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Department of Mathematics, IIT Bombay
MA 515 Partial Differential Equations
Semester I – 20072008
Prerequisites :
MA 417 Ordinary Differential Equations.
MA 448 Multivariable Calculus
Credit Structure: 3108
Objectives: To provide an understanding of, and methods of solution for, the most importanttypes of partial differential equations that arise in Mathematical Physics. On completionof this course, the students should be able to: a) use the method of characteristics to solve first order partial differential equations; b) classify a second order PDE as elliptic, parabolic or hyperbolic;
c) derive representation formulas for the solutions of Laplace equation, heat equation and wave equation;d) to solve heat and wave equations using the method of separation of variables.
Syllabus:
● Cauchy Problems for First Order Hyperbolic Equations: Method of Characteristics,
Monge Cone.
● Classification of Second Order Partial Differential Equations: Normal forms and Characteristics.
● Formulation of Initial and Boundary Value Problems: LagrangeGreen's identity and uniqueness by energy methods. Stability theory, Energy conservation and dispersion.
● Laplace equation: Fundamental Solution, Mean value property, Weak and Strong Maximum principle, Green's function, Poisson's formula, Dirichlet's principle, existence of solution using Perron's method (without proof).
● Heat equation: initial value problem, fundamental solution, Mean value formulae, weak and strong maximum principle and uniqueness results.
● Wave equation: uniqueness, D'Alembert's method, method of spherical means and Duhamel's principle.
● Methods of separation of variables for heat and wave equations.
Texts / References:
F. John, Partial Differential Equations, 3rd ed., Narosa Publ. Co., New Delhi,1979. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd ed., John Wiley and Sons, , 1989. E.Di Benedetto, Partial Differential Equations, Birkhauser , 1995. L.Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, 1998.
Lecture Schedule:
Venue: 105, Department of Mathematics.
Lectures Tutorials
Monday : 8.30 a.m. – 9.25 a.m.Tuesday : 10.35 a.m. – 11.30 a.m. 4.00 p.m.4.55 p.m.Friday : 9.30 a.m. 10.25 a.m.
Total number of lecture hours : 42
Total number of tutorial hours : 13
Form of assessment:
Duration Weightage Date
Quiz 1 45 mts 10% 21st August
Quiz 2 45 mts 10% 12th October
Quiz 3 45 mts 10% 2nd November
Mid semester examination 2 hours 30% Third week of September
End Semester examination 3 hours 40% Fourth week of November
Instructor:
Prof. Neela Nataraj, Department of Mathematics, IIT Bombay
email:[email protected]
Telephone number: +91 22 2576 7468
Office hours: Tuesdays 5 p.m. 6 p.m.