This course covers numerical methods used to solve engineering and social problems. It includes 4 units: (1) solving algebraic and transcendental equations using Newton-Raphson and Regula-Falsi methods and solving linear systems using Gaussian elimination; (2) interpolation and approximation using finite differences, Newton's formulas, and Lagrange interpolation; (3) numerical differentiation and integration including Simpson's rule and Gaussian quadrature; (4) solving ordinary differential equations using methods like Euler, Runge-Kutta, and predictor-corrector approaches. The course aims to help students understand basic numerical concepts and apply them to problems.
This course covers numerical methods used to solve engineering and social problems. It includes 4 units: (1) solving algebraic and transcendental equations using Newton-Raphson and Regula-Falsi methods and solving linear systems using Gaussian elimination; (2) interpolation and approximation using finite differences, Newton's formulas, and Lagrange interpolation; (3) numerical differentiation and integration including Simpson's rule and Gaussian quadrature; (4) solving ordinary differential equations using methods like Euler, Runge-Kutta, and predictor-corrector approaches. The course aims to help students understand basic numerical concepts and apply them to problems.
This course covers numerical methods used to solve engineering and social problems. It includes 4 units: (1) solving algebraic and transcendental equations using Newton-Raphson and Regula-Falsi methods and solving linear systems using Gaussian elimination; (2) interpolation and approximation using finite differences, Newton's formulas, and Lagrange interpolation; (3) numerical differentiation and integration including Simpson's rule and Gaussian quadrature; (4) solving ordinary differential equations using methods like Euler, Runge-Kutta, and predictor-corrector approaches. The course aims to help students understand basic numerical concepts and apply them to problems.
This course covers numerical methods used to solve engineering and social problems. It includes 4 units: (1) solving algebraic and transcendental equations using Newton-Raphson and Regula-Falsi methods and solving linear systems using Gaussian elimination; (2) interpolation and approximation using finite differences, Newton's formulas, and Lagrange interpolation; (3) numerical differentiation and integration including Simpson's rule and Gaussian quadrature; (4) solving ordinary differential equations using methods like Euler, Runge-Kutta, and predictor-corrector approaches. The course aims to help students understand basic numerical concepts and apply them to problems.
Objective: The Objective of this course is to help student to understand basic concepts in numerical methods and able to solve engineering as well as social problems.
COURSE DESCRIPTION:
UNIT I: Introduction to Numerical solutions of algebraic and transcendental equations
Newton-Raphson and Regula-Falsi methods. Solution of linear simultaneous equations: Gauss elimination and Gauss Jordon methods. Iterative methods: Gauss Jacobi and Gauss-Seidel methods. 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.
UNIT II: Interpolation and Approximation
Finite differences (Forward and Backward differences) Interpolation, Newton’s forward and backward interpolation formulae. Newton’s divided difference formula. Lagrange’s interpolation and inverse interpolation formulae.
UNIT III: Numerical differentiation and Numerical Integration
Numerical differentiation using Newton’s forward and backward interpolation formulae. Numerical Integration: Simpson’s one third and three eighth’s value, Boole’s rule, Weddle’s rule, Romberg’s method, two and three point Gaussian quadrature formulae, Double integrals using trapezoidal and Simpson’s rules.
UNIT IV: Ordinary Differential Equations
Single-step explicit methods: Taylor series method, Euler and modified Euler methods, Solution of first order and simultaneous equations by Euler’s and Picard’s method, Fourth order Runge Kutta method for solving first and second order equations, single-step implicit methods, multistep linear methods, Milne’s and Adam’s predictor and corrector methods.
Text Books: 1. Balagurusamy, E., “Numerical Methods”, Tata McGraw-Hill Pub.Co.Ltd, New Delhi, 1999. 2. S.S. Sastry, “Introductory Methods of Numerical Analysis”, PHI learning Pvt Ltd. 3. Curtis F. Gerald and Patrick O.Wheatley, “Applied Numerical Analysis”, Pearson Education Ltd 4. Gerald, C.F, and Wheatley, P.O, “Applied Numerical Analysis”, Sixth Edition, Pearson Education Asia, New Delhi, 2002. 5. Higher Engineering Mathematics by Dr. B.S. Grewal, 36th Edition, Khanna Publishers. 6. S.C. Chapra and R.P. Canale, “Numerical Methods for Engineers with Programming and Software Applications”, McGraw-Hill, Newyork – 1998 7. Kandasamy, P., Thilagavathy, K. and Gunavathy, K., “Numerical Methods”, S.Chand Co. Ltd., New Delhi, 2003. 8. Burden, R.L and Faires, T.D., “Numerical Analysis”, Seventh Edition, Thomson Asia Pvt. Ltd., Singapore, 2002.
Course Plan: Week Unit Topics Hours Theory Practical Tutorial 1 I Introduction to Numerical solutions of algebraic and 3 0 1 transcendental equations: Newton-Raphson and Regula-Falsi methods. 2 I Solution of linear simultaneous equations: Gauss 3 0 1 elimination and Gauss Jordon methods. 3 I Iterative methods: Gauss Jacobi and Gauss-Seidel 3 0 1 methods. 4 I Definition of Eigen values and Eigen vectors of a 3 0 1 square matrix. Computation of largest Eigen value and the corresponding Eigen vector by Rayleigh’s power method. 5 II Finite differences (Forward and Backward 3 0 1 differences) Interpolation, Newton’s forward and backward interpolation formulae. 6 II Divided differences: Newton’s divided difference 3 0 1 formula. Lagrange’s interpolation and inverse interpolation formulae. 7 III Numerical differentiation using Newton’s forward 3 0 1 and backward interpolation formulae. 8 III Numerical Integration: Simpson’s one third and 3 0 1 three eighth’s value, Boole’s rule, Weddle’s rule 9 III Romberg’s method, two and Three point Gaussian 3 0 1 quadrature formulas. 10 III Double integrals using trapezoidal and Simpson’s 3 0 1 rules. 11 IV Single-step explicit methods: Taylor series method, 3 0 1 Euler and modified Euler methods, 12 IV simultaneous equations by Euler’s and Picard’s method 13 IV Fourth order Runge Kutta method for solving first 3 0 1 and second order equations. 14 IV Single-step implicit methods, multistep linear 3 0 1 methods, 15 IV Milne’s and Adam’s predictor and corrector 3 0 1 methods. 16 Revision 3 0 1
