MAT3003 Complex- Variables-AND- Partial- Differential- Equations TH 1 PDF

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Download MAT3003 Complex- Variables-AND- Partial- Differential- Equations TH 1 PDF

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MAT3003 Pre-requisite

COMPLEX VARIABLES AND PARTIAL DIFFERENTIAL EQUATION MAT2002 Applications of Differential and Difference Equations

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3 2 0 0 4 Syllabus Version 1.1

Course Objectives: The aim of this course is to present a comprehensive, compact and integrated treatment of two most important branches of applied mathematics for engineers and scientists namely the functions of complex variable and Partial differential equations in finite and infinite domains Expected Course Outcome: By the end of the course, the students are expected to 1. Construct analytic functions and find complex potential of fluid flow and electric fields 2. Find the image of straight lines by elementary transformations 3. Express analytic functions in power series 4. Evaluate real integrals using techniques of contour integration 5. Analyze partial differential equations, and its applications, design the boundary value problems (one dimensional heat and wave equations) and find Fourier series, Fourier transform techniques in their respective engineering problems Student Learning Outcomes (SLO): Module: 1

Analytic Functions

1, 2, 9 6 hours

Complex variable-Analytic functions and Cauchy – Riemann equations - Laplace equation and Harmonic functions - Construction of Harmonic conjugate and analytic functions - Applications of analytic functions to fluid-flow and Field problems. Module: 2 Conformal and Bilinear transformations 5 hours Conformal mapping - Elementary transformations-translation, magnification, rotation, inversion. Exponential and Square transformations (w = ez, z2) - Bilinear transformation - Cross-ratio-Images of the regions bounded by straight lines under the above transformations. Module: 3 Power series 4 hours Functions given by Power Series – Taylor and Laurent series – singularities – poles – Residues. Module: 4

Complex Integration

5 hours

Integration of a complex function along a contour – Cauchy-Goursat theorem- Cauchy’s integral formula -Cauchy’s residue theorem - Evaluation of real integrals - Indented contour integral. Module: 5 Partial Differential equations of first order 6 hours Formation and solution of partial differential equation - General, Particular, Complete and Singular integrals - Partial Differential equations of first order of the forms: F(p,q)=0, F(z,p,q)=0, F(x,p)=G(y,q) and Clairaut’s form - Lagrange’s equation: Pp+Qq = R. Module: 6 Applications of Partial Differential equations 10 hours Linear partial differential equations of higher order with constant coefficients. Solution of a partial differential equation by separation of variables - Boundary Value Problems-one dimensional wave and heat equations- Fourier series solution. Module: 7 Fourier transforms 7 hours Complex Fourier transform and properties - Relation between Fourier and Laplace transforms Fourier sine and cosine transforms – Convolution Theorem and Parseval’s identity.

Module: 8

Contemporary Issues

2 hours

Industry Expert Lecture Total Lecture hours

45 hours

A minimum of 10 problems to be worked out by students inventory Tutorial Class 30 hours  Another 5 problems per Tutorial Class to be given as home work. Text Book(s) 1. Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition, John Wiley & Sons (Wiley student Edison) (2015) Reference Books 1. B. S. Grewal, Higher Engineering Mathematics, 42nd Edition (2013), Khanna Publishers, New Delhi 2. G.DennisZill, Patrick D. Shanahan, A first course in complex analysis with applications, 3rd Edition, 2013, Jones and Bartlett Publishers Series in Mathematics: 3. Michael, D. Greenberg, Advanced Engineering Mathematics, 2nd Edition, Pearson Education (2002) 4. Peter V. O’ Neil, Advanced Engineering Mathematics, 7th Edition, Cengage Learning (2011) 5. JH Mathews, R. W. Howell, Complex Analysis for Mathematics and Engineers, Fifth Edition (2013), Narosa Publishers Mode of Evaluation: Digital Assignments, Quiz, Continuous Assessments, Final Assessment Test. Recommended by Board of Studies 16.08.2017 Tutorial

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Approved by Academic Council

47th ACM

Date

05.10.2017...