## 21MAT31

## Module - 1

**Laplace Transform**

Definition and Laplace transforms of elementary functions (statements only). Problems on

Laplace’sTransform of 𝑒

𝑎𝑡𝑓(𝑡), 𝑡𝑛𝑓(𝑡) ,

𝑓(𝑡)

𝑡

. Laplace transforms of Periodic functions (statement

only) and unit-step function – problems.

Inverse Laplace transforms definition and problems, Convolution theorem to find the inverse

Laplace transforms (without Proof) problems.Laplace transforms of derivatives, solution

ofdifferential equations.

## Module - 2

**Fourier Series**

Introduction toinfinite series, convergence and divergence. Periodic functions, Dirichlet’s condition.

Fourier series of periodic functions with period 2𝜋 and arbitrary period. Half range Fourier series.

Practical harmonic analysis.

## Module - 3

**Infinite Fourier Transforms and Z-Transforms**

Infinite Fourier transforms definition, Fourier sine and cosine transforms. Inverse Fourier transforms,

Inverse Fourier cosine and sine transforms. Problems.

Difference equations, z-transform-definition, Standard z-transforms, Damping and shifting rules,

Problems. Inverse z-transform and applications to solve difference equations

## Module - 4

**Numerical Solution of Partial Differential Equations**

Classifications of second-order partial differential equations, finite difference approximations to

derivatives, Solution of Laplace’s equationusing standard five-point formula. Solution of heat equation

by Schmidt explicit formula and Crank- Nicholson method, Solution of the Wave equation. Problems.

## Module - 5

**Numerical Solution of Second-Order ODEs and Calculus of Variations**

Second-order differential equations – Runge-Kutta method and Milne’s predictor and corrector

method. (No derivations of formulae).

Calculus of Variations:Functionals, Euler’s equation, Problems on extremals of functional.

Geodesics on a plane,Variationalproblems