Note: This syllabus is common for
- R18 - B.TECH II Year II Sem. - ECE, EEE
MA401BS: LAPLACE TRANSFORMS, NUMERICAL METHODS AND COMPLEX VARIABLES
B.Tech. II Year II Sem. L T P C
3 1 0 4
Pre-requisites: Mathematical Knowledge at pre-university level
Course Objectives: To learn
- Concept, properties of Laplace transforms
- Solving ordinary differential equations using Laplace transforms techniques.
- Various methods to the find roots of an equation.
- Concept of finite differences and to estimate the value for the given data using interpolation.
- Evaluation of integrals using numerical techniques
- Solving ordinary differential equations using numerical techniques.
- Differentiation and integration of complex valued functions.
- Evaluation of integrals using Cauchy’s integral formula and Cauchy’s residue theorem.
- Expansion of complex functions using Taylor’s and Laurent’s series.
Course outcomes: After learning the contents of this paper the student must be able to
- Use the Laplace transforms techniques for solving ODE’s
- Find the root of a given equation.
- Estimate the value for the given data using interpolation
- Find the numerical solutions for a given ODE’s
- Analyze the complex function with reference to their analyticity, integration using Cauchy’s integral and residue theorems.
- Taylor’s and Laurent’s series expansions of complex Function
UNIT - I
Laplace Transforms 10 L
Laplace Transforms; Laplace Transform of standard functions; first shifting theorem; Laplace transforms of functions when they are multiplied and divided by‘t’. Laplace transforms of derivatives and integrals of function; Evaluation of integrals by Laplace transforms; Laplace transforms of Special functions; Laplace transform of periodic functions.
Inverse Laplace transform by different methods, convolution theorem (without Proof), solving ODEs by Laplace Transform method.
UNIT - II
Numerical Methods – I 10 L
Solution of polynomial and transcendental equations – Bisection method, Iteration Method, Newton- Raphson method and Regula-Falsi method.
Finite differences- forward differences- backward differences-central differences-symbolic relations and separation of symbols; Interpolation using Newton’s forward and backward difference formulae. Central difference interpolation: Gauss’s forward and backward formulae; Lagrange’s method of interpolation
UNIT - III
Numerical Methods – II 08 L
Numerical integration: Trapezoidal rule and Simpson’s 1/3rd and 3/8 rules. Ordinary differential equations: Taylor’s series; Picard’s method; Euler and modified Euler’s methods; Runge-Kutta method of fourth order.
UNIT - IV
Complex Variables (Differentiation) 10 L
Limit, Continuity and Differentiation of Complex functions. Cauchy-Riemann equations (without proof), Milne- Thomson methods, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties.
UNIT - V
Complex Variables (Integration) 10 L
Line integrals, Cauchy’s theorem, Cauchy’s Integral formula, Liouville’s theorem, Maximum-Modulus theorem (All theorems without proof); zeros of analytic functions, singularities, Taylor’s series, Laurent’s series; Residues, Cauchy Residue theorem (without proof)
TEXT BOOKS:
- B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010.
- S.S. Sastry, Introductory methods of numerical analysis, PHI, 4th Edition, 2005.
- J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Ed., Mc-Graw Hill, 2004.
REFERENCE BOOKS:
- M. K. Jain, SRK Iyengar, R.K. Jain, Numerical methods for Scientific and Engineering Computations , New Age International publishers.
- Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons,2006.
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CreatedDec 11, 2020
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UpdatedDec 13, 2020
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