Unit 1

Existence and Uniqueness: Picard’s method of successive approximation, Problems of existence and uniqueness, Lipschitz condition, Existence and Uniqueness Theorem, Cauchy-Peano’s Theorem. Linear equations: Basic theory, Homogeneous equation, Wronskian, method of Variation of parameters, equations with constant coefficients and method of undetermined coefficients, Cauchy Euler equation.

Unit 2

Power series solution: Ordinary and Singular points, Gauss’s Hypergeometric Equation, Chebyshev Polynomials, Frobenius’s method, Bessel equation and Bessel functions, Legendre Polynomials, Gamma Functions.

Unit 3

Systems of Linear Differential equations: Differential operators, Operator method for linear systems with constant coefficients, Matrix method for homogeneous linear systems with constant coefficients.

Unit 4

Sturm-Liouville Boundary value problems: Definition and examples, Characteristic values and characteristic functions, Orthogonality of characteristic functions, series of orthonormal functions. Calculus of Variations: Introduction, Variation and its properties, Variational problems with the fixed boundaries, Euler’s equation, the fundamental lemma of the calculus of variations, Functionals involving more than one dependent variables.

Unit 5

Variational problems in parametric form, Isoperimetric problems,Variational problems with moving boundaries, Moving boundary problems with more than one dependent variables, One-sided variations, Field of extremals, central field of extremals, Jacobi’s condition, The Weierstrass function, The Legendre condition, weak extremum, strong extremum.

Course Outcomes
CO1: To understand variational problems and the necessary condition for extremal namely Euler equation. To apply these conditions in evaluations of extremal of functionals for several variables.
CO2: To apply the variational problems in solving physical problems which involves the Principle of Least Action, Conservation Laws, The Hamilton-Jacobi Equation.
CO3: To understand the concept of weak and strong extremum. To apply in the Field of a Functional, Hilbert’s Invariant Integral, The Weierstrass E-Function.
CO4: To apply these techniques in solving differential equations by the Ritz Method and the Method of Finite Differences. To solve the Sturm-Liouville Problem using variational method.
CO5: To understand the idea of solving various integral equations and to apply these tools to solve Fredholm and Volterra Integro – Differential equation by the methods of the Green’s function. Decomposition, direct computation, Successive approximation, series solution, successive approximation.

Textbooks

1. Shepley L. Ross, Differential Equations, Wiley.
2. M.D. Raisinghania, Advanced Differential equations, S Chand Publications.

References

• G.F. Simmons, Differential Equations with Applications and Historical Notes, McGraw-Hill. I.M.Gelfand, S.V.Fomin, Calculus of Variations, Dover.
• L.D.Elsgolc, Calculus of Variations, Dover.