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Dr. Rajen Kumar Sinha
IIT Guwahati

 

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First order linear and quasi-linear PDEs, The Cauchy problem, Second order PDEs, Classification of PDEs, Characteristics, Well-posed problems, Fourier Series, Solutions of hyperbolic, parabolic and elliptic equations, Dirichlet and Neumann problems, Maximum principles, Fourier Transform methods for PDEs, The method of Green's functions for Laplace, Heat and wave equations.

 

Sl. No.

Module/ Lecture Topics

No. of (Total) Hours

1

Mathematical Preliminaries: A Review of Multivariable Calculus, Essential Ordinary Differential Equations, Integral Curves and Surfaces of Vector Fields, Solving Equations of the form: dx/P=dy/Q=dz/R.

4

2

First-Order Partial Differential Equations(PDEs)– Formation and  classification of  first-order PDEs,  Linear  and  Quasi-linear first-order PDEs, Cauchy's problem for first order PDEs, The Cauchy Kowalevski Theorem, Integral surfaces passing through a given curve, Nonlinear first-order PDEs, The method of characteristics, Compatible systems,   Charpit’s method, Jacobi’s method for nonlinear PDEs.

7

3

Second-Order  PDEs  -  Classification,  Canonical  forms,  Well-posed problems, Superposition principle.

3

4

Fourier Series (FS) – Introduction to FS, Convergence of FS for continuous  and  piecewise  continuous  functions,  Differentiation and integration of FS, Fourier cosine and sine series.

3

5

The  Heat   Equation  -  Derivation  of  the  heat  equation,  The maximum and minimum principles, Uniqueness, Continuous dependence, Method of separation of variables, Time-independent boundary conditions, Time-dependent boundary conditions, Duhamel’s principle.

5

6

The  Wave  Equation  -  Derivation of  the  wave  equation, The infinite string problem, The D'Alembert solution of the wave equation, The semi-infinite string problem, The finite vibrating string problem, The method of separation variables, The inhomogeneous wave equation.

5

7

Laplace's  Equation – Basic concepts, Types of boundary value problems, The maximum and minimum principle, Green’s identity and fundamental solution, The Poisson integral formula, The method of separation of variables, The Dirichlet problem for the rectangle,  The  Dirichlet  problem  for  Annuli  and  Disk,  The exterior Dirichlet problem.

6

8

The Fourier Transform Methods for PDEs –Fourier transform, Fourier  sine  and  cosine  transform,  Heat  flow  problem  in  an infinite and semi-infinite rod, Infinite string problem, Laplace equation in a half-plane.

5

9

The  Method  of Green’s  Functions – Integral formulation, The method of Green’s functions for  the  Laplace, Heat  and  Wave equations.

3

 

Total

41

Basic Multivariable Calculus and ODEs


  1. D. Bleecker and G. Csordas, Basic Partial Differential Equations, Van Nostrand
  2. Reinhold, New York, 1992.
  3. C. Constanda, Solution Techniques for Elementary Partial Differential Equations,
  4. Chapman & Hall/CRC, New York, 2002.
  5. L. J. Crowin and R. H. Szczarba, Multivariable Calculus, Marcel Dekker, Inc, New
  6. York, 1982.
  7. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Birkh auser,
  8. New York, 1993.
  9. F. John, Partial Differential Equations, Springer-Verlag, New York, 1982.
  10. E. Kreyszig, Advanced Engineering Mathematics, Wiley, 2011.
  11. J. Marsden and A. Weinstein, Calculus III, Springer-Verlag, New York, 1985.
  12. T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and
  13. Engineers, Birkh auser, Boston, 2007.
  14. R. K. Nagle and E. B. Saff, Fundamentals of Differential Equations and Boundary
  15. Value Problems, Addison-Wesley, New York, 1996.
  16. I.N. Sneddon, Elements of Partial Differential Equations, Dover Publications, New
  17. York, 2006.
  18. E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations
  19. with Applications, Dover Publication, Inc., New York, 1986.
  20. E. Zauderer, Partial Differential Equations of Applied Mathematics, Second Edition,
  21. John Wiley & Sons, New York, 1989.


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