MA602 Applied Mathematics

(for M. Tech. STRUCTURAL ENGINEERING)

Number of credits: 4

Course Learning Objectives

1. To develop students with knowledge in Laplace and Fouriertransform.

2. To familiarize the students in the field of differential equations to solve boundary value problems associated with engineering applications.

3. To expose the students to calculus of variation, conformal mappings and tensor analysis.

4. To familiarize students in the field of bilinear transformations.

5. To expose students to the concept of vector analysis.

Course Content. Laplace transform: Definitions, properties -Transform of error function, Bessel’s function, Dirac Delta function, Unit Step functions –Convolution theorem –InverseLaplace Transform: Complex inversion formula –Solutions to partial differential equations: Heat equation, Wave equation.

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.

Calculus of Variation: 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 –Problems with constraints –Direct methods –Ritz and Kantorovich methods.

Complex Variables: Introduction to conformal mappings and bilinear transformations –Schwarz Christoffel transformation –Transformation of boundaries in parametric form –Physical applications: Fluid flow and heat flow problems.

Tensors: Polar co-ordinates - Expressions of gradient of scalar point function – divergence and curl of a vector point function in orthogonal curvilinear co-ordinates - Summation convention - Contra-variant and covariant vectors –Contraction of tensors –Innerproduct –Quotient law – Metric tensor – Christoffel symbols – Covariant differentiation.

Reference Books

[1] K. Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India, New Delhi, 1997.

[2] A.S. Gupta, Calculus of Variations with Applications, Prentice Hall of India Pvt. Ltd., New Delhi, 1997.

[3] M.R. Spiegel, Theory and Problems of Complex Variables and its Application (Schaum’s Outline Series), McGraw Hill Book Co., Singapore,1981.

[4] G. James, Advanced Modern Engineering Mathematics, Pearson Education, Third Edition, 2004.

[5] Lev. D. Elsgolc, Calculus of Variations, Dover Publications, NewYork, 2012.

Course outcomes. At the end of the course student will be able

1. To solve boundary value problems using Laplace and Fourier transform techniques.

2. To solve fluid flow and heat flow problems using conformal mapping.

3. To develop the mathematical methodsof applied mathematics and mathematical physics with an emphasis on calculus of variation and integral transforms.

4. To apply vector calculus in linear approximations, optimization, physics and engineering.

5. To solve physical problems such as elasticity, fluid mechanics and general relativity.