TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES (21MAT31)

TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES

Course Code 21MAT31 
CIE Marks 50
Teaching Hours/Week (L:T:P:S) 2:2:0:0 
SEE Marks 50
Total Hours of Pedagogy 40 
Total Marks 100
Credits 03 
Exam Hours 03

Module-1

Laplace Transform Definition and Laplace transforms of elementary functions (statements only). Problems on Laplace'sTransform of  𝑎𝑡𝑓(𝑡), 𝑡𝑛𝑓(𝑡) , 𝑓(𝑡) 𝑡 . Laplace transforms of Periodic functions (statement only) and unit-step function – problems. Inverse Laplace transforms definition and problems, Convolution theorem to find the inverse Laplace transforms (without Proof) problems.Laplace transforms of derivatives, solution ofdifferential equations.(8 Hours)

Self-study: Solution of simultaneous first-order differential equations.

Module-2

Fourier Series Introduction toinfinite series, convergence and divergence. Periodic functions, Dirichlet’s condition. Fourier series of periodic functions with period 2𝜋 and arbitrary period. Half range Fourier series. Practical harmonic analysis.(8 Hours)

Self-study: Convergence of series by D’Alembert’s Ratio test and, Cauchy’sroot test.

Module-3

Infinite Fourier Transforms and Z-Transforms Infinite Fourier transforms definition, Fourier sine and cosine transforms. Inverse Fourier transforms, Inverse Fourier cosine and sine transforms. Problems. Difference equations, z-transform-definition, Standard z-transforms, Damping and shifting rules, Problems. Inverse z-transform and applications to solve difference equations.(8 Hours) 

Self Study: Initial value and final value theorems, problems.

Module-4

Numerical Solution of Partial Differential Equations Classifications of second-order partial differential equations, finite difference approximations to derivatives, Solution of Laplace’s equationusing standard five-point formula. Solution of heat equation by Schmidt explicit formula and Crank- Nicholson method, Solution of the Wave equation. Problems. (8 Hours) Self Study: Solution of Poisson equations using standard five-point formula. (RBT Levels: L1, L2 and L3)

Module-5

Numerical Solution of Second-Order ODEs and Calculus of Variations Second-order differential equations - Runge-Kutta method and Milne’s predictor and corrector method. (No derivations of formulae). Calculus of Variations:Functionals, Euler’s equation, Problems on extremals of functional. Geodesics on a plane,Variationalproblems.(8 Hours)

Self Study: Hanging chain problem (RBT Levels: L1, L2 and L3)

Suggested Learning Resources:
Text Books:

1. B.S.Grewal:“HigherEngineeringMathematics”,Khannapublishers,44thEd.2018

2. E.Kreyszig:“AdvancedEngineeringMathematics”,JohnWiley&Sons,10thEd.(Reprint),2016.

Reference Books

1. V.Ramana:“HigherEngineeringMathematics”McGraw-HillEducation,11thEd.

2. SrimantaPal&SubodhC.Bhunia:“EngineeringMathematics”OxfordUniversityPress,3rdReprint, 2016.

3. N.P Bali and Manish Goyal: “A textbook of Engineering Mathematics” Laxmi Publications, Latest edition.

4. C. Ray Wylie, Louis C. Barrett: “Advanced Engineering Mathematics” McGraw – Hill Book

Co.Newyork, Latested.

5. Gupta C.B, Sing S.R and Mukesh Kumar: “Engineering Mathematic for Semester I and II”, Mc- Graw Hill Education(India) Pvt. Ltd2015.

6. H.K.DassandEr.RajnishVerma:“HigherEngineeringMathematics”S.ChandPublication(2014).

7. JamesStewart:“Calculus”Cengagepublications,7thedition,4thReprint2019.

Web links and Video Lectures (e-Resources):

 http://.ac.in/courses.php?disciplineID=111

 http://www.class-central.com/subject/math(MOOCs)

 http://academicearth.org/

 http://www.bookstreet.in.

 VTU e-ShikshanaProgram

 VTU EDUSATProgram

Activity-Based Learning (Suggested Activities in Class)/ Practical Based learning

 Quizzes

 Assignments

 Seminars