FIFTH SEMESTER MATHEMATICS
Integral transforms with applications-7B
Unit – 1: Laplace transforms- I
1. Definition of Laplace transform, linearity property-piece wise continuous function.
2. Existence of Laplace transform, functions of exponential order and of class A.
3. First shifting theorem, second shifting theorem and change of scale property.
Unit – 2: Laplace transforms- II
1. Laplace Transform of the derivatives, initial value theorem and final value theorem. Laplace transforms of integrals.
2. Laplace transform of tn. f (t), division by t, evolution of integrals by Laplace transforms.
3. Laplace transform of some special functions-namely Dirac delta function, error function, Bessel function and Laplace transform of periodic function.
Unit – 3: Inverse Laplace transforms
1. Definition of Inverse Laplace transform, linear property, first shifting theorem, second shifting theorem, change of scale property, use of partial fractions.
2. Inverse Laplace transforms of derivatives, inverse, Laplace transforms of integrals, multiplication by powers of ‘p’, division by ‘p’.
3. Convolution, convolution theorem proof and applications.
Unit – 4: Applications of Laplace transforms
1. Solutions of differential equations with constants coefficients, solutions of differential equations with variable coefficients.
2. Applications of Laplace transforms to integral equations- Abel’s integral equation.
3. Converting the differential equations into integral equations, converting the integral equations into differential equations.
Unit – 5: Fourier transforms
1. Integral transforms, Fourier integral theorem (without proof), Fourier sine and cosine integrals.
2. Properties of Fourier transforms, change of scale property, shifting property, modulation theorem. Convolution.
3. Convolution theorem for Fourier transform, Parseval’s Identify, finite Fourier transforms.