Mathematical Methods Notes (MM)
UNIT – I : Solution for linear systems
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.
Mathematical Methods Notes (MM) 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.
Mathematical Methods Notes (MM) 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.
6. A text Book of KREYSZIG’S Mathematical Methods, Dr .A. Ramakrishna Prasad. WILEY publications.
