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# Mathematical Methods for Boundary Value Problems

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## Overview

This course is intended to provide methods to solve linear and nonlinear boundary value problems involving ordinary as well as partial differential equations. The course will start providing mathematical tools based on integral transformation, Fourier series solution and Greens function for obtaining analytic solutions for BVPs. The applicability of the BVP in several practical contexts, e.g. boundary layer flow, transport phenomena and population balance models will be made. Numerical solutions based on the shooting methods will be introduced. Finite difference methods for linear BVP of second-order and higher orders will be discussed. Iterative techniques to solve nonlinear BVP are included in this course. Algorithms for block tri-diagonal system to handle higher order and system of BVPs will be discussed. Computation of elliptic type of PDEs arises in diffusion dominated process will be described. All the methods will be illustrated by working out several examples. This course, apart from being a part of regular undergraduate/ postgraduate mathematics course, will provide a guidance to solve BVPs arise in mathematical modeling of several transport phenomena. Pre-requisite for this course should be the basic knowledge of undergraduate calculus.
INTENDED AUDIENCE: Undergraduates of any Engineering course, Mathematics, Physics and Postgraduate student
of Mathematics/ Mechanical/ Aerospace/Chemical EngineeringPREREQUISITES: Basic UG course in Mathematics/ Undergraduate Calculus

## Syllabus

### COURSE LAYOUT

Week 1:Boundary Value Problems ( BVP); Strum-Liouville Problems; Eigen Values, Eigen Functions. Solution of homogeneous/ non-homogeneous BVPs by Eigen function expansion.Week 2:Eigen function expansion techniques for PDEs; Green’s function; Dirichlet Problems; Maximum Principle.Week 3:Numerical Techniques for BVP: Shooting Method; Finite Difference Method; Block tri-diagonal system of equations; Numerical Methods for Non-linear BVPsWeek 4:Finite difference method for PDEs; Stability analysis; Crank-Nicolson Scheme; ADI scheme; Elliptic type of Partial Differential Equations; Successive-Over-Relaxation Method.

#### Taught by

Prof.Somnath Bhattacharyya

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