Syllabus
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Course Code: MMATH21-402 Course Name: Partial Differential Equations |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Partial Differential Equations (PDE) of kth order: Definition, examples and classifications. Initial value problems. Transport equations homogeneous and non-homogeneous, Radial solution of Laplace’s Equation: Fundamental solutions, harmonic functions and their properties, Mean value Formula. Poisson’s equation and its solution, strong maximum principle, uniqueness, local estimates for harmonic functions, Liouville’s theorem, Harnack’s inequality. |
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| 2 | Green’s function and its derivation, representation formula using Green’s function, symmetry of Green’s function, Green’s function for a half space and for a unit ball. Energy methods: uniqueness, Drichlet’s principle. Heat Equations: Physical interpretation, fundamental solution. Integral of fundamental solution, solution of initial value problem, Duhamel’s principle, non-homogeneous heat equation, Mean value formula for heat equation, strong maximum principle and uniqueness. Energy methods. |
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| 3 | Wave equation- Physical interpretation, solution for one dimentional wave equation, D’Alemberts formula and its applications, Reflection method, Solution by spherical means Euler-Poisson_Darboux equation. Kirchhoff’s and Poisson’s formula (for n=2, 3 only). Solution of non –homogeneous wave equation for n=1,3. Energy method. Uniqueness of solution, finite propagation speed of wave equation. |
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| 4 | Non-linear first order PDE- complete integrals, envelopes, Characteristics of (i) linear, (ii) quasilinear, (iii) fully non-linear first order partial differential equations. Hamilton Jacobi equations. Other ways to represent solutions: Method of Separation of variables for the Hamilton Jacobi equations, Laplace, heat and wave equations. Similarity solutions (plane waves, traveling waves, solitons, similarity under scaling). Fourier Transform, Laplace Transform, Convertible non-linear into linear PDE, Cole-Hop Transform, Potential functions, Hodograph and Legendre transforms. Lagrange and Charpit methods. |
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