Syllabus (PHYSICS)

Course Type: MAJ-8

Semester: 6

Course Code: BPHSMAJ08C

Course Title: Mathematical Methods II

(L-P-Tu): 4-2-0

Credit: 6

Practical/Theory: Combined

Course Objective: Mathematical Methods II

Learning Outcome: Mathematical Methods II

CC 8: Mathematical Methods II (6 Credits)

Course Objective:

Theory (4 Credits)

Frobenius Method and Special Functions: Singular Points of Second Order Linear Differential Equations and their importance. Frobenius method and its applications to differential equations. Legendre, Bessel, Hermite and Laguerre Differential Equations. Legendre Polynomials: Rodrigues Formula, Generating Function, Orthogonality. Simple recurrence relations. Expansion of function in a series of Legendre Polynomials. Bessel Functions of the First Kind: Generating Function, simple recurrence relations. Zeros of Bessel Functions (J_o(x) and J₁(x)) and Orthogonality. (16 Lectures)

Matrices: Addition and Multiplication of Matrices. Null Matrices. Diagonal, Scalar and Unit Matrices. Transpose of a Matrix. Symmetric and Skew-Symmetric Matrices. Conjugate of a Matrix. Hermitian and Skew- Hermitian Matrices. Singular and Non-Singular matrices. Orthogonal and Unitary Matrices. Trace of a Matrix. Inner Product. (7 Lectures)

Eigenvalues and Eigenvectors: Cayley-Hamiliton Theorem. Diagonalization of Matrices. Functions of a Matrix.Solution of linear equations by matrix method. (7 Lectures)

Complex Analysis: Complex Numbers and their Graphical Representation. Euler's formula, DeMoivre's theorem, Roots of Complex Numbers. Functions of Complex Variables, Analyticity and Cauchy-Riemann Conditions. Examples of analytic functions. Harmonic functions. Analytic functions. Entire functions. Multiple-valued functions. Singular functions: poles and branch points, order of singularity, branch cuts. Integration of a function of a complex variable. Cauchy's Inequality. Cauchy’s Integral formula. Simply and multiply connected region. Laurent and Taylor’s series expansion. Residues, Residue Theorem, and its applications. (30 Lectures)

Practical (2 Credits)

Introduction to Numerical computation using numpy and scipy: Introduction to the python numpy module. Arrays in numpy, array operations, array item selection, slicing, shaping arrays. Basic linear algebra using the linalg submodule. Introduction to online graph plotting using matplotlib. Introduction to the scipy module. Uses in optimization and solution of differential equations.

Numerical solution of Ordinary differential equations- Euler and Runge-Kutta (RK) second and fourth order methods. Numerical solution of partial differential equations, First order Differential equation.

First order differential equation

• Radioactive decay
• Current in RC, LC circuits with DC source
• Newton’s law of cooling
• Classical equations of motion
• Harmonic oscillator (no friction)
• Damped Harmonic oscillator a) Over damped b) Critical damped
• Oscillatory Motion
• Forced Harmonic oscillator
• Transient and Steady state solution
• Apply above to LCR circuits also

Partial differential equations

• Wave equation
• Heat equation
• Poisson equation
• Laplace equation

Reading References

Theory

1. Mathematical Methods for Physicists, G.B. Arfken, H.J. Weber and F.E. Harris, Elsevier
2. Mathematical Methods for Physics and Engineers, K.F Riley, M.P. Hobson and S. J. Bence, Cambridge University Press
3. Mathematical Physics, V Balakrishnan, Ane Books Pvt Ltd
4. Complex Variables and Applications, J.W. Brown & R.V. Churchill, Tata McGraw-Hill
5. Complex Variables, M R Spiegel, TMH
6. Higher Engineering Mathematics, B S Grewal, Khanna Publishers, 44^th Edn
7. Mathematical Physics, H K Dass and R Verma, S Chand & Co

Practical

1. A first course in Numerical Methods, U.M. Ascher & C. Greif, PHI Learning.
2. Elementary Numerical Analysis, K.E. Atkinson, Wiley India Edition, 3^rd Edn.,
3. Numerical Methods for Scientists & Engineers, R.W. Hamming, Courier Dover Pub.
4. An Introduction to Computational Physics, T. Pang, Cambridge Univ. Press, 2^nd Edn.
5. Computational Physics, Darren Walker, Scientific International Pvt. Ltd.
6. Numpy beginners guide, Idris Alba, Packt Publishing
7. Computational Physics, D.Walker, 1st Edn., Scientific International Pvt. Ltd.
8. Simulation of ODE/PDE Models with MATLAB®, OCTAVE and SCILAB: Scientific and
9. Engineering Applications: A.V. Wouwer, P. Saucez, C.V. Fernández., Springer