Bachelor of Engineering (Mechanical Engineering)

Unit I
Laplace Transform – Definition and its properties, Transform of derivatives and integrals,
Evaluation of integrals by LT, inverse and its properties, convolution theorem. LT of periodic
functions and unit step function, Applications of Laplace transform to solve ordinary
differential equations and partial differential equations (one dimensional wave and diffusion
equations) (8 Hours)
Unit II
Z- Transform – Definition and properties, inversion, relation with Laplace Transform.
Applications of z-transform to solve difference equation with constant coefficient. (5 Hours)
Unit III
Fourier Transform – Definition, Fourier Integral Theorem, Fourier Sine and Cosine Integrals,
Finite Fourier Sine and Cosine Transform, Parse Val’s Identity, Convolution Theorem
(5 Hours)
Unit IV
Complex Variable – Analytic function Cauchy-Riemann Conditions, conjugate functions,
singularities, Cauchy’s Integral Theorem and Integral Formula (statement only). Taylor’s and
Laurent’s Theorem (statement only) Residue Theorem, Contour Integration, Evaluation of
real and complex integrals by Residue Theorem, Conformal mapping, Mapping by linear and
inverse Transformation. (10 Hours)
Unit V
Special Functions and Series solution. – Series solution of differential equation by Frobaniu’s
method, Bessel’s Function, Legendre’s Polynomials, Recurrence Relations, Rodrigue’s
Formula, Generating Functions, Orthogonal Properties of Jn (x) and Pn (x). (8Hours)
Unit VI
Fourier Series – Periodic Function and their Fourier series expansion, Fourier Series for even
and odd function, Change of interval, half range expansions.
Partial Differential Equations – PDE of first order first degree i.e. Lagrange’s form, linear
homogeneous equations of higher order with constant coefficient. Method of separations of
variables, applications to one-dimensional heat and diffusion equation. Two-dimensional
Heat Equation (only steady state).