Convergence and divergence of infinite series of positive terms, definition
and illustrative examples*
Periodic functions, Dirichlet's conditions, Fourier series of periodic functions
of period 2π and arbitrary period, half range Fourier series. Complex form of
Fourier Series. Practical harmonic analysis.
Infinite Fourier transform, Fourier Sine and Cosine transforms, properties,Inverse transforms
Various possible solutions of one dimensional wave and heat equations, two dimensional Laplace’s equation by the method of separation of variables,Solution of all these equations with specified boundary conditions.D’Alembert’s solution of one dimensional wave equation.
Curve fitting by the method of least squares- Fitting of curves of the form y=ax+b, y=ax2+bx+c, y=aebx , y=axb Optimization: Linear programming, mathematical formulation of linear programming problem (LPP), Graphical method and simplex method.
Numerical Solution of algebraic and transcendental equations: Regula-falsimethod, Newton - Raphson method. Iterative methods of solution of a systemof equations: Gauss-seidel and Relaxation methods. Largest eigen value andthe corresponding eigen vector by Rayleigh’s power method.
Finite differences: Forward and backward differences, Newton’s forward and backward interpolation formulae. Divided differences - Newton’s divided difference formula, Lagrange’s interpolation formula and inverse interpolation formula. Numerical integration: Simpson’s one-third, three-eighth and Weddle’s rules (All formulae/rules without proof)
Numerical solutions of PDE – finite difference approximation to derivatives,Numerical solution of two dimensional Laplace’s equation, one dimensional heat and wave equations
Difference equations: Basic definition; Z-transforms – definition, standard Z-transforms, damping rule, shifting rule, initial value and final value theorems. Inverse Z-transform. Application of Z-transforms to solve difference equations. Note: * In the case of illustrative examples, questions are not to be set.