Calculus and Numerical Method

Linear differential equation of nth order with constant coefficients, Cauchy-Euler equation, solution of second order linear differential equations with variable coefficient by the method of variation of parameters, change of independent variable, normal form. Applications: Modeling heat transfer during textile drying and curing processes, Analysis of fiber stretching and yarn tension during spinning.

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 textile machinery and drive systems, Modeling transient heat flow in dyeing and finishing processes.

Complex 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: Modeling fluid flow around textile fibers during coating and dye penetration, Analysis of electrostatic fields in electrospinning processes.

Fourier Series & Fourier Transform 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: Fabric defect detection using frequency-domain image processing, Signal analysis in textile quality monitoring systems.

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. Application: Principal Component Analysis (PCA) for textile quality inspection, Machine learning for textile defect classification using SVD.