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HD1
: MATHEMATICS III
1
PARTIAL
DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATION
Introduction,
Limit , Partial derivatives , Partial derivatives of Higher orders,
Which variable is to be treated as constant, Homogeneous function,
Euler’s Theorem on Homogeneous Functions, Introduction, Total
Differential Coefficient, Important Deductions, Typical cases,
Geometrical Interpretation of
 , ,
Tangent plane to a surface, Error determination, Jacobians,
Properties of Jacobians, Jacobians of Implicit Functions, Partial
Derivatives of Implicit Functions by Jacobian, Taylor’s series,
Conditions for F(x,y) to be of two variables maximum or minimum,
Lagrange’s method of undermined Multipliers.
2
PARTIAL DIFFERENTIAL
EQUATIONS
Partial
Differential Equations, Order, Method of Forming Partial
Differential Equations, Solution of Equation by direct Integration,
Lagrange’s Linear equation, Working Rule, Method of Multipliers,
Partial Differential Equations non- Linear in p,q , Linear
Homogeneous Partial Diff. Eqn., Rules for finding the complimentary
function, Rules for finding the particular Integral, Introduction,
Method of Separation of Variables, Equation of Vibrating Strain,
Solution of Wave Equation, One Dimensional Heat Flow, Two
dimensional Heat Flow.
3
FOURIER SERIES
Periodic
Functions, Fourier Series, Dirichlet’s Conditions, Advantages of
Fourier Series, Useful Integrals, Determination of Fourier constants
(Euler’s Formulae), Functions defined in two or more sub spaces,
Even Functions, Half Range’s series, Change of Interval, Parseval’s
Formula, Fourier series in Complex Form, Practical Harmonic
Analysis.
4
LAPLACE
TRANSFORMATION
Introduction,
Laplace Transform, Important Formulae, Properties of Laplace
Transforms, Laplace Transform of the Derivative of f (t), Laplace
Transform of Derivative of order n, Laplace Transform of Integral of
f (t), Laplace Transform of t.f (t) (Multiplication by t), Laplace
Transform of f(t)
(Diversion by t), Unit step function, second shifting theorem,
Theorem, Impulse Function, Periodic Functions, Convolution Theorem,
Laplace Transform of Bessel function, Evaluation of Integral,
Formulae of Laplace Transform, properties of Laplace Transform,
Inverse of Laplace Transform, Important formulae, Multiplication by
s, Division of s (Multiplication by 1/s), First shifting properties,
second shifting properties, Inverse Laplace Transform of
Derivatives, Inverse Laplace Transform of Integrals,Partial Fraction
Method, Inverse Laplace Transform, Solution of Differential
Equations, Solution of simultaneous equations, Inversion Formulae
for the Laplace Transform.
5
NUMERICAL TECHNIQUES
Solution of
Ordinary Differential Equations, Taylor’s Series Method, Picard’s
method of successive approximations, Euler’s method, Euler’s
Modified formula, Runge’s Formula, Runge’s Formula (Third only),
Runge’s Kutta Formula (Fourth order), Higher order Differential
Equations.
6
NUMERICAL METHODS
FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATION
General
Linear partial differential equations, Finite-Difference
Approximation to Derivatives, Solution of Partial Differential
equation(Laplace’s method), Jacobi’s Iteration Formula, Guass-Seidal
method, Successive over-Relanation or S.O.R. method, Poisson
Equation, Heat equation(parabolic equations), Wave equation
(Hyperbolic Equation).
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