**MA4104 Applied Mathematics for Mechatronics Syllabus:**

MA4104 Applied Mathematics for Mechatronics Syllabus – Anna University PG Syllabus Regulation 2021

**COURSE OBJECTIVES:**

1. Mathematical foundations of numerical techniques for solving linear systems, eigenvalue problems and generalized inverse.

2. To expose the students to variational formulation and numerical integration techniques and demonstrate solution methodology for the variational problems.

3. To understand the basics of random variables with emp0hasis on the standard discrete and continuous distributions.

4. To make the students appreciate the purpose of using Laplace transforms to solve the partial differential equation.

5. To introduce the Fourier transforms and its properties.

**UNIT – I MATRIX THEORY**

Matrix representation of Linear Transformation – Eigen values – Generalized Eigenvectors – Rank of Matrix – The Cholesky decomposition – Canonical basis – QR factorization – Least squares method – Singular value decomposition.

**UNIT – II CALCULUS OF VARIATIONS**

Concept of variation and its properties – Euler’s equations – Functional dependent on first and higher order derivatives – Functional dependent on functions of several independent variables – Variational problems with moving boundaries – Isoperimetric problems – Direct methods: Ritz and Kantorovich methods – Taylor polynomials and Taylor series.

**UNIT – III PROBABILITY AND RANDOM VARIABLES**

Probability – Axioms of probability – Conditional probability – Bayes’ theorem – Random variables – Probability function – Moments – Moment generating functions and their properties – Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions – Function of a random variable.

**UNIT – IV LAPLACE TRANSFORM TECHNIQUES FOR PDE**

Laplace transform – Definitions – Properties – Transform error function – Bessel’s functions – Dirac delta function – Unit step functions – Convolution theorem – Inverse Laplace transform: Complex inversion formula – Solutions to Partial Differential Equations (PDE): Heat equations – Wave equation.

**UNIT – V FOURIER TRANSFORM TECHNIQUES FOR PDE**

Fourier transform: Definitions, properties – Transform of elementary functions – Dirac Delta function – Convolution theorem – Parseval’s identity – Solutions to partial differential equations: Heat equation – Wave equation – Laplace and Poisson’s equations.

**TOTAL: 60 PERIODS**

**COURSE OUTCOMES:**

At the end of the course, students will be able to

1. apply various methods in matrix theory to solve system of linear equations.

2. maximizing and minimizing the functional that occur in various branches of Engineering disciplines.

3. computation of probability and moments, standard distributions of discrete and continuous random variables and functions of a random variable.

4. application of Laplace transforms to initial value, initial- boundary value and boundary value problems in Partial Differential Equations.

5. obtain Fourier transforms for the functions which are needed for solving application problems.

**REFERENCES:**

1. Andrews, L. C. and Shivamoggi, B., “Integral Transforms for Engineers”, Prentice Hall of India, New Delhi, 2003.

2. Bronson, R.,” Matrix Operations”, Schaum’s outline series, 2nd Edition, McGraw Hill, 2011.

3. James, G., “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education, 2004.

4. Johnson, R.A., Miller, I and Freund J., “Miller and Freund’s Probability and Statistics for Engineers”, Pearson Education, Asia, 8th Edition, 2015.

5. O’Neil P.V., “Advanced Engineering Mathematics”, Thomson Asia Pvt. Ltd., Singapore, 2003.

6. Sankara Rao,K., “ Introduction to Partial Differential Equations”, Prentice Hall of India Pvt. Ltd., New Delhi, 1997.