Partial Differential Equations

Code: MA567 | L-T-P-C: 3-1-0-8

Prerequisites: MA541 Real Analysis MA548 ODE

First-Order PDE: Cauchy problem, Method of Characteristics, Lagrange Method, Compatible systems of first-order PDEs, Charpit’s method. Classification of second order PDEs, Canonical Transformations, Characteristics, Canonical forms, characteristics, Propagation of Singularities. Wave Equation: Wave equation in one dimension; Infinite, Semi-infinite, and finite String Problem, D'Alembert's formula, Wave equation in higher dimension; Solutions by spherical means, Inhomogeneous Problems; Duhamel’s principle. Laplace Equation: Fundamental solution, Mean value property, maximum principle, Green's function, Poisson's integral formula, Harnack's inequality, Dirichlet's principle. Heat equation: One dimensional homogeneous heat equation; infinite and semi-infinite rod, Inhomogeneous problems; Duhamel's Principle, Fundamental solution, Maximum principle. Separation of variables: Fourier series expansions, Solutions by separation of Variables method for Wave Equation, Heat Equation and Laplace Equation, BVP in different coordinate systems. Transform methods: Fourier Transforms, Properties of Fourier Transforms, Convolution, Fourier Sine and Cosine Transforms, Application to Initial BVP; Laplace Transforms: Properties of Laplace Transforms, Convolution, Application to initial BVP. Uniqueness ofsolutions of hyperbolic, elliptic, and parabolic equations by Energy Methods.

Texts:

1. I. N. Sneddon, Elements of Partial Differential Equations, Dover Publications, 2006.
2. R. McOwen, Partial Differential Equations: Methods and Applications, Pearson, 2002.

References:

1. Phoolan Prasad and R. Ravindran, Partial Differential equations, New Age International, 2011.
2. Qing Han, A Basic Course in Partial Differential Equations, AMS, 2011.
3. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998
4. Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Birkhauser, 2007.