Note: This syllabus is same for AE, Bio-Tech, Chem Engg, CE, CEE, Metallurgy and Material Engg., PE.

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
II Year B.Tech. CE - I Sem L T/P/D C
4 -/-/- 4

MATHEMATICS - II

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Objectives:

• The objective is to find the relation between the variables x and y out of the given data (x,y).
• The aim to find such relationships which exactly pass through data or approximately satisfy the data under the condition of least sum of squares of errors.
• The aim of numerical methods is to provide systematic methods for solving problems in a numerical form using the given initial data.
• This topic deals with methods to find roots of an equation and solving a differential equation.
• The numerical methods are important because finding an analytical procedure to solve an equation may not be always available.
• In the diverse fields like electrical circuits, electronic communication, mechanical vibration and structural engineering, periodic functions naturally occur and hence their properties are very much required.
• Indeed, any periodic and non-periodic function can be best analyzed in one way by Fourier series and transforms methods.
• The aim at forming a partial differential equation (PDE) for a function with many variables and their solution methods. Two important methods for first order PDE’s are learnt. While separation of variables technique is learnt for typical second order PDE’s such as Wave, Heat and Laplace equations.
• In many Engineering fields the physical quantities involved are vector-valued functions.
• Hence the unit aims at the basic properties of vector-valued functions and their applications to line integrals, surface integrals and volume integrals.

UNIT – I

Vector Calculus: Vector Calculus: Scalar point function and vector point function, Gradient- Divergence- Curl and their related properties. Solenoidal and irrotational vectors – finding the Potential function. Laplacian operator. Line integral – work done – Surface integrals -Volume integral. Green’s Theorem, Stoke’s theorem and Gauss’s Divergence Theorems (Statement & their Verification).

UNIT – II

Fourier series and Fourier Transforms: Definition of periodic function. Fourier expansion of periodic functions in a given interval of length 2 . Determination of Fourier coefficients – Fourier series of even and odd functions – Fourier series in an arbitrary interval – even and odd periodic continuation – Half-range Fourier sine and cosine expansions.
Fourier integral theorem - Fourier sine and cosine integrals. Fourier transforms – Fourier sine and cosine transforms – properties – inverse transforms – Finite Fourier transforms.

UNIT – III

Interpolation and Curve fitting

Interpolation: Introduction- Errors in Polynomial Interpolation – Finite differences- Forward Differences- Backward differences –Central differences – Symbolic relations of symbols. Difference expressions – Differences of a polynomial-Newton’s formulae for interpolation - Gauss Central Difference Formulae –Interpolation with unevenly spaced points-Lagrange’s Interpolation formula.
Curve fitting: Fitting a straight line –Second degree curve-exponential curve-power curve by method of least squares.

UNIT – IV : Numerical techniques

Solution of Algebraic and Transcendental Equations and Linear system of equations: Introduction – Graphical interpretation of solution of equations .The Bisection Method – The Method of False Position – The Iteration Method – Newton-Raphson Method .
Solving system of non-homogeneous equations by L-U Decomposition method (Crout’s Method). Jacobi’s and Gauss-Seidel iteration methods.

UNIT – V

Numerical Integration and Numerical solutions of differential equations:

Numerical integration - Trapezoidal rule, Simpson’s 1/3rd and 3/8 Rule , Gauss-Legendre one point, two point and three point formulas.
Numerical solution of Ordinary Differential equations: Picard’s Method of successive approximations. Solution by Taylor’s series method – Single step methods-Euler’s Method-Euler’s modified method, Runge-Kutta (second and classical fourth order) Methods.

Boundary values & Eigen value problems: Shooting method, Finite difference method and solving eigen values problems, power method

TEXT BOOKS:

1. Advanced Engineering Mathematics by Kreyszig, John Wiley & Sons.
2. Higher Engineering Mathematics by Dr. B.S. Grewal, Khanna Publishers.

REFERENCES:

1. Mathematical Methods by T.K.V. Iyengar, B.Krishna Gandhi & Others, S. Chand.
2. Introductory Methods by Numerical Analysis by S.S. Sastry, PHI Learning Pvt. Ltd.
3. Mathematical Methods by G.Shankar Rao, I.K. International Publications, N.Delhi
4. Advanced Engineering Mathematics with MATLAB, Dean G. Duffy, 3rd Edi, 2013, CRC Press Taylor & Francis Group.
5. Mathematics for Engineers and Scientists, Alan Jeffrey, 6ht Edi, 2013, Chapman & Hall/ CRC
6. Advanced Engineering Mathematics, Michael Greenberg, Second Edition. Person Education
7. Mathematics For Engineers By K.B.Datta And M.A S.Srinivas, Cengage Publications

Outcomes:

From a given discrete data, one will be able to predict the value of the data at an intermediate point and by curve fitting, can find the most appropriate formula for a guessed relation of the data variables. This method of analysis data helps engineers to understand the system for better interpretation and decision making

• After studying this unit one will be able to find a root of a given equation and will be able to find a numerical solution for a given differential equation.
• Helps in describing the system by an ODE, if possible. Also, suggests to find the solution as a first approximation.
• One will be able to find the expansion of a given function by Fourier series and Fourier Transform of the function.
• Helps in phase transformation, Phase change and attenuation of coefficients in acoustics.
• After studying this unit, one will be able to find a corresponding Partial Differential Equation for an unknown function with many independent variables and to find their solution.
• Most of the problems in physical and engineering applications, problems are highly non-linear and hence expressing them as PDEs’. Hence understanding the nature of the equation and finding a suitable solution is very much essential.
• After studying this unit, one will be able to evaluate multiple integrals (line, surface, volume integrals) and convert line integrals to area integrals and surface integrals to volume integrals.
• It is an essential requirement for an engineer to understand the behavior of the physical system.