Course Outcomes:
1.Solve Volterra integral equations using resolvent kernels, successive approximations, and Laplace transforms.
2.Analyze and apply methods for Fredholm equations, Green’s functions, and eigenvalue problems in integral equations.
3.Apply variational principles to derive Euler’s equation and solve classical optimization problems in calculus of variations.
4.Evaluate multivariable variational problems and apply Hamilton’s and Lagrange’s equations to physical systems.
Volterra Integral Equations: Basic concepts - Relationship between Linear differential equations and Volterra Integral equations - Resolvent Kernel of Volterra Integral equation. Differentiation of some resolvent kernels - Solution of Integral equation by Resolvent Kernel - The method of successive approximations - Convolution type equations - Solution of Integro-differential equations with the aid of the Laplace Transformation – Volterra integral equation of the first kind-Euler integrals-Abel’s problem-Abel’s integral equation and its generalizations.
Fredholm Integral Equations : Fredholm integral equations of the second kind – Fundamentals – The Method of Fredholm Determinants - Iterated Kernels constructing the Resolvent Kernel with the aid of Iterated Kernels - Integral equations with Degenerated Kernels. Hammerstein type equation – Characteristic numbers and Eigen function and its properties.
Green’s function :Construction of Green’s function for ordinary differential equations-Special case of Green’s function –Using Green’s function in the solution of boundary value problem.
CALCULS OF VARIATIONS:
Introduction – The Method of Variations in Problems with fixed Boundaries: Definitions of Functionals –Variation and Its properties - Euler’s’equation- Fundamental Lemma of Calculus of Variation – The problem of minimum surface of revolution - Minimum Energy Problem Brachistochrone Problem - Variational problems involving Several functions - Functional dependent on higher order derivatives - Euler Poisson equation.
Functional dependent on the functions of several independent variables - Euler’s equations in two dependent variables – Variational problems in parametric form-Applications of Calculus of Variation-Hamilton’s principle - Lagrange’s Equation,Hamilton’s equations.
Problems and Exercises in Integral Equations by M.KRASNOV, A.KISELEV, G.MAKARENKO, (1971).
Integral Equations by S.Swarup, (2008).
Differential Equations and The Calculus of Variations by L.ELSGOLTS, MIR Publishers, MOSCOW.
Analytical Mechanics by Grant R. Fowles and George L. Cassiday, 7Th Edition.
Course Outcomes:
1.Solve first-order nonlinear PDEs using characteristics, Charpit’s method, and special equation forms.
2.Classify higher-order PDEs and transform them into canonical forms for further analysis.
3.Apply Fourier transform techniques to evaluate transforms and use their properties in solving PDE-related problems.
4.Solve diffusion, wave, and Laplace equations using Fourier transforms and separation of variables.
First order Nonlinear Equations, Cauchy’s method of Characteristics, compatible systems of first order equations, Charpit’s method, Special types of first order equations.
Higher order Linear Partial Differential Equations with constant coefficients, Homogeneous Partial Differential Equations with constant coefficients, Classification of second order Partial Differential Equations, Canonical forms, Canonical form for hyperbolic, parabolic and elliptic equations.
Fourier Transforms : Fourier Integral Representations, Fourier Transforms Pairs, Fourier Transform
of Elementary Functions, Properties of Fourier Transform, Convolution theorem, Parseval’s Relation,Transform of Dirac Delta Function, Finite Fourier Transforms.
Solution of diffusion, wave and Laplace equations by using Fourier transforms and Separation of Variables Methods, D’Alembert’s solution of wave equation, Dirichlet problem and Neumann problem.
Introdction to Partial Differential Equations by K. Shankar Rao, PHI, Third Edition.
Elements of Partial Differential Equations by Ian Sneddon, Mc.Graw-Hill International Edition.
Partial Differential Equations by Lawrence C. Evans, American Mathematical Society.
Course Outcomes:
1.Apply iterative methods to find numerical solutions of nonlinear equations with known convergence behavior.
2.Solve systems of linear equations using direct and iterative numerical methods with convergence analysis.
3.Construct interpolation polynomials and approximate functions using interpolation and least-squares methods.
4.Compute numerical integrals and solve ordinary differential equations using single-step and multi-step numerical schemes.
Transcendental and Polynomial Equations: Introduction, Bisection Method - Iteration Methods Based on First Degree Equation: Secant Method, RegulaFalsi Method, Newton-Raphson Method
- Iteration Methods Based on Second Degree Equation: Muller’s Method, Chebyshev Method, Multipoint Iteration Methods, Rate of convergence - Iteration Methods.
System of Linear Algebraic Equations: Introduction - Direct Methods: Gauss Elimination Method, Gauss Jordan Elimination Method, Triangularization Method, Cholesky Method, Partition Method - Iteration Methods: Jacobi Iteration Method, Gauss Seidel Iteration Method, SOR Method, Convergence Analysis for iterative Methods.
Interpolation and Approximation: Interpolation: Introduction - Lagrange and Newton Interpolations, Finite Difference Operators - Interpolating Polynomials using Finite Differences - Hermite Interpolations, Piecewise and Spline Interpolations. Approximation: Least Squares Approximation.
Differentiation : Methods based on interpolation, Methods based on finite differences.
Numerical Integration: Methods Based on Interpolation: Newton- Cotes Methods - Methods Based on Undetermined Coefficients: Guass- Legendre Integration Methods - Composite Integration Methods.
Numerical Solution of ODEs: Introduction - Numerical Methods: Euler Methods-Mid point Method Single Step Methods: Taylor series method, Runge-Kutta Method (2nd and 4th orders). Multistep Methods: Adams Bashforth Method - Adams Moulton Method, Milne-Simpson Method
- Predictor Corrector Methods.
Numerical Methods for Scientific and Engineering computation by M.K. Jain,
S.R.K. Iyengar, R.K. Jain, 7th Edition, New Age International Publishers,2019.