# JNTU 2007-2008 1ST YEAR B.Tech -MATHEMATICAL METHODS

Dec 23, 2010

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY

I Year B.Tech. M.E. T P C

3+1* 0 6

MATHEMATICAL METHODS

UNIT - I

Matrices and Linear systems of equations: Elementary row transformations-Rank-Echelon form, Normal form - Solution of Linear Systems - Direct Methods- LU Decomposition- LU Decomposition from Gauss Elimination -Solution of Tridiagonal Systems-Solution of Linear Systems

UNIT - II

Eigen values, eigen vectors - properties - Cayley-Hamilton Theorem - Inverse and powers of a matrix by Cayley-Hamilton theorem - Diagonolization of matrix. Calculation of powers of matrix - Modal and spectral matrices.

UNIT - III

Real matrices - Symmetric, skew - symmetric, orthogonal, Linear Transformation - Orthogonal Transformation. Complex matrices: Hermitian, Skew-Hermitian and Unitary - Eigen values and eigen vectors of complex matrices and their properties. Quadratic forms- Reduction of quadratic form to canonical form - Rank - Positive, negative definite - semi definite - index - signature - Sylvester law.

UNIT - IV

Solution of Algebraic and Transcendental Equations: Introduction - The Bisection Method - The Method of False Position - The Iteration Method - Newton-Raphson Method.

Interpolation: Introduction- Errors in Polynomial Interpolation - Finite differences- Forward Differences- Backward differences -Central differences - Symbolic relations and separation of symbols-Differences of a polynomial-Newton's formulae for interpolation - Central difference interpolation Formulae - Gauss Central Difference Formulae -Interpolation with unevenly spaced points-Lagrange's Interpolation formula.

UNIT - V

Curve fitting: Fitting a straight line -Second degree curve-exponentional curve-power curve by method of least squares. Numerical Differentiation and Integration- Trapezoidal rule - Simpson's 1/3 Rule -Simpson's 3/8 Rule.

UNIT - VI

Numerical solution of Ordinary Differential equations: Solution by Taylor's series-Picard's Method of successive Approximations-Euler's Method-Runge-Kutta Methods -Predictor-Corrector Methods- Adams- Moulton Method -Milne's Method.

UNIT - VII

Fourier Series: Determination of Fourier coefficients - Fourier series - 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 (only statement)- Fourier sine and cosine integrals. Fourier transform - Fourier sine and cosine transforms - properties - inverse transforms - Finite Fourier transforms.

UNIT - VIII

Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions -solutions of first order linear (Lagrange) equation and nonlinear (standard type) equations. Method of separation of variables. z-transform - inverse z-transform - properties - Damping rule - Shifting rule - Initial and final value theorems. Convolution theorem - Solution of difference equation by z-transforms.