Course

Unit 1

Formation of PDEs, Classification of First order PDEs, Complete, general and Singular integrals, Lagrange’s or quasi linear equations, Integral surfaces through a given curve, Orthogonal surfaces to a given system of surfaces, Characteristic curves.

Unit 2

Pfaffian differential equations, Compatible systems, Charpit’s method, Jacobi’s method, Linear equations with constant coefficients, Reduction to canonical forms.

Unit 3

Classification of second order PDEs, Method of separation of variables: Laplace, Diffusion and Wave equations in Cartesian, Cylindrical and Spherical polar coordinates.

Unit 4

Boundary value problems for transverse vibrations in a string of finite length and heat diffusion in a finite rod, Maximum and Minimum Principles, Integral equations: Fredholm and Volterra integral equations of first, second and third kind, conversion of ODEs into integral equations.

Unit 5

Fredholm equations of second kind with separable kernels, Fredholm alternative theorem, eigen values and eigen functions, Method of successive approximation for Fredholm and Volterra equations, Resolvent kernel.

Course outcomes
CO1: Develops an understanding for the construction of proofs and an appreciation for deductive logic.
CO2: Explore the already familiar properties of the derivative and the Riemann Integral, set on a more rigorous and formal footing which is central to avoiding inconsistencies in engineering applications.
CO3: Explore new theoretical dimensions of uniform convergence, completeness and important consequences as interchange of limit operations.
CO4: Develop an intuition for analyzing sets of higher dimension (mostly of the Rn type) space.
CO5: Solve the most common PDEs, recurrent in engineering using standard techniques and understanding of an appreciation for the need of numerical techniques.

Textbooks:

1. T. Amaranath, An Elementary Course in Differential equations, Narosa.
2. M.D. Raisinghania, Integral equations and Boundary value problems, S Chand Publications.

References:

• I.N. Sneddon, Elements of Partial Differential equations, Dover.
• M.D. Raisinghania, Advanced Differential equations, S Chand Publications.