MA4153 Advanced Mathematical Methods Syllabus:
MA4153 Advanced Mathematical Methods Syllabus – Anna University PG Syllabus Regulation 2021
OBJECTIVES:
To provide the student with a repertoire of mathematical methods that are essential to the solution of advanced problems encountered in the fields of applied physics and engineering. This course covers a broad spectrum of mathematical techniques such as Laplace Transform, Fourier Transform, Calculus of Variations, Conformal Mapping and Tensor Analysis. The application of these topics to the solution of problems in physics and
engineering is stressed.
UNIT I LAPLACE TRANSFORM TECHNIQUES FOR PARTIAL DIFFERENTIAL EQUATIONS
Laplace transform – Definitions – Properties – Transform error function – Bessel’s function – Dirac delta function – Unit step functions – Convolution theorem – Inverse Laplace transform – Complex inversion formula – Solutions to partial differential equations – Heat equation – Wave equation.
UNIT II FOURIER TRANSFORM TECHNIQUES FOR PARTIAL DIFFERENTIAL EQUATIONS
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.
UNIT III CALCULUS OF VARIATIONS
Concept of variation and its properties – Euler’s equation – Functional dependent on first and higher order derivatives – Functionals dependent on functions of several independent variables – Variational problems with moving boundaries – Isoperimetric problems – Direct methods – Ritz and Kantorovich methods.
UNIT IV CONFORMAL MAPPING AND APPLICATIONS
Introduction to conformal mappings and bilinear transformations – Schwarz Christoffel transformation – Transformation of boundaries in parametric form – Physical applications – Fluid flow and heat flow problems.
UNIT V TENSOR ANALYSIS
Summation convention – Contravariant and covariant vectors – Contraction of tensors – Inner product – Quotient law – Metric tensor – Christoffel symbols – Covariant differentiation – Gradient – Divergence and curl.
OUTCOMES:
After completing this course, students should demonstrate competency in the following skills:
CO1 Application of Laplace and Fourier transforms to the initial value, initial–boundary value and boundary value problems in Partial Differential Equations.
CO2 Maximizing and minimizing the functions that occur in various branches of Engineering Disciplines.
CO3 Construct conformal mappings between various domains and use conformal mapping in studying problems in physics and engineering, particularly fluid flow and heat flow problems.
CO4 Understand tensor algebra and its applications in applied sciences and engineering and develops the ability to solve mathematical problems involving tensors.
CO5 Competently use tensor analysis as a tool in the field of applied sciences and related fields.
REFERENCES:
1. Andrews L.C. and Shivamoggi, B., “Integral Transforms for Engineers”, Prentice Hall of India Pvt. Ltd., New Delhi, 2003.
2. Elsgolc, L.D., “Calculus of Variations”, Dover Publications Inc., New York, 2007.
3. Mathews, J. H., and Howell, R.W., “Complex Analysis for Mathematics and Engineering”, 6th Edition, Jones and Bartlett Publishers, 2011.
4. Kay, D. C., “Tensor Calculus”, Schaum’s Outline Series, Tata McGraw Hill Edition, 2014.
5. Naveen Kumar, “An Elementary Course on Variational Problems in Calculus “, Narosa Publishing House, 2005.
6. Saff, E.B and Snider, A.D, “Fundamentals of Complex Analysis with Applications in Engineering, Science and Mathematics”, 3rd Edition, Pearson Education, New Delhi, 2014.
7. Sankara Rao, K., “Introduction to Partial Differential Equations”, 3rd Edition, Prentice Hall of India Pvt. Ltd., New Delhi, 2010.
8. Spiegel, M.R., “Theory and Problems of Complex Variables and its Applications”, Schaum’s Outline Series, McGraw Hill Book Co., 1981.
9. Ramaniah. G. “Tensor Analysis”, S. Viswanathan Pvt. Ltd., 1990.