D.E**Derivation of one dimensional wave and heat equations. Various possible solutions of these by the method of separation of variables. D’Alembert’s solution of wave equation. Two dimensional Laplace’s equation – various possible solutions. Solution of all these equations with specified boundary conditions. (Boundary value problems).

7 Hours

PART – B

UNIT 5:

**Numerical Methods**Introduction, Numerical solutions of algebraic and transcendental equations:- Newton-Raphson and Regula-Falsi methods. Solution of linear simultaneous equations : - Gauss elimination and Gauss Jordon methods. Gauss - Seidel iterative method. Definition of eigen values and eigen vectors of a square matrix. Computation of largest eigen value and the corresponding eigen vector by Rayleigh’s power method.

6 Hours

UNIT 6:

Finite differences (Forward and Backward differences) Interpolation, Newton’s forward and backward interpolation formulae. Divided differences – Newton’s divided difference formula. Lagrange’s interpolation and inverse interpolation formulae. Numerical differentiation using Newton’s forward and backward interpolation formulae. Numerical Integration – Simpson’s one third and three eighth’s value, Weddle’s rule.

(All formulae / rules without proof)

7 Hours

UNIT 7:

**Calculus of Variations**Variation of a function and a functional Extremal of a functional, Variational problems, Euler’s equation, Standard variational problems including geodesics, minimal surface of revolution, hanging chain and Brachistochrone problems.

6 Hours

UNIT 8:

**Difference Equations and Z-transforms**Difference equations – Basic definitions. Z-transforms – Definition, Standard Z-transforms, Linearity property, Damping rule, Shifting rule, Initial value theorem, Final value theorem, Inverse Z-transforms. Application of Z-transforms to solve difference equations.

**Reference Books:**0. Text Book: Higher Engineering Mathematics by Dr. B.S. Grewal (36th Edition – Khanna Publishers

1. Higher Engineering Mathematics by B.V. Ramana (Tata-Macgraw Hill).

2. Advanced Modern Engineering Mathematics by Glyn James – Pearson Education.

**Note:**1. One question is to be set from each unit.

2. To answer Five questions choosing atleast Two questions from each part.

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