**Jntu Mathematical methods syllabus for first year**

**MATHEMATICAL METHODS**

**UNIT I**: Solution for linear systems TOP

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

Eigen values, eigen vectors properties Condition number of rank, Cayley-Hamilton Theorem (without Proof) - Inverse and powers of a matrix by Cayley-Hamilton theorem Diagonolization of matrix. Calculation of powers of matrix Modal and spectral matrices.

**UNIT III**: Linear Transformations

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, Singular value decomposition.

**UNIT IV :**Solution of Non-linear Systems

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-Difference Equations - 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. B. Spline interpolation - Cubic spline.

**UNIT V :**Curve fitting & Numerical Integration

Curve fitting: Fitting a straight line Second degree curve-exponentional curve-power curve by method of least squares. Numerical Differentiation Simpson s 3/8 Rule , Gaussian Integration, Evaluation of principal value integrals, Generalized Quadrature.

**UNIT VI :**Numerical solution of IVP s in ODE

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-Bashforth Method.

**UNIT VII**Fourier Series

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.

**UNIT VIII**Partial differential equations

Introduction and Formation of partial differential equation 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 for second order equations -Two dimensional wave equation.

**TEXT BOOKS:**

1.Mathematical Methods by P.B.Bhaskara Rao, S.K.V.S. Rama Chary, M.Bhujanga Rao, B.S.Publications.

2.Mathematical Methods by K.V.Suryanarayana Rao by Scitech Publications.

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.Higher Engineering Mathematics by B.S. Grewal, Khanna Publications.

5.Mathematical Methods by V. Ravindranath, Etl, Himalaya Publications.

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