# Applied Mathematics 3 Semester 3 (SE Second Year) BE Automobile Engineering University of Mumbai Topics and Syllabus

CBCGS [2017 - current]
CBGS [2013 - 2016]
Old [2000 - 2012]

## Syllabus

100 Laplace Transform
101 Function of Bounded Variation
102 Linearity Property of Laplace Transform
200 Inverse Laplace Transform
201 Linearity Property

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

202 Applications to Solve Initial and Boundary Value Problems Involving Ordinary Differential Equations with One Dependent Variable

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

300 Complex Variables
301 Functions of Complex 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

302 Milne- Thomson Method

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

303 Mapping

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

400 Complex Integral
401 Line Integral of a Function

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.

402 Singularities and Poles
403 Taylor’S and Laurent’S Series Development (Without Proof)
404 Residue at Isolated Singularity and Its Evaluation
405 Residue Theorem

Residue theorem, application to evaluate real integral of type.

500 Fourier Series
501 Orthogonal and Orthonormal Functions

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

502 Dirichlet’s Theorem

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

503 Complex Form of Fourier Series
600 Partial Differential Equations
601 Numerical Solution of Partial Differential Equations

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

602 Partial Differential Equations

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

603 Heat Equation

Heat equation, steady-state configuration for heat flow

604 Two and Three Dimensional Laplace Equations

Two and Three dimensional Laplace equations.