**FACULTY MEMBER:** Prof. A. K. BANERJEE

**OBJECTIVE :** To make the students understand the principles and applications of Laplace and Fourier transforms, Bessel's functions and Legendre polynomials. The students will also be taught the formations of Partial differential equations and their solutions,

**1. LAPLACE TRANSFORMS:**

Laplace Transforms:Transform of standard functions-linearity property-Transform of e^{at}f(t),tf(t)and f(t)/t-Transform of unit step function and Dirac delta function-Transform of derivatives and integrals-periodic function-inverse Laplace transform using standard methods-convolution theorem-Application to solve ordinary differential equations-simultaneous equations with constant coefficients and integral equation.

**2. FOURIER TRANSFORMS**

Fourier Transforms-Fourier sine and cosine transforms-evaluation of definite integrals using these transforms-convolutions theorem-Z transforms-Inverse Z transforms-Solution of Difference equation with constant coefficients using Z transforms.

**3. SPECIAL FUNCTIONS:**

Special Functions: Series solution of Bessel's equation-Bessel functions-Recurrence relations-Generating function for Bessel functions-Series solution of Legendre's equations-Legendre polynomials-Rodrigue's formulae-generating function and recurrence relations for Legendre polynomials-orthogonality property of legendre polynomials.

**4. PARTIAL DIFFERENTIAL EQUATIONS:**

Formations of PDE by eliminating arbitrary constants and arbitrary functions-solution of first order PDE in standard form and Lagrange's linear equations-solution of homogeneous and non-homogeneous linear higher order PDE with constant coefficients.

**5. APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS**

One and two dimensional heat flow equations: Derivation and solution of one dimensional heat flow equation and two dimensional heat flow equation(Cartesian and Polar form) in steady state by the method of separation of variables-solution of boundary values problems involving these equations using Fourier series.

**TEXTBOOKS:**

1. M. K. Venkataraman, "Engineering Mathematics", 3^{rd} year Part A, 9^{th} Edn.,1984 NPC (for unit 1) and part B, 11th Edn.l990 NPC (for unit 2, 4,and 5)

2. I. N. Sneddon, "Elements of Partial Differential Equations", Mc Graw Hill (for Units 4 and 5) ,

3. M. K. Venkataraman, "Higher Mathematics for Engineering and Science",NPC(for unit 3)

**REFERENCES:-**

1. B. S. Grewal," Higher Engineering Mathematics", Khanna Publishers(for unit 1 to 5)Edn.,36,2001

2. H. K. Das, "Advanced Engineering Mathematics","Advanced Engineering Mathematics",3^{rd} Edn., 1992 Re print 1994.S.Chand and Company Ltd., (For Units I to 5).