Advanced Differential Equations
advanced-differential-equations
Syllabus
Faculty: Mrs. Saubhagyalaxmi Singh
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
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/STUDY182@741319.pdf
Classification of linear partial differential equation of second order
Classification of linear partial differential equation of second order
Cauchy's problem
https://www.youtube.com/watch?v=aa1mk20eM1w
https://www.slideshare.net/PratikGadhiya/higher-order-ode-with-applications
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
Diffusion equation
https://www.youtube.com/watch?v=P9qar8mv3Tk
https://www.slideshare.net/MartaGerid/172080c34b4?qid=133d6c08-c26e-416e-bba7-3b9431729846&v=&b=&from_search=2
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
Heat equation
https://www.youtube.com/watch?v=ToIXSwZ1pJU
https://www.slideshare.net/IJESMJOURNAL/derivation-and-solution-of-the-heat-equation-in-1d
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
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.
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 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
Session 13
Classification of linear partial differential equation of second order
Classification of linear partial differential equation of second order
Session 14
Cauchy's problem
https://www.youtube.com/watch?v=aa1mk20eM1w
https://www.slideshare.net/PratikGadhiya/higher-order-ode-with-applications
Session 15
Method of separation of variables
https://www.youtube.com/watch?v=aolvx8bWFDA
https://www.slideshare.net/JuanApolinarioReyes/separation-of-variables
Session 16
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
Session 17
One- dimensional Laplace equation by method of separation of variables and Fourier series
https://www.youtube.com/watch?v=XbCvGRjjzgg
Session 18
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
Session 19 & 20
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
Session 21
one- dimensional Wave equation
https://www.youtube.com/watch?v=0Rfushlee0U
https://www.slideshare.net/DhavalJalalpara/the-wave-equation
Session 22
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
Session 23
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
Session 24 & 25
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
Session 26 & 27
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
Session 28 & 29
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 30
Diffusion equation
https://www.youtube.com/watch?v=P9qar8mv3Tk
https://www.slideshare.net/MartaGerid/172080c34b4?qid=133d6c08-c26e-416e-bba7-3b9431729846&v=&b=&from_search=2
Session 31
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
Session 32
Solution of one- dimensional Diffusion equation by method of separation of variables and Fourier series
https://www.youtube.com/watch?v=WwS3CYltxqY
Session 33 & 34
Project 4(2 hrs)
Solution of Diffusion equation in n-dimensional
http://www.math.utk.edu/~freire/teaching/m435s14/HigherDimHeatEqn.pdf
Session 35 & 36
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 37 & 38
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
Session 39
Heat equation
https://www.youtube.com/watch?v=ToIXSwZ1pJU
https://www.slideshare.net/IJESMJOURNAL/derivation-and-solution-of-the-heat-equation-in-1d
Session 40
one- dimensional Heat equation by method of separation of variables
https://www.youtube.com/watch?v=ToIXSwZ1pJU&t=147s
Session 41 & 42
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 43 & 44
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
Case Studies
Case Studies
Case Studies
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