Mathematical Science Syllabus Paper 2 Unit 15 To 21

Mathematical Science Syllabus Paper 2 Unit 15 To 21

15. Calculus of Variations: Linear functionals, minimal functional theorem, general variation of a functional, Euler- Lagrange equation; Variational methods of boundary value problems in ordinary and partial differential equations.

16. Linear Integral Equations: Linear Integral Equations of the first and second kind of Fredholm and Volterra type; solution by successive substitutions and successive approximations; Solution of equations with separable kernels; The Fredholm Alternative; Holbert-Schmidt theory for symmetric kernels.

17. Numerical analysis: Finite differences, Interpolation; Numerical solution of algebraic equation; Iteration; Newton- Raphson method; Solution on linear system; Direct method; Gauss elimination method; Matrix - Inversion, eigenvalue problems; Numerical differentiation and integration.

Numerical solution of ordinary differential equation; iteration method, Picard's method, Euler's method and improved Euler's method.

18. Integral Transform: Laplace transform; Transform of elementary functions, Transform of Derivatives, Inverse Transform, Convolution Theorem, Applications, Ordinary and Partial differential equations; Fourier transforms; sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations.

19. Mathematical Programming: Revised simplex method, Dual simplex method, Sensitivity analysis and parametric linear programming. Kuhn-Tucker conditions of optimality. Quadratic programming; methods due to Beale, Wofle and Vandepanne, Duality in quadratic programming, self-duality, Integer programming.

20. Measure Theory: Measurable and measure spaces; Extension of measures, signed measures, Jordan-Hahn decomposition theorems. Integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem. Absolute continuity, Radon Nikodym theorem, Product measures, Fubini's theorem.

21. Probability: Sequences of events and random variables; Zero- one laws of Borel and Kolmogorov. Almost sure convergence, convergence in mean square, Khintchine's weak law of large numbers; Kolmogorov's inequality, strong law of large numbers.

Convergence of series of random variables, three-series criterion. Central limit theorems of Liapounov and Lindeberg- Feller. Conditional expectation, martingales.