**Course Name : Advanced Differential Equations
**

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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.

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

**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

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

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

Non-linear ordinary differential equations of particular forms

Youtube: Non-linear ordinary differential equations of particular forms

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

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

Partial differential equation of second order

https://www.youtube.com/watch?v=AXH9Xm6Rbfc

https://www.slideshare.net/3820170034/introduction-of-partial-differential-equations

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

Monge's method and its properties

https://www.youtube.com/watch?v=AXH9Xm6Rbfc

https://www.slideshare.net/alibhalli43/differential-equations-48242801

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/[email protected]

Classification of linear partial differential equation of second order

Classification of linear partial differential equation of second order

Method of separation of variables

https://www.youtube.com/watch?v=aolvx8bWFDA

https://www.slideshare.net/JuanApolinarioReyes/separation-of-variables

one- dimensional Laplace equation

https://www.youtube.com/watch?v=EW08rD-GFh0

https://nptel.ac.in/content/storage2/courses/115101005/downloads/lectures-doc/Lecture-14.pdf

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

https://www.youtube.com/watch?v=XbCvGRjjzgg

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

https://www.youtube.com/watch?v=6cMn1F6bNPU

https://www.youtube.com/watch?v=WWphCZkdByA

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

Project-2 (2 hrs)

Solution of Laplace’s Equation for a Disk

https://math.okstate.edu/people/binegar/4263/4263-l15.pdf

https://www.youtube.com/watch?v=MCoe3HFu9Yk

one- dimensional Wave equation

https://www.youtube.com/watch?v=0Rfushlee0U

https://www.slideshare.net/DhavalJalalpara/the-wave-equation

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

https://www.youtube.com/watch?v=9TQCKWWAVjM&t=17s

https://www.youtube.com/watch?v=imdFhDbWDyM

https://www.slideshare.net/neerajparmar68/1-d-wave-equation

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

https://www.youtube.com/watch?v=-k2TuJfNQ9s

https://www.youtube.com/watch?v=ck-r_qmNNG0

https://www.slideshare.net/8laddu8/fourier-series

Project-3 (2 hrs)

D’ Alembert’s solution of the wave equation

https://www.youtube.com/watch?v=j2G91naZ8bo

D’ Alembert’s solution of the wave equation

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

https://in.mathworks.com/matlabcentral/answers/440084-pdepe-help-boundry-condition-in-pde-solver

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

one- dimensional Diffusion equation by method of separation of variables

https://www.youtube.com/watch?v=jco8ygBA8LY

https://www.slideshare.net/EnriqueValderrama4/partial-differential-equations-3-simple-examples

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

https://www.youtube.com/watch?v=WwS3CYltxqY

Project 4(2 hrs)

Solution of Diffusion equation in n-dimensional

http://www.math.utk.edu/~freire/teaching/m435s14/HigherDimHeatEqn.pdf

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

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

https://in.mathworks.com/matlabcentral/answers/440084-pdepe-help-boundry-condition-in-pde-solver

one- dimensional Heat equation by method of separation of variables

https://www.youtube.com/watch?v=ToIXSwZ1pJU&t=147s

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

Project 6 (2 hrs)

Temperature distribution in Rectangular plate

http://www.math.ttu.edu/~gilliam/ttu/s10/m3351_s10/c14_2d_disk_heat.pdf

Mrs. Saubhagyalaxmi Singh works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. She has over 6 years of teaching experience with exposure to Utkal University, Bhubaneswar, Department of Mathematics. She has published 6 papers communicated to different International Journals. She has guided 7 M.Sc. projects in her area. She has […]

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