Advanced Differential Equations

Teacher

Dr. Swarnalata Jena

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Course Name : Advanced Differential Equations

Code(Credit) : CUTM1530(2-1-1)

Course Objectives

1. Working with systems of ordinary differential equations and non-linear ordinary differential equations
is also stressed.
2. Developing and understanding and appreciation of the qualitative behavior of the solution
3. To introduce wave equations, Laplace equations, Heat equations, Diffusion equations.

Course Outcomes

COs	Course outcomes	Mapping Cos with POs (High-3, Medium-2, Low-1)
CO1	Solve wave equation and understand significance of transverse waves.	PO1(3), PO2(2)
CO2	Identify classes of non-linear ordinary differential equations.	PO2(2), PO4(2)
CO3	Apply an appropriate method for the solution of non-linear ordinary differential equations.	PO9(3)
CO5	Competence in solving applied problems which are linear and nonlinear form, Solve Laplace equation, Diffusion equation, heat equation.	PO2(2), PO1(3)

Course Syllabus

Module I:
Introduction to Ordinary Differential Equations and Partial Differential Equations, First Order Non- linear Ordinary differential equations such as Equations solvable for x, Equations solvable for y, Equations solvable for p.

Practice- 1: Solve Ordinary Differential Equations in Python
Practice-2: Solve Partial differential Equations by python

Module II:
Partial differential equation of second order with variable coefficients- Monge's method and its properties
Project 1: Monge’s Method of Solution of Non-linear Partial Differential Equations of Order Two.

Module III:

Classification of linear partial differential equation of second order, Cauchy's problem, Method of separation of variables.

Module IV

Solution of one- dimensional Laplace equation by method of separation of variables and Fourier series
Project 2 : Solution of Laplace’s Equation for a Disk

Module V:
Solution of one- dimensional Wave equation by method of separation of variables and Fourier series

Project 3: D’ Alembert’s solution of the wave equation
Practice 3: Solution of wave equation associated condition
u(x,0)=φ(x),u_t (x,0)=ψ(x),u(0,t)=0,xϵ(0,∞),t>0
Practice 4: Solution of wave equation associated condition
u(x,0)=φ(x),u(0,t)=a,xϵ(0,∞),t≥0

Module VI
Solution of one- dimensional Diffusion equation by method of separation of variables and Fourier series
Project 4: Solution of Diffusion equation in n-dimensional
Practice 5: Solution of one-dimensional diffusion equation by using boundary conditions u(x,0)=φ(x),u(0,t)=a,xϵ(0,∞),t≥0
Practice 6:Solution of one-dimensional diffusion equation
u(x,0)=φ(x),u(0,t)=a,u(1,t)=b,0<x0

Module VII
Solution of one- dimensional Heat equation by method of separation of variables and Fourier series
Project 5: Two dimensional Heat equations- Polar form
Project 6: Temperature distribution in Rectangular plate

Text Books

1. A COURSE ON ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS, by J. Sinha Roy, S. Padhy, Kalyani Publishers, ISBN 81-272-1091-9.
Chapter 3 (3A, 3B,3C), Chapters 13, 14,15,16,17
2. Differential Equations and Their Applications, by Martin Braun, Springer, 4e, ISBN 9781111827052 (1993).
3. S. L. Ross: Differential Equations, Blaisdell Publishing Company, Londan, 1964.

Reference books:
1. S.J. Farlow: An Introduction to Ordinary Differential Equations, PHI
2. M.D. Raisinghania: Ordinary and Partial Differential Equations, S. Chand & Co.
3. V. Sundarapandian: Ordinary and Partial Differential Equations, McGraw-Hill

Session Plan

Session 1

Introduction to Ordinary Differential Equations and Partial Differential Equations
https://www.youtube.com/watch?v=ly4S0oi3Yz8

https://www.slideshare.net/EMDADULHAQUE13/introduction-to-differential-equations

Session 2

First Order Non- linear Ordinary differential equations such as Equations solvable for x
https://www.youtube.com/watch?v=p_di4Zn4wz4
https://www.slideshare.net/myaukish/formulation-of-first-order-linear-and-nonlinear-2nd-order-differential-equation

Session 3

Non-linear ordinary differential equations of particular forms
Youtube: Non-linear ordinary differential equations of particular forms

Session 4 & 5

Practice-1 (2 hrs)
Solve Ordinary Differential Equations in Python
https://www.youtube.com/watch?v=JLtRK4Z1tEg
https://physics.nyu.edu/pine/pymanual/html/chap9/chap9_scipy.html

Session 6 & 7

Practice-2 (2 hrs)

Solve Partial differential Equations by python
http://csc.ucdavis.edu/~cmg/Group/readings/pythonissue_3of4.pdf
https://www.youtube.com/watch?v=mMdIxa5qC9Y

Session 8

Partial differential equation of second order

https://www.youtube.com/watch?v=AXH9Xm6Rbfc
https://www.slideshare.net/3820170034/introduction-of-partial-differential-equations

Session 9

Partial differential equation of second order with variable coefficients
https://www.youtube.com/watch?v=3lHbafzc6a8
https://www.slideshare.net/aman1894/partial-differential-equations-42054204

Session 10

Monge's method and its properties
https://www.youtube.com/watch?v=AXH9Xm6Rbfc
https://www.slideshare.net/alibhalli43/differential-equations-48242801

Session 11 & 12

Project-1 (2 hrs)
Monge’s Method of Solution of Non-linear Partial Differential Equations of Order Two
http://182.18.165.51/Fac_File/STUDY182@741319.pdf