Module 1
Green’s Function
Heavisides, unit step function – Derivative of unit step function – Dirac delta function -properties of delta function – Derivatives of delta function – testing functions – symbolic function – symbolic derivatives – inverse of differential operator – Green’s function -initial value problems – boundary value problems – simple cases only.
Green’s Function
Heavisides, unit step function – Derivative of unit step function – Dirac delta function -properties of delta function – Derivatives of delta function – testing functions – symbolic function – symbolic derivatives – inverse of differential operator – Green’s function -initial value problems – boundary value problems – simple cases only.
Module 2
Integral Equations
Definition of Volterra and Fredholm Integral equations – conversion of a linear differential equation into an integral equation – conversion of boundary value problem into an integral equation using Green’s function – solution of Fredhlom integral equation with separable Kernels – Integral equations of convolution type – Neumann series solution.
Integral Equations
Definition of Volterra and Fredholm Integral equations – conversion of a linear differential equation into an integral equation – conversion of boundary value problem into an integral equation using Green’s function – solution of Fredhlom integral equation with separable Kernels – Integral equations of convolution type – Neumann series solution.
Module 3
Gamma, Beta functions
Gamma function, Beta function – Relation between them – their transformations – use of them in the evaluation certain integrals – Dirichlet’s integral – Liouville’s extension, of Dirichlet’s theorem – Elliptic integral – Error function.
Gamma, Beta functions
Gamma function, Beta function – Relation between them – their transformations – use of them in the evaluation certain integrals – Dirichlet’s integral – Liouville’s extension, of Dirichlet’s theorem – Elliptic integral – Error function.
Module 4
Power Series solution of differential equation
The power series method – Legendre’s Equation – Legendre’s polynomial – Rodrigues formula – generating function – Bessel’s equation – Bessel’s function of the first kind -Orthogonality of Legendre’s Polynomials and Bessel’s functions.
Power Series solution of differential equation
The power series method – Legendre’s Equation – Legendre’s polynomial – Rodrigues formula – generating function – Bessel’s equation – Bessel’s function of the first kind -Orthogonality of Legendre’s Polynomials and Bessel’s functions.
Module 5
Numerical solution of partial differential equations.
Classification of second order equations- Finite difference approximations to partial derivatives – solution of Laplace and Poisson’s equations by finite difference method -solution of one dimensional heat equation by Crank – Nicolson method – solution one dimensional wave equation.
Numerical solution of partial differential equations.
Classification of second order equations- Finite difference approximations to partial derivatives – solution of Laplace and Poisson’s equations by finite difference method -solution of one dimensional heat equation by Crank – Nicolson method – solution one dimensional wave equation.
References
Linear Integral Equation: Ram P.Kanwal, Academic Press, New York
A Course on Integral Equations: Allen C.Pipkin, Springer – Verlag
Advanced Engg. Mathematics: H.K.Dass, S.Chand
Advanced Engg. Mathematics: Michael D.Greenberge, Pearson Edn. Asia
Numrical methods in Engg. &Science: B.S.Grewal, Khanna Publishers
Generalized functions: R.F. Hoskins, John Wiley and Sons.
Principles and Techniques of Bernard Friedman: John Wiley and sons Applied Mathematics
Principles of Applied Mathematics: James P.Keener, Addison Wesley.
Numerical methods: P.Kandasamy, K.Thilagavathy, K.Gunavathy
A Course on Integral Equations: Allen C.Pipkin, Springer – Verlag
Advanced Engg. Mathematics: H.K.Dass, S.Chand
Advanced Engg. Mathematics: Michael D.Greenberge, Pearson Edn. Asia
Numrical methods in Engg. &Science: B.S.Grewal, Khanna Publishers
Generalized functions: R.F. Hoskins, John Wiley and Sons.
Principles and Techniques of Bernard Friedman: John Wiley and sons Applied Mathematics
Principles of Applied Mathematics: James P.Keener, Addison Wesley.
Numerical methods: P.Kandasamy, K.Thilagavathy, K.Gunavathy