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