Course Offered

2012

• Mathematical and Numerical Methods
• Course Contents:
1. 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
2. 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
3. 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
4. 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)
5. Integral Transforms
6. 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
7. 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
8. Numerical Solution of Partial Differential Equations

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

• 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

• 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)

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