p01 Lecture _ Math - III(A) - (Syllabus and Introduction class) - Dr. T. Singh -D16.11.2021.pdf

Department of Mathematics MTH21103: MATHEMATICS – IIIA Dr. Trailokyanath Singh Associate Professor Contents 01 SYLLABUS (MATHEMATICS – IIIA) 02 MATHEMATICS – IIIA (INTRODUCTION)...

Department of Mathematics MTH21103: MATHEMATICS – IIIA Dr. Trailokyanath Singh Associate Professor Contents 01 SYLLABUS (MATHEMATICS – IIIA) 02 MATHEMATICS – IIIA (INTRODUCTION) 03 SIGNIFICANT FIGURES 04 SIGNIFICANT FIGURES RULES 05 EXAMPLES 16-11-2021 1 C. V. Raman Global University, Odisha Syllabus MTH21103: MATHEMATICS–IIIA (3-0-1) Unit – 1: NUMERICAL SOLUTION OF TRANSCENDENTAL AND LINEAR SYSTEM OF EQUATIONS: [07 hours] Numerical methods: Approximation and round of errors. Roots of equation: Newton-Raphson method, Secant Method, Muller’s method. [T1] Interpolation: Lagrange Interpolation, Newton’s forward and backward interpolation. Solution of System of Linear equations: by Gauss-Seidel method. [T2] Unit – 2: NUMERICAL INTEGRATION AND NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS: [07 hours] Numerical integration: The trapezoidal rule, The Simpson’s rules, Gauss quadrature. Numerical Solutions of Ordinary Differential Equations: Euler’s method, Improvement of Euler’s method, Runge-Kutta methods. Adam-Bashforth and Adam-Moulton Multistep Methods to solve Ordinary Differential Equations. [T1]2 16-11-2021 C. V. Raman Global University, Odisha Syllabus (Continued…) Unit – 3: COMPLEX ANALYSIS: [07 hours] Complex Analytic functions: Complex Analytic functions, Cauchy-Riemann equations, Laplace equation. Complex Integration: Cauchy’s Integral Theorem and formula, Derivatives of analytic functions. Laurent Series and Residue Integration: Laurent’s series, Singularities and zeros, Residue integration method, Evaluation of real integrals. Uniform Convergence of Complex Power Series. Unit – 4: PROBABILITY THEORY: [07 hours] Probability: Probability, Random variables, Probability distributions, Mathematical Expectation, Mean and variance of distribution, Binomial, Poisson, and Normal distributions. [T2], Rayleigh distribution, Rician distribution, Central Limit Theorem. [T3] Random Processes – Temporal Characteristics: Introduction, The Random Process Concept, Stationary and Independence, Correlation Functions. Power spectral density. 16-11-2021 2 C. V. Raman Global University, Odisha Syllabus (Continued…) Unit – 5: MATHEMATICAL STATISTICS: [07 hours] Mathematical Statistics: Mean and Variance of a data, Random sampling, Estimation of Parameters, Confidence Intervals, Testing of hypothesis, Acceptance sampling, Chi square test for goodness of fit , Regression Analysis, Fitting Straight Lines, Correlation analysis. Nonparametric Tests. Text Books: T1. “Numerical methods for Engineers, Steven C. Chapra and Raymond P. Canale, Tata McGraw-Hill Publishing Company Limited, New Delhi”, Fifth Edition,2007. Chapters 2, 5 (5.2, 5.3), 6 (6.4), 16(16.1, 16.2), 20(20.1, 20.2, 20.3) T2. “Advanced Engineering Mathematics, Erwin Kreyszig, John Willy and Sons”, 8th Edition, 1999. Chapters: 12(12.3, 12.4), 13(13.2 – 13.4), 15, 17(17.3), 22 (22.5-22.8), 23(23.1- 23.4, 23.6, 23.7, 23.9, 23.10) 16-11-2021 2 C. V. Raman Global University, Odisha Syllabus (Continued…) Text Books: T3. “Probability, Random Variables and Random Signal Principles, Peyton Z. Peebles, JR., Tata Mc-Graw – Hill”, 4th Edition, 2010. Chapters: 1(1.1 – 1.5), 2(2.0 – 2.5), 3(3.0 – 3.2), 6(6.0 – 6.3) Reference Books: R1. “Numerical Methods For Scientific and Engineering Computation”, M. K. Jain, S. R. K. Iyengar, R.K. Jain, New Age International Publishers, Sixth Edition, 2014 R2. “Probability & Statistics for Engineers & Scientists”, Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers & Keying Ye, Pearson Education Inc., New Delhi, Eighth Edition, 2007 16-11-2021 2 C. V. Raman Global University, Odisha  MATHEMATICS – IIIA Introduction & Applications : Numerical Methods: 1) Estimation of ocean currents 2) Modeling combustion flow in a coal power plant. 3) Airflow patterns in the respiratory tract. 4) Transport and disposition of chemicals through the body (ODEs + PDEs) etc. 16-11-2021 2 C. V. Raman Global University, Odisha  MATHEMATICS – IIIA INTRODUCTION TO MATHEMATICS - IIIA The limitations of analytical methods in practical applications have led scientists and engineers to evolve numerical methods. Understanding the concept of error is very important for the effective use of numerical methods. In professional practice errors can be costly and sometimes catastrophic. If a structure or device fails, lives can be lost. How much error is present in our calculations and is it tolerable? Identification, quantification and minimization of the errors. 16-11-2021 2 C. V. Raman Global University, Odisha  MATHEMATICS – IIIA DEFINITION (ACCURACY OF NUMBERS): Approximate Numbers: Numbers which represent the given numbers to a certain degree of accuracy are called approximate numbers. Examples: 1  0.3333, 3 2  1.4142,   3.1415, etc. 16-11-2021 2 C. V. Raman Global University, Odisha  MATHEMATICS – IIIA DEFINITION (ACCURACY OF NUMBERS): (Continued…) Significant figures: 1. Concept developed to formally designate the reliability of a numerical value. 2. The significant digits of a numbers are those that can be used with confidence. Significant figure rules: 1. Any non-zero numbers are significant figures. Example -1: Numbers Significant figures 4257.13 6 425257.2 7 16-11-2021 2 C. V. Raman Global University, Odisha  MATHEMATICS – IIIA *DEFINITION (ACCURACY OF NUMBERS): (Continued…) Significant figures: 1. Concept developed to formally designate the reliability of a numerical value. 2. The significant digits of a numbers are those that can be used with confidence. Significant figure rules: 1. Any non-zero numbers are significant figures. Example -1: Numbers Significant figures 4257.13 6 425257.2 7 16-11-2021 2 C. V. Raman Global University, Odisha  Thank You D r. Tra i l o k y a n a t h S i n g h Mathematics [email protected] 9 8 6 1 2 8 7 9 1 3 w w w. c g u - o d i s h a. e d u. i n