Transport Equation And Its Solution

Transport Equation in
Partial Differential Equations
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Y.K.Sahu Sir -7999897824
Unit­-I
Examples of PDE.
Classification.
Transport Equation‐Initial value
Problem. Non‐homogeneous Equation. Laplace’s Equation‐
Fundamental Solution, Mean Value Formulas, Properties of
Harmonic Functions, Green’s Function, Energy Methods.
Unit­-II*
Heat Equation‐Fundamental Solution, Mean Value Formula,
Properties of Solutions, Energy Methods. Wave Equation‐Solution by
Spherical Means, Non‐homogeneous Equations, Energy Methods.
Analytical Dynamics:
Unit­III Generalized coordinates. Holonomic and Non‐holonomic systems.
Scleronomic and Rheonomic sytems. Generalized potential.
Lagrange’s equations of first kind. Lagrange’s equations of second
kind. Uniqueness of solution. Energy equation for conservative
fields. Hamilton’s variables. Donkin’s theorem. Hamilton canonical
equations. Cyclic coordinates. Routh’s equations.
Unit­IV Poisson’s Bracket. Poisson’s Identity. Jacobi‐Poisson Theorem.
Motivating problems of calculus of variations, Shortest distance.
Minimum surface of revolution. Brachistochrone problem.
Isoperimetric problem. Geodesic. Fundamental lemma of calculus of
variations. Euler’s equation for one dependent function and its
generalization to (1) ‘n’ dependent functions, (ii) higher order
derivatives. Conditional extremum under geometric constraints and
under integral constraints.