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##### Mrs. Saubhagyalaxmi Singh Category

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# 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.

## Learning Outcomes

After successfully completing this course the expectation is that students will be able to:
1. Solve wave equation and understand significance of transverse waves.
2. Identify classes of non-linear ordinary differential equations.
3. Apply an appropriate method for the solution of non-linear ordinary differential equations.
4. solve the problems choosing the most suitable method.

5. Solve laplace equation, Diffusion equation, heat equation

5. competence in solving applied problems which are linear and nonlinear form

## 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. Differential Equations and Their Applications, by Martin Braun, Springer, 4e, ISBN 9781111827052 (1993).
2. 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 1

Introduction to Ordinary Differential Equations and Partial Differential Equations

## Session 3

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

## Session 6

Partial differential equation of second order

## Session 9

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

## Session 10

Classification of linear partial differential equation of second order
Classification of linear partial differential equation of second order

## Session 14

One- dimensional Laplace equation by method of separation of variables and Fourier series

## Session 15

Solution of one- dimensional Laplace equation by method of separation of variables and Fourier series

https://www.slideshare.net/alexkhan129/laplace-equation-78729248

## Session 18

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

## Session 19

one- dimensional Wave equation by method of separation of variables and Fourier series

## Session 20

Project-3 (2 hrs)
D’ Alembert’s solution of the wave equation
D’ Alembert’s solution of the wave equation

## Session 21

Practice 3 (2 hrs)
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

## Session 22

Practice 4 (2 hrs)
Solution of wave equation associated condition
u(x,0)=φ(x),u(0,t)=a,xϵ(0,∞),t≥0
https://www.slideshare.net/AmrMousa12/2-dimensional-wave-equation-analytical-and-numerical-solution

## Session 25

Solution of one- dimensional Diffusion equation by method of separation of variables and Fourier series

## Session 26

Project 4(2 hrs)
Solution of Diffusion equation in n-dimensional
http://www.math.utk.edu/~freire/teaching/m435s14/HigherDimHeatEqn.pdf

## Session 27

Practice 5 (2 hrs)
Solution of one-dimensional diffusion equation by using boundary conditions u(x,0)=φ(x),u(0,t)=a,xϵ(0,∞),t≥0
https://www.slideshare.net/AmrMousa12/2-dimensional-wave-equation-analytical-and-numerical-solution

## Session 28

Practice 6 (2 hrs)
Solution of one-dimensional diffusion equation
u(x,0)=φ(x),u(0,t)=a,u(1,t)=b,0<x0

## Session 30

one- dimensional Heat equation by method of separation of variables

## Session 31

Project 5 (2 hrs)
Two dimensional Heat equations- Polar form
http://www.math.ttu.edu/~gilliam/ttu/s10/m3351_s10/c14_2d_disk_heat.pdf

## Session 32

Project 6 (2 hrs)
Temperature distribution in Rectangular plate
http://www.math.ttu.edu/~gilliam/ttu/s10/m3351_s10/c14_2d_disk_heat.pdf

## Case Studies 