Engineering Mathematics (ODE/PDE)

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Description:

Syllabus :

Ordinary Differential Equations (ODEs):

First order linear ODEs: Exact differential equation, integrating factor method, orthogonal trajectories. Second order linear ODEs: Equations with constant coefficients, Euler-Cauchy form, non- homogeneous equations, linear independence, methods of undetermined coefficients and variation of parameters. Higher order linear ODEs: Systems of first order linear ODEs, matrix method, eigenvalues and eigenvectors. Laplace transform: Application to Heaviside function, Dirac delta function. Uniqueness and existence of ODEs, Lipschitz condition, Wronskian. Sturm-Liouville theory, characteristic function, orthogonality, eigenfunction expansion, properties of Bessel, Legendre and Chebyshev functions. Stability of second-order systems: phase plane, critical point, paths, centre, saddle point, node, spiral.

Partial Differential Equations (PDEs):

First-order PDEs, linear and quasi-linear first-order PDEs. Quasi-linear second- order PDEs, canonical form, method of characteristics. Fourier Series: half-range series, Fourier sine and cosine series, Fourier integral, complex form, Fourier transform, time and frequency shift, duality, differentiation and integration, convolution. Parabolic PDEs, unsteady heat conduction equation, use of separation of variables, use of Fourier series and transform. Elliptic PDEs, steady state heat conduction equation, use of Fourier series. Hyperbolic PDEs, second-order wave equation, D'Alembert's construction, use of Fourier series and transform.

Text Books :

1. Shepley L. Ross (2007). Differential Equations (3rd ed.). Wiley
2. I. N. Sneddon (2013). Elements of Partial Differential Equations. Courier Corporation

Reference Books :

1. Peter V. O'Neil (2012). Advanced Engineering Mathematics. Cengage Learning
2. George B. Arfken, Hans J. Weber & Frank E. Harris (2012). Mathematical Methods for Physicists. Elsevier
3. Bhaskar Dasgupta (2006). Applied Mathematical Methods. Pearson Education
4. Erwin Kreyszig (2020). Advanced Engineering Mathematics. Wiley
5. R. K. Jain & S. R. K. Iyenger (2006). Advanced Engineering Mathematics. Narosa