Multivariable Calculus and Numerical Methods

Ordinary and singular point, Series solution about an ordinary point, Solution of equation by Frobenius method to solve second order linear differential equation, formation of Indicial equation, Determination of indicial roots, cases of indicial roots: Distinct roots, equal roots, roots differ by an integer and not differ by an integer. Applications: Radial heat conduction in cylindrical and spherical reactors, Diffusion in porous catalyst particles.

Laplace Transform: Definition of Laplace Transform, Existence Conditions, Linearity Property, Laplace Transform of some standard functions, I and II Shifting theorems (without proof), Change of Scale Property, Laplace transform of derivatives and integrals, Periodic Functions, Unit Step Function, Inverse Laplace Transform, Partial Fraction Method, Convolution theorem (without proof), Application of Laplace transform to solve ordinary differential equation. Applications : Dynamic analysis of stirred tank reactors (CSTR), Process control and transfer function analysis

Periodic function, even and odd function, Dirichlet’s Condition for the Existence of a Fourier Series, Fourier series expansion of a function in the interval c to c+2l, Change of Interval, Half Range Series: Half range sine and cosine series in the interval -l to l, Parseval’s Identity, Fourier Integral theorem, Fourier transform, Inverse Fourier Transform, Finite Fourier Sin and Cosine transform. Applications: Heat conduction in solids, Mass diffusion analysis.

Functions of Complex Variables, Limit Continuity and Differentiability, Analytic Function, Cauchy-Riemann Equations (Cartesian and Polar Form), Harmonic Function, Milne- Thomson Method, Complex integrals, Cauchy- Integral theorem (without proof), Cauchy integral formula (without proof),Taylor’s series and Laurent’s series, Singularities, Residue of a function, Cauchy Residue Theorem, Evaluating the Real definite integrals. Applications: Potential flow around pipes and reactors, Two-dimensional heat conduction.

Applied Linear Algebra: Eigen Value Decomposition and Singular Value Decomposition, Matrix factorization, LU Decomposition (Dolittle method), QR Decomposition, Linear transformation, Matrix representation of Linear Transformation, Kernel and Range, Introduction to inner product space, Norm of a vector. Applications: Material balance calculations, Process simulation and optimization.