Page 85 - CatalogNEP-PS

 Solve the Laplace equation (elliptic equation), Heat equation (Parabolic equation) and
Wave equation (hyperbolic equation) by variable separable method and solve some
boundary value problems by some standard methods.
 Learn the use of theory and solutions/tools in solving the dynamical problems arising in
engineering and physical sciences.

THEORY (45 Hours)

UNIT I (15 Hours)
Classification of Second Order Partial Differential Equations. Canonical Forms: Canonical Form
for Hyperbolic Equation, Canonical Form for Parabolic Equation, Canonical form for elliptic
equation. Adjoint Operators. Occurrence of the Laplace and Poisson Equations: Derivation of
Laplace Equation, Derivation of Poisson Equation. Boundary Value Problems (BVPs). Some
Important Mathematical Tools. Properties of Harmonic Functions. Separation of Variables,
Dirichlet problem for a rectangle, the Neumann problem for rectangle.
UNIT II (15 Hours)

Occurrence of the Diffusion Equation. Boundary Conditions. Elementary Solutions of the
Diffusion Equation. Dirac Delta Function. Separation of Variables Method. Solution of Diffusion
Equation in Cylindrical Coordinates. Solution of Diffusion Equation in Spherical Coordinates.
Maximum-Minimum Principle and its Consequences.

UNIT III (15 Hours)
Occurrence of the Wave Equation. Derivation of One-dimensional Wave Equation. Solution of
One-dimensional Wave Equation by Canonical Reduction. The Initial Value Problem;
D‘Alembert‘s Solution. Vibrating String – Variables Separable Solution. Forced Vibrations –
Solution of Nonhomogeneous Equation. Boundary and Initial Value Problem for Two-
dimensional Wave Equation – Method of Eigenfunction. Periodic Solution of One- dimensional
Wave Equation in Cylindrical Coordinates. Periodic Solution of One-dimensional Wave Equation
in Spherical Polar Coordinates.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READING:

1. K. Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India
Private Limited, New Delhi, 1997.
2. Ian Sneddon, Elements of Partial Differential Equations, McGraw-Hill Book Company,
1985.
3. K.S. Bharma, Partial Differential Equations, An Introductory Treatment with Applications,
PHI, N. Delhi, 2010.
4. Purna Chandra Biswal, Partial Differential Equations, PHI, Pvt. Ltd, New Delhi, 2015.

Math.423 Linear Algebra and Matrix Analysis 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 The basic idea of operators on finite dimensional vector spaces and the basic properties of
Normal operators in the context of spectral theory.
 The diagonalizable matrices and have the basic properties of these matrices.
 The basic concept of matrix norms, their examples, and the unitarily invariant norm.

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