## Details of MA5102 (Autumn 2018)

Level: 5 |
Type: Theory |
Credits: 4.0 |

Course Code | Course Name | Instructor(s) |
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MA5102 |
Partial Differential Equations |
Ratikanta Behera |

Syllabus |
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MA5102 : Partial Differential Equations
First-order Equations : Method of Characteristics and existence of local solutions; Hamilton-Jacobi equation, Hopf-Lax formula, weak solution of Hamilton-Jacobi equation and its uniqueness; introduction to conservation laws, weak solutions, Rankine-Hugoniot condition, shocks, Lax-Oleinik formula, entropy condition and uniqueness of entropy solution. Characteristic Manifolds and Cauchy Problem: Non-characteristic surfaces, Cauchy-Kowalevski theorem and uniqueness theorem of Holmgren. Laplace Equation: Fundamental solution, harmonic function and its properties; Poissons equation, Dirichlet problem and Greens function; existence of solution of the Dirichlet problem using Perrons method; introduction to variational method. Heat Equation: Fundamental solution and initial-value problem; mean value formula, maximum principle, uniqueness and regularity; nonnegative solutions. Wave Equation: dAlemberts formula, method of spherical means, Hadamards method of descent, Dumahels principle and Cauchy problem, initial-boundary-value problem. |

References |
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Suggested Texts / Reference Books:
1. Evans, L. C., Partial Differential Equations (2nd Edition), American Mathematical Society, 2010. 2. Folland, G., Introduction to Partial Differential Equations, Princeton University Press, 1976. 3. Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order (2nd Edition), Springer-Verlag, 1983. 4. John, F., Partial Differential Equations (4th Edition), Springer-Verlag, 1995. 5. Renardy, M. and Rogers, R., An Introduction to Partial Differential Equations (2nd Edition), Springer-Verlag, 2003. |

#### Course Credit Options

Sl. No. | Programme | Semester No | Course Choice |
---|---|---|---|

1 | IP | 1 | Not Allowed |

2 | IP | 3 | Elective |

3 | IP | 5 | Not Allowed |

4 | MR | 1 | Not Allowed |

5 | MR | 3 | Not Allowed |

6 | MS | 9 | Core |

7 | RS | 1 | Elective |

8 | RS | 2 | Not Allowed |