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Programmes → I.M.Sc. in Mathematical Sciences; M.Sc. in Mathematics/Applied Mathematics

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Partial Differential Equations-I

Course Code: MA 402; MM 502; AM 502  |  Credits: 4  |  Prerequisite: Real Analysis-I, Real Analysis-II, Ordinary Differential Equations-I

Syllabus Details

  • Unit I:(First Order Partial Differential Equations) Surfaces and curves; classification of first order P.D.E.; classification of solutions; Pfaffian differential equations; quasi-linear equations; Lagrange’s method; compatible systems; Charpit’s method; Jacobi’s method; integral surfaces passing through a given curve; method of characteristics for quasi-linear and nonlinear P.D.E.; Monge cone; characteristic strip.
  • Unit II:(Second Order Partial Differential Equations) Classification of second order semi-linear P.D.E.; Hadamard’s definition of well-posedness.
  • Unit III:(Wave Equation) D’Alembert’s solution; vibrations of a finite string; existence and uniqueness of solution; Riemann method.
  • Unit IV:(Laplace’s Equation) Boundary value problems; maximum and minimum principles; uniqueness and continuity theorems; Dirichlet problem for a circle; Dirichlet problem for a circular annulus; Neumann problem for a circle; theory of Green’s function for Laplace’s equation.
  • Unit V:(Heat Equation) Heat conduction problem for an infinite rod; heat conduction in a finite rod; existence and uniqueness of the solution.
  • Unit VI:(Duhamel’s Principle and Higher Dimensions) Duhamel’s principle for wave and heat equations; variable separable methods for second order linear partial differential equations; classification of semi-linear partial differential equations in higher dimensions; Kelvin’s inversion theorem; equipotential surfaces.

Course Learning Objective

  1. Apply a suitable technique to solve a given first order/second order partial differential equation based on the type of the equation.
  2. Explain the limitations of the characteristic method that is used to solve first order PDEs and the method of separation of variables which is used to solve second order linear PDEs.
  3. Construct Green’s function for the Laplacian with Dirichlet/Neumann boundary conditions when the geometry of the domain is simple.
  4. Analyze the nature of the solutions to the heat equation and the wave equation by observing phenomena like finite speed of propagation and propagation regularizing effect.
  5. Identify various results/tools from the prerequisite courses that are used in the proofs of the theorems given in this course.
  1. John, Fritz. Partial Differential Equations. 2nd ed. Applied Mathematical Sciences, Vol. 1, Springer-Verlag, 1978
  2. Weinberger, H. F. A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Wiley, 1965
  3. Sneddon, Ian. Elements of Partial Differential Equations. McGraw-Hill, NY, 1957; Dover, 2006
  4. Han, Qing. A Basic Course in Partial Differential Equations. ATM, Vol. 120, Indian ed., 2013
  5. McOwen, Robert C. Partial Differential Equations: Methods and Applications. 2nd ed. Pearson India, 2006
  6. Evans, Lawrence C. Partial Differential Equations. AMS-GTM, Vol. 19, Indian ed., 2010
  7. Amaranath, T. An Elementary Course in Partial Differential Equations. 2nd ed. Narosa Publishing House, 2012
  8. Zauderer, Erich. Partial Differential Equations of Applied Mathematics. 3rd ed. Wiley, 2011
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