Course | Undergraduate |
Semester | Sem. IV |
Subject Code | MA221 |
Subject Title | Integral Transforms, PDE, and Calculus of Variations |
Integral Transforms: The Fourier transform pair – algebraic properties of Fourier transform – convolution, modulation, and translation – transforms of derivatives and derivatives of transform – inversion theory. Laplace transforms of elementary functions – inverse Laplace transforms – linearity property – first and second shifting theorem – Laplace transforms of derivatives and in- tegrals – Laplace transform of Dirac delta function – applications of Laplace transform in solving ordinary differential equations.
Partial Differential Equations: introduction to PDEs – modeling problems related and general second order PDE – classification of PDE: hyperbolic, elliptic and parabolic PDEs – canonical form – scalar first order PDEs – method of characteristics – Charpits method – quasi-linear first order equations – shocks and rarefactions – solution of heat, wave, and Laplace equations using separable variable techniques and Fourier series.
Calculus of Variations: optimization of functional – Euler-Lagrange equations – first variation – isoperimetric problems – Rayleigh-Ritz method.
• Kreyszig, E., Advanced Engineering Mathematics , 10 th ed., John Wiley (2011)
1. Wylie, C. R. and Barrett, L. C., Advanced Engineering Mathematics , McGraw-Hill (2002).
2. Greenberg, M. D., Advanced Engineering Mathematics , Pearson Education (2007).
3. James, G., Advanced Modern Engineering Mathematics , 3 rd ed., Pearson Education (2005).
4. Sneddon, I. N., Elements of Partial Differential Equations , McGraw-Hill (1986).
5. Renardy, M. and Rogers, R. C., An Introduction to Partial Differential Equations , 2 nd ed., Springer-Verlag (2004).
6. McOwen, R. C., Partial Differential Equations: Methods and Applications , 2 nd ed., Pearson Education (2003).
7. Borelli, R. L., Differential Equations: A Modelling Perspective , 2 nd ed., Wiley (2004)