## Units and Topics

# | Unit/Topic | Marks |
---|---|---|

100 | Laplace Transform | - |

200 | Inverse Laplace Transform | - |

300 | Complex Variables | - |

400 | Complex Integral | - |

500 | Fourier Series | - |

600 | Partial Differential Equations | - |

Total | - |

## Syllabus

Linearity property, use of theorems to find inverse Laplace Transform, Partial fractions method and convolution theorem.

Applications to solve initial and boundary value problems involving ordinary differential equations with one dependent variable.

Functions of complex variable, Analytic function, necessary and sufficient conditions for f (z) to be analytic (without proof), Cauchy-Riemann equations in polar coordinates

Milne- Thomson method to determine analytic function f (z) when it’s real or imaginary or its combination is given. Harmonic function, orthogonal trajectories

Mapping: Conformal mapping, linear, bilinear mapping, cross ratio, fixed points and standard transformations such as Rotation and magnification, invertion and reflection, translation

Line integral of a function of a complex variable, Cauchy’s theorem for analytic function, Cauchy’s Goursat theorem (without proof), properties of line integral,Cauchy’s integral formula and deductions.

Residue theorem, application to evaluate real integral of type.

Orthogonal and orthonormal functions, Expressions of a function in a series of orthogonal functions. Dirichlet’s conditions. Fourier series of periodic function with period

Dirichlet’s theorem(only statement), even and odd functions, Half range sine and cosine series, Parsvel’s identities (without proof)

Numerical Solution of Partial differential equations using Bender-Schmidt Explicit Method, Implicit method( Crank- Nicolson method) Successive over relaxation method.

Partial differential equations governing transverse vibrations of an elastic string its solution using Fourier series.

Heat equation, steady-state configuration for heat flow

Two and Three dimensional Laplace equations.