Faculty  Incharge:Shri. J. Suriya Praksh, Dr. Madan Kumar Lakshmanan. 


Week  Course Details 
Module I. Linear Algebra  
Week 1  Introduction to LA Vector & Linear combination., Length & Dot Product, Determinant – Properties, Cofactor, Cramers Rule, Volume Linear Equation – Elimination, A=LU form, Transpose & Permutation 
Week 2  Vector Spaces Spaces & Subspaces, Null space Ax=0; Rank & Row Reduced Form, Solution to Ax=b; Independence, basis and Dimension, Dimension of four spaces 
Week 3  Orthogonality, Projection Gram Schmidt Orthogonality in four spaces, Projection & Least square Approximation, Gram Schmidt Process 
Week 4  Eigen Values and Vectors, Differential Eqn solution, Matrix Diagonalization 
Week 5  Singular Value Decomposition, Other Matrices Introduction to Singular Value Decomposition, Other important Matrices – Symetric, Positive Matrices, 
Module II. Differential Equations  
Week 6  Fundamentals of differential equations: How differential equations have been applied in diverse fields to understand physical/biological phenomenon; Examples such as Maxwell's equations, heat equations, progress of Avian influenza virus where DE has found applicability to solve practical problems will be highlighted; Types of differential equations  ODE/PDE, linear/nonlinear, constant/variable coefficient; Solution of a differential equation  general solutions vs actual/particular solutions, initial value problems, existence and uniqueness of a solution; What are direction fields and how they can be used to obtain a geometric view of the differential equations. 
Week 7  First order differential equations General form of expression; Methods to solve 1st order differential equations of the form (all cases handled with examples); Problems which are linear in nature Problems where the differentials are separable Bernoulli's differential equations Homogeneous differential equations 
Week 8  Numerical methods to solve first order differential equations An introduction to numerical techniques for DEs; Euler's method to solve ODE's numerically (uniform and nonuniform steps); Runge Kutta method and backward Euler methods as extensions to the Euler's method to solving DEs numerically 
Week 9  Second Order differential equations General form of expression; Types of second order ODEs (homogeneous/nonhomogeneous, constant/variable coefficients, linear/nonlinear); Solving 2nd order differential equations which are linear, homogenous and have constant coefficients (all cases handled with examples): I General form of solution What are characteristic equations Solution for the cases when (a)the roots are real and distinct, (b)the roots are Complex and (c) double roots. I Reduction of order of differential equations to solve cases when the coefficients are not constants. I Wronskian condition and fundamental sets of solutions  the necessary conditions between the roots under which a valid solution can exist 
Week 10  Partial differential equations An introduction to PDE + general form of expression; Types of PDEs  elliptic, parabolic and hyperbolic; The procedure to solve PDEs illustrated with 2 examples, namely, solving the 1st order heat equation and solving the 1st order wave equation I Solving the heat equation (1st order) General form of expression Expatiation of the initial value and boundary conditions Explanation of terminologies such as specific heat, mass density and heat flux Simplification of the heat equation (moving from 2 unknowns to 1 unknown and considering the case of a uniform surface) Moving from PDE to ODE through separation of variables Finding out the general solutions to the heat equation. I Solving the wave equation (1st order) General form of expression Initial value and boundary conditions Separation of variables to simplify problem Derivation of the general solution to the wave equation The relationship between the solutions to the heat and wave equations and the connection to the derivation of the Fourier series representation A brief introduction to solving 2nd and 3rd order heat and wave equations. 
Module III. Mathematical transforms  
Week 11  What are mathematical transforms, their significance and why they are useful; An introduction to Fourier series and Fourier transform (FFT, DFT, DTFT) 
Week 12  Gabor (Short time fourier) transform ; Wavelet (discrete and continuous) transforms; KarhunenLoeve transform 
Week 13  Laplace and Ztransform (with examples) 
Module IV. Mathematical methods for data analysis  
Week 14  Review of Probability Probability function; Conditional Probability; Independence, Random Variable; Expectation, Variance, Standard Deviation; Distributions – Binomial, Poisson, Exponential, Normal 
Week 15  Statistics Basics Research methods, Histograms, Summary statistics; Correlation analysis, Sampling and standard Error, 
Week 16  Statistics Methods Regression Analysis; Central Limit Theorem; Multiple regression; T test and ANOVA; ChiSquare Test; logistic Regression; Non parametric Tests 
Week 17  Data analysis Methods Descriptive, Exploratory, Inferential, predictive, Causal Data Analysis; Clustering – Hierarchical, Kmean; Dimensionality reduction – SVD, PCA 
Week 18  Prediction and Data classification Data collection, feature creation, training and test sets for Data Analysis; Methods – Naive Bayes, Random Forest, Classification trees 
Books
Calculus  
Text Book  Thomas’ Calculus, Pearson education 11th ed, 2007, by G.B. Thomas, M.D. Weir, J. Hass and F.R. Giordano. 
Reference Books  1. James Stewart – Calculus, 5e, Cengage learning, 2003. 2. Erwin Kreyszig – Advanced Engineering. Mathematics, 8th Edition WileyIndia,1999. 
Linear algebra  
Text Book  Introductory Linear Algebra with Applications by B. Kolman and D.R. Hill, 8th edition, 2005, Pearson Education, Inc 
Reference Books  1. Linear Algebra and its Applications by D.C. Lay, 3rd edition, Pearson Education, Inc. 2. Introduction to Linear Algebra, by Gilbert Strang, WellesleyCambridge Press (1993). 3. Linear Algebra: A Modern Introduction, by David Poole. 4. Linear Algebra and Its Applications, by Gilbert Strang. 
Sequences and Series  
Text Book  Complex Variables and applications by R.V. Churchill and J.W. Brown, 8th edition, 2008, McGrawHill. 
Reference Books  Complex Variables with Applications by A.D. Wunsch, 3rd edition, Pearson Education, Inc. 
Differential Equations  
Text Book  Simmons G.F., Differential Equations with Applications and Historical Notes, TMH, 2nd ed.,1991. 
Reference Books  1. Edwards & Penney: Differential Equations and Boundary value problems, Pearson Education, 3rd ed. 2. Shepley L. Ross: Differential Equations, Willy India Pvt. Ltd, 3rd ed. 3. Birkhoff & Rota: Ordinary Differential Equations, Wiley India Pvt. Ltd., 4th ed. 4. Zill, Differential Equation, Thomson Learning, 5th ed., 2004 5. R.K. Patnaik: Differential Equation, PHI, 2009. 
Transforms  
Text Book  Oppenheim, Alan V., and A. S. Willsky. Signals and Systems. Prentice Hall, 1982.ISBN: 9780138097318. 
Reference Books  B.Burke, The World According to Wavelets: The Story of a Mathematical Technique in the Making. Upper Saddle River,New Jersey: A K Peters, May 1998. 
Links
Matlab for differential equations