Course Offered

2012

- Mathematical and Numerical Methods
- Course Contents:
- Complex Variables: Definition; Geometric Representation; De Moivre’s Theorem; Roots of a Complex Number; Series Expansion; Hyperbolic Functions; Continuity of Function; Derivative of f(z); Cauchy-Riemann Equations; Analytic Functions; Harmonic Functions; Orthogonal System; Stream Function; Equipotential Function; Application to 2-D Steady Flow Problems-Fluid Flow, Electrostatics, Gravitational Fields and Heat Flow; Plotting of f(z); Conformal Transformation; Translation; Magnification and Rotation; Inversion and Reflection; Bilinear Transformation; Special Transformations; Schwarz-Christoffel Transformation; Integration of Complex Functions; Cauchy’s Theorem; Cauchy’s Integral Formula; Morera’s Theorem; Series of Complex Terms; Weirstrass’s M-test; Taylor’s Series; Laurent’s Series; Schwarz Reflection Principle; Analytic Continuation; Meromorphic Function; Permanence of Algebraic Form; Singular Points; Residues; Residue Theorem; Calculation of Residues; Cauchy Principal Value; Pole Expansion of Meromorphic Functions; Partial Fraction Expansion; Product Expansion of Entire Functions; Rouche’s Theorem; Evaluation of Definite Integrals-Integration around circle, semi-circle and rectangular contours, Singularity on real axis; Dispersion Relations; Symmetry Relations; Optical Dispersion; Parseval Relation; Causality
- Matrix Algebra: Definition; Complex Matrix; Special Matrices; Addition; Subtraction; Multiplication; Transpose, Adjoint, Elementary Transformation, Normal Form, Inverse, Rank, Characteristic Equation, Eigen Values; Degenerate Eigen Values; Eigen Vectors; Cayley-Hamilton Theorem; Similarity Transformation (Including a Case for Diagonalization); Row Space; Column Space (Range or Image); Row Reduced Echelon Form; Kernel (or Null Space); Rank-Nullity Theorem; Matrix Method of Solution of Homogeneous and Linear Simultaneous Equations-Underdetermined, Square and Overdetermined Systems; Data Fitting by Least Square Solution; Discrete Fourier Transform by Matrix Method; Singular Value Decomposition; Pseudo Inverse of a Matrix by SVD
- Vector Analysis: Scalars and Vectors; Unit Vector; Addition; Subtraction; Multiplication by Scalar; Position Vector; Resolution of Vector; Coplanar Vectors; Space Vectors; Direction Cosines; Point of Division; Scalar or Dot Product of Two Vectors; Vector or Cross Product of Two Vectors; Applications-Work done, Normal flux, Monent of a force, Angular velocity of a rigid body; Scalar Tripple Product; Volume of Tetrahedron; Vector Tripple Product; Scalar Prodcut of Four Vectors; Vector Product of Four Vectors; Reciprocal Vector Triads; Application to Analytic Geometry-Equations of line and plane; Cartesian Coordinate System; Cylindrical Coordinate System; Spherical Coordinate System; Differentiation of Vectors; Velocity and Acceleration; Relative Velocity and Relative Acceleration; Scalar Point Function; Vector Point Function; Vector Operator Del; Gradient; Divergence; Curl; Laplacian; Del Applied Twice to Point Functions; Del Applied to Products of Point Functions; Integration of Vectors; Differential Length, Area and Volume; Line Integral; Surface Integral; Green's Theorem in Plane; Stoke's Theorem; Volume Integral; Divergence Theorem; Green's Theorem; Classification of Vector Fields; Solenoidal Field; Irrotational Field; Scalar Potential; Vector Potential; Helmholtz Theorem; Introduction to Poisson and Laplace Equations; Dirac Delta Function; Helmholtz Decompostion of Vector Field; Critical Points like Sources, Sinks and Rotational Centres
- Ordinary and Partial Differential Equations
- Differential Equations of First Order and First Degree: Separable variables, Homogeneous, Linear (Leibnitz linear, Bernoulli’s equation, Equations reducible to linear) and Exact (Equations reducible to exact)
- Equations of the First Order and Higher Degree: Equations solvable for p, Equations solvable for y and Equations solvable for x, Clairut’s Equation

- Linear Differential Equations: CF, PI and CS, Method of Variation of Parameters for PI, Equations Reducible to Linear Equation with Constant Coefficients-Cauchy’s Homogeneous Linear Equation and Legendre’s Linear Equation, Simultaneous Linear Equation with Constant Coefficients, Equations of the Form y''=f(x), Equations of the Form y''=f(y), Equations with One Known Solution
- Total Differential Equations, Simultaneous Total Differential Equations-Method of grouping and Method of multipliers
- Series Solution of Differential Equations, Validity of Series Solution (Fuchs’s Theorem), Frobenius Method, Forms of Series Solution
- Wronskian, Dependent and Independent Solutions, Second Solution
- Singular Points
- Singular Solution: p-discriminant, C-discriminant, Envelope, Tac Locus, Cusp Locus and Node Locus
- Partial Differential Equations: Formation, CI, PI, GI, SI
- Direct Integration
- (Lagrange's) Linear Equation of First Order
- Non-linear Equation of First Order: f(p, q) = 0, f(z, p, q) = 0, f(x, p) = F(y, q), z = p x + q y + f(p, q), Charpit's Method
- Homogeneous Linear Equations with Constant Coefficients
- Non-homogeneous Linear Equations
- Non-linear Equations of Second Order (Monge's Method)
- Integral Transforms
- Equation Solving Numerical Methods: Solution of Algebraic and Transcendental Equations-Bisection method, Secant method, Method of false position, Newton’s method; Complex root; Multiple roots-Discriminant, Resultant, Sylvester matrix; Descartes’ rule of signs; Approximate Solution of Equations-Horner’s Method; Solution of Linear Simultaneous Equations; Direct Method of Solution-Gauss elimination method, Gauss-Jordan method, Crout’s method; Iterative Methods of Solution-Jacobi’s method, Gauss-Seidal method, Relaxation method; Solution of Non-linear Simultaneous Equations-Newton-Raphson method; Muller’s Method; Finite Differences-Forward differences, Backward differences, Central differences; Differences of a Polynomial; Factorial Notation; Relations Between the Operators; Newton’s Forward Interpolation Formula; Newton’s Backward Interpolation Formula; Central Differences Interpolation Formulae-Stirling’s formula, Bessel’s formula; Lanrange’s Formula for Unequal Intervals; Divided Differences; Newton’s Formula for Unequal Intervals; Numerical Differentiation-For forward differences, For backward differences, For central differences; Numerical Integration-Newton-Cotes integration formula, Trapezoidal rule, Simpson’s one-third rule, Simpson’s three-eighth rule, Weddle’s rule; Extrapolation; Richardson Extrapolation
- Numerical Solution of Ordinary Differential Equations: Picard’s Method; Taylor’s Series Method; Euler’s Method; Modified Euler’s Method; Runge’s Method; Runge-Kutta Method; Predictor-Corrector Methods-Milne’s Method, Adams-Bashforth Method; Simultaneous First Order Differential Equations; Second Order Differential Equations
- Numerical Solution of Partial Differential Equations

- Books:

- George B. Arfken and Hans J. Weber, “Mathematical Methods for Physicists”, PRISM Books Pvt. Ltd., Bangalore.
- Mary L. Boas, “Mathematical Methods in the Physical Sciences”, Wiley.
- Curtis F. Gerald and Patrick O. Wheatley, “Applied Numerical Analysis”, Pearson Education.
- Alfio Quarteroni, Riccardo Sacco and Fausto Saleri, “Numerical Mathematics”, Springer.

- Additional: Introduction to MATLAB

- Group of Student
- Assignment 1 and Attempt by Student
- Assignment 2 and Attempt by Student
- Assignment 3 and Attempt by Student
- Assignment 4 and Attempt by Student
- Assignment 5 and Attempt by Student
- Assignment 6 and Attempt by Student
- Assignment 7 and Attempt by Student
- Assignment 8 and Attempt by Student
- Assignment 9 and Attempt by Student
- Assignment 10 and Attempt by Student
- Assignment 11 and Attempt by Student
- Assignment 12 and Attempt by Student

- Instructions:
- Please, kindly write group number, problem number and problem definition in the solution.

- Test-1:
- Conducted on 27th September 2012 Thursday between 14:00 to 15:30 hrs (90 min).
- 20 appeared out of 22. Two were absent.
- Chapters covered...1st and 2nd
- Expected/Indicative time for solutions...(A. 1-a) 5 min, (A. 1-b) 5 min, (A. 2) 10 min, (A. 3-a) 5 min, (A. 3-b) 5 min, (A. 4) 20 min, (A. 5) 35 min
- Spare time 5 min
- Question Paper
- Solution by the Instructor
- Result

- Test-2:
- Conducted on 30th October 2012 Tuesday between 14:00 to 15:30 hrs.
- 22 appeared out of 22.
- Chapters covered...3rd and 4th
- Question Paper
- Solution by the Instructor
- Result

- Test-3:
- Conducted on 4th December 2012 Tuesday from 15:15 to 16:45 hours at Thaltej in room no. 119.
- 21 appeared out of 22
- Chapter covered...6th
- Question Paper
- Solution by the Instructor
- Result

- End Semester Exam:
- It will be conducted in two parts.

- First Part of End Semester Exam:
- It will be conducted on 18th December 2012 Tuesday from 10:00 to 12:30 hrs.
- Chapters to be covered...1st (Complex Variables), 3rd (Vector Analysis) and 4th (Ordinary and Partial Differential Equations)

- Second Part of End Semester Exam:
- It will be conducted on 19th December 2012 Wednesday from 10:00 to 12:30 hrs.
- Chapters to be covered...2nd (Matrix Algebra), 6th (Equation Solving Numerical Methods) and 7th (Numerical Solution of Ordinary Differential Equations)