3130005,Complex Variables and Partial Differential Equations (CVPD)

 

3130005,Complex Variables and Partial Differential Equations  (CVPD) Mechanical Engineering Semester-3 MC Graw Hill Publication Written  by Mukul Bhatt AND Ravish R Singh As per  GTU revised 2019 syllabus download a books and note in this site.

Syllabus:-

Sr. No.	Content	Total Hrs
1	Polar Form of Complex Numbers, Powers and Roots, Complex Variable – Differentiation : Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties; Conformal mappings, Mobius transformations and their properties	12
2	Complex Variable - Integration : Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (without proof), Liouville’s theorem and Maximum-Modulus theorem (without proof); Sequences, Series, Convergence Tests, Power Series, Functions Given by Power Series, Taylor and Maclaurin Series, Uniform Convergence	08
3	Laurent’s series; Zeros of analytic functions, singularities, Residues, Cauchy Residue theorem (without proof), Residue Integration Method, Residue Integration of Real Integrals.	08
4	First order partial differential equations, solutions of first order linear and nonlinear PDEs, Charpit’s Method	06
5	Solution to homogeneous and nonhomogeneous linear partial differential equations second and higher order by complementary function and particular integral method. Separation of variables method to simple problems in Cartesian coordinates, second-order linear equations and their classification, Initial and boundary conditions, Modeling and solution of the Heat, Wave and Laplace equations	10

Suggested Specification table with Marks (Theory):

Distribution of Theory Marks

R Level	U Level	A Level	N Level	E Level	C Level
07	28	35	0	0	0

Legends: R: Remembrance; U: Understanding; A: Application, N: Analyze and E: Evaluate C: Create and above Levels (Revised Bloom’s Taxonomy)

Course Outcome:

Sr. No.	CO statement	Marks % weightage
CO-1	convert complex number in a polar form, plot the roots of a complex number in complex plane, find harmonic conjugate of analytic functions and apply conformal mapping in geometrical transformation	28%
CO-2	evaluate complex integration by using various result, test convergence of complex sequence and series and expand some analytic function in Taylor’s series	20%
CO-3	find Laurent’s series and pole of order, and apply Cauchy Residue theorem in evaluating some real integrals	14%
CO-4	form and solve first order linear and nonlinear partial differential equations	14%
CO-5	apply the various methods to solve higher order partial differential equations, modeling and solve some engineering problems related to Heat flows, Wave equation and Laplace equation	24%

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This book is essentially two books in one. Namely, it is an introduction to two large areas of mathematics complex analysis and differential equations and the material is naturally divided into two parts. This includes holomorphic analytic functions, functions, ordinary differential equations, Fourier series, and partial differential equations. Moreover, half of the book consists of approximately 200 worked-out problems plus 200 exercises of variable level of difficulty. The worked-out problems fill the gap between the theory and the exercises.

To a considerable extent, the parts of complex analysis and differential equations can be read independently. In the second part, some special emphasis is given to the applications of complex analysis to differential equations. On the other hand, the material is still developed with sufficient detail in order that the book contains an ample introduction to differential equations, and not strictly related to complex analysis.

The text is tailored to any course giving a first introduction to complex analysis or to differential equations, assuming as prerequisite only a basic knowledge of linear algebra and of differential and integral calculus. But it can also be used for independent study. In particular, the book contains a large number of examples illustrating the new concepts and results.

I N D E X

UNIT 1 – COMPLEX FUNCTION AND CONFORMAL MAPPING ......... 1

METHOD – 1: BASIC EXAMPLES ........................................................................................ 3

METHOD – 2: SQUARE ROOT OF COMPLEX NUMBER .................................................. 7

METHOD – 3: N^TH ROOT OF COMPLEX NUMBER ........................................................... 8

METHOD – 4: TRIGONOMETRIC FUNCTION OF COMPLEX NUMBER ....................... 12

METHOD – 5: LOGARITHM OF COMPLEX NUMBER ..................................................... 12

METHOD – 6: DIFFERENTIBILITY OF COMPLEX FUNCTION ...................................... 15

METHOD – 7: ANALITICITY OF COMPLEX NUMBER .................................................. 18

METHOD – 8: TO FIND HARMONIC FUNCTION ............................................................ 21

METHOD – 9: FIXED POINT, CRITICAL POINT, ORDINARY POINT............................. 23

METHOD – 10: ELEMENTARY TRANSFORMATION .................................................... 24

METHOD – 11: BILINEAR TRANSFORMATION ............................................................... 2

UNIT-2 » COMPLEX INTEGRAL, SEQUENCE AND SERIES .................... 29

METHOD – 1: LINE INTEGRAL ................................................................................................. 30

METHOD – 2: MAXIMUM MODULUS THEOREM .................................................................. 33

METHOD – 3: CAUCHY INTEGRAL THEOREM ...................................................................... 34

METHOD – 4: CAUCHY INTEGRAL FORMULA ...................................................................... 36

METHOD – 5: CONVERGENCE OF A SEQUENCE ................................................................... 39

METHOD – 6: CONVERGENCE OF SERIES ............................................................................... 41

METHOD – 7: REDIUS OF CONVERGENCE .............................................................................. 42

METHOD – 8: TAYLOR’S SERIES AND MACLAURIN’S SERIES ............................................ 44

UNIT-3 » LAURENT’S SERIES AND RESIDUES ...................................... 47

METHOD – 1: LAURENT’S SERIES .................................................................................................. 47

METHOD – 2: RESIDUES OF FUNCTION ........................................................................................ 50

METHOD – 3: CAUCHY’S RESIDUE THEOREM ............................................................52

METHOD – 4: CONTOUR INTEGRATION BY USING RESIDUE THEOREM .............................. 55

UNIT-4 » FIRST ORDER PARTIAL DIFFERENTIAL EQUATION ........................ 59

METHOD – 1: EXAMPLE ON FORMATION OF PARTIAL DIFFERENTIAL EQUATION ........ 60

METHOD – 2: EXAMPLE ON LAGRANGE’S DIFFERENTIAL EQUATION .................... 63

METHOD – 3: EXAMPLE ON NON-LINEAR PDE ............................................................. 64

METHOD – 4: EXAMPLE ON CHARPIT’S METHOD.......................................................... 66

UNIT-5 » HIGHER ORDER PARTIAL DIFFERENTIAL EQUATION ..................... 67

METHOD – 1: EXAMPLE ON SOLUTION OF HOMO. HIGHER ORDERED PDE .......................... 68

METHOD – 2: EXAMPLE ON SOLUTION OF NON-HOMO. HIGHER ORDERED PDE .............. 70

METHOD – 3: EXAMPLE ON SEPARATION OF VARIABLES...................................... 72

METHOD – 4: EXAMPLE ON CLASSIFICATION OF 2^ND ORDER PDE ................ 74

METHOD – 5: EXAMPLE ON WAVE, HEAT AND LAPLACE EQUATION ...................... 82

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