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VTU NOTES
Subject code
21MAT31
Semester
3rd

## Module - 1

Laplace Transform Definition and Laplace transform of elementary functions (statements only). Problems on Laplace’sTransform of 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 derivatives, solutions of differential equations.

## Module - 2

Fourier Series Introduction to infinite 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, and 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 equation using standard five-point formula. Solution of heat equation by Schmidt explicit formula and Crank- Nicholson method, Solution of the Wave equation. Problems. (8 Hours) Self Study: Solution of Poisson equations using standard five-point formula.

## 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