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: Modeling enzyme kinetics using differential equations, Population growth and microbial growth models

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 bioreactors, Pharmacokinetic and pharmacodynamic modeling.

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, Fourier Integral theorem, Fourier transform, Inverse Fourier Transform, Finite Fourier Sin and Cosine transform. Applications: ECG and EEG signal analysis, DNA and protein sequence signal processing.

Zeroes of transcendental and polynomial equations using Bisection method, Regula-falsi method and Newton-Raphson method, Rate of convergence of above methods. Interpolation: Finite differences, Newton’s forward and backward interpolation, Lagrange’s and Newton’s divided difference formula for unequal intervals. Applications: Enzyme Kinetics, Drug Concentration Estimation, Microbial Growth Analysis, Bioprocess Data Analysis

Solution of system of linear equations, Matrix Decomposition method, Jacobi method, Gauss- Seidel method. Numerical differentiation, Numerical integration, Trapezoidal rule, Simpson’s one third and three-eight rules, Solution of ordinary differential equations by Picard’s and fourth-order Runge-Kutta methods. Applications: Pharmacokinetic Modeling, Population Growth Models, Nutrient Transport, Numerical Solution of Biological ODEs.