21MAT31 Transform Calculus, Fourier Series and Numerical Techniques syllabus for IM

Laplace Transform

Definition and Laplace transforms of elementary functions (statements only). Problems on Laplace'sTransform of 𝑒^𝑎𝑡𝑓(𝑡), 𝑡^𝑛𝑓(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.

Fourier Series

Introduction to infinite 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.

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.

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.

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.

Course outcomes:

After successfully completing the course, the students will beable :

• To solve ordinary differential equations using Laplace transform.
• Demonstrate the Fourier series to study the behaviour of periodic functions and their applications in system communications, digital signal processing and field theory.
• To use Fourier transforms to analyze problems involving continuous-time signals and to apply Z-Transform techniques to solve difference equations
• To solve mathematical models represented by initial or boundary value problems involving partial differential equations
• Determine the extremals of functionals using calculus of variations and solve problems arising in dynamics of rigid bodies and vibrational analysis.

Assessment Details (both CIE and SEE)

• The weightage of Continuous Internal Evaluation (CIE) is 50% and for Semester End Exam (SEE) is 50%.
• The minimum passing mark for the CIE is 40% of the maximum marks (20 marks out of 50).
• A student shall be deemed to have satisfied the academic requirements and earned the credits allotted to each subject/ course if the student secures not less than 35% ( 18 Marks out of 50)in the semester-end examination(SEE), and a minimum of 40% (40 marks out of 100) in the sum total of the CIE (Continuous Internal Evaluation) and SEE (Semester End Examination) taken together

Continuous Internal Evaluation:

1. Three Unit Tests each of 20 Marks (duration 01 hour)

2. First test at the end of 5th week of the semester

3. Second test at the end of the 10th week of the semester

4. Third test at the end of the 15th week of the semester

Two assignments each of 10 Marks

5. First assignment at the end of 4th week of the semester

6. Second assignment at the end of 9th week of the semester Group discussion/Seminar/quiz any one of three suitably planned to attain the COs and POs for 20 Marks (duration 01 hours)

7. At the end of the 13th week of the semester The sum of three tests, two assignments, and quiz/seminar/group discussion will be out of 100 marks and will be scaled down to 50 marks (to have less stressed CIE, the portion of the syllabus should not be common /repeated for any of the methods of the CIE. Each method of CIE should have a different syllabus portion of the course).

CIE methods /question paper is designed to attain the different levels of Bloom’s taxonomy as per the outcome defined for the course.

Semester End Examination:

Theory SEE will be conducted by University as per the scheduled timetable, with common question papers for the subject (duration 03 hours)

• The question paper will be set for 100 marks and marks scored will be proportionally scaled down to 50 marks
• The question paper will have ten questions. Each question is set for 20 marks.
• There will be 2 questions from each module. Each of the two questions under a module (with a maximum of 3 sub-questions), should have a mix of topics under that module.
• The students have to answer 5 full questions, selecting one full question from each module.

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”, McGraw Hill Education(India) Pvt. Ltd2015.

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

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