Applied Mathematics – II

Course Objectives:

1. To understand Complex series methods.
2. To understand Complex analysis
3. To understand Fourier and Laplace methods
4. To understand how to solve specific formulated engineering problems using PDE methods.

Course Outcomes (CO):

• CO1 Ability to use Complex series methods.
• CO2 Ability to use Complex analysis to solve formulated engineering problems
• CO3 Ability to use Fourier and Laplace methods to solve formulated engineering problems
• CO4 Ability to solve specific formulated engineering problems using PDE methods.

Unit I

Complex Analysis – I : Complex Numbers and Their Geometric Representation, Polar Form of Complex Numbers. Powers and Roots, Derivative. Analytic Function, Cauchy–Riemann Equations.

Laplace’s Equation, Exponential Function, Trigonometric and Hyperbolic Functions. Euler’s Formula, de’Moivre’s theorem (without proof), Logarithm. General Power. Principal Value. Singularities and Zeros. Infinity,

Line Integral in the Complex Plane, Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Derivatives of Analytic Functions, Taylor and Maclaurin Series.

Unit II

Complex Analysis – II: Laurent Series, Residue Integration Method. Residue Integration of Real Integrals,

Geometry of Analytic Functions: Conformal Mapping, Linear Fractional Transformations (Möbius Transformations), Special Linear Fractional Transformations, Conformal Mapping by Other Functions, 

Applications: Electrostatic Fields, Use of Conformal Mapping. Modeling, Heat Problems, Fluid  Flow. Poisson’s Integral Formula for Potentials 

Unit III

Laplace Transforms: Definitions and existence (without proof), properties, First Shifting Theorem (sShifting), Transforms of Derivatives and Integrals and ODEs, Unit Step Function (Heaviside Function).

Second Shifting Theorem (t-Shifting), Short Impulses. Dirac’s Delta Function. Partial Fractions, Convolution. Integral Equations, Differentiation and Integration of Transforms. Solution of ODEs with Variable Coefficients, Solution of Systems of ODEs. Inverse Laplace transform and its properties.

Fourier Analysis: Fourier Series, Arbitrary Period. Even and Odd Functions. Half-Range Expansions, Sturm–Liouville Problems. Fourier Integral, Fourier Cosine and Sine Transforms, Fourier Transform. Usage of fourier analysis for solution of ODEs. Inverse Fourier transform and its properties.

Unit IV

Partial Differential Equations (PDEs): Basic Concepts of PDEs. Modeling: Vibrating String, Wave Equation. Solution by Separating Variables. Use of Fourier Series. D’Alembert’s Solution of the Wave Equation. Characteristics. Modeling: Heat Flow from a Body in Space. Heat Equation:Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem. Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms. Modeling: Membrane, Two-Dimensional Wave Equation. Rectangular Membrane. Laplacian in Polar Coordinates. Circular Membrane.

Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential. Solution of PDEs by Laplace Transforms.

Textbooks:

1. Advanced Engineering Mathematics by Erwin Kreyszig, John Wiley, 10th Ed., 2011.

References:

1. Engineering Mathematics by K.A. Stroud withDexter J. Booth, Macmillan, 2020.
2. Advanced Engineering Mathematics by Larry Turyn, Taylor and Francis, 2014.
3. Advanced Engineering Mathematics by Dennis G. Zill, Jones & Bartlett Learning, 2018.
4. Advanced Engineering Mathematics with MATLAB by Dean G. Duffy, Taylor and Francis, 2017.
5. Advanced Engineering Mathematics by Merle C. Potter, Jack L. Lessing, and Edward F. Aboufadel, Springer (Switzerland), 2019.
6. Mathematical Methods for Physics and Engineering, by K. F. Riley, M. P. Hobson and S. J. Bence, CUP, 2013.