The Department of Electrical Engineering (EED) at Shiv Nadar Institution of Eminence (Deemed to be University), Delhi NCR invites applications for admission to its Full-Time and Part-Time Ph.D. program (Duration: 5 Years). Students admitted for full-time, completely residential Ph.D. program will receive a fellowship stipend of ₹45,000/- per month for the first two years and ₹50,000/- per month for the last three years, subject to the candidate fulfilling the departmental and university academic requirements. Conference and publication support will be provided for deserving Ph.D. students. Students can get a Research Grant of INR 1,50,000 (One Lakh fifty thousand) for conference (Scopus-indexed) travel (local and/or international) during their five-year Ph.D. program.

Research Areas:

• Reliability-aware design of advanced computing architecture; VLSI for communication, VLSI Architecture, Hardware Accelerators, Mixed-signal VLSI, CAD algorithms for VLSI.
• TCAD-based modeling of advanced transistor architectures and on-chip interconnects and their circuit implications, simulation-based study of novel materials for advanced transistors.
• Perovskite solar cells, fundamental studies on charge transport in organic semiconducting materials and devices, organic solar cells, bioelectronics, 3D-printed wearable sensors, next-generation organ-on-chips, brain-on-a-chip, and semiconductor/MEMS sensors and devices for various applications like RF, biomedical, IoT, etc.
• IoT system architecture, IoT enabling technologies, IoT communication & networking IoT for smart agriculture, smart health, and smart cities.
• Antennas and devices for 5G/6G applications, sub-terahertz and terahertz antennas and devices, optoelectronic antennas, microwave devices, microwave and millimeter wave antennas, active and passive circuits, radar based non-invasive sensors for medical.
• Signal processing for Cognitive Robotics, Cyber Physical Systems, Signal processing communication and machine learning, computer vision using machine learning and artificial intelligence, brain signal prediction, graph-based signal modeling and prediction for 6G multi-antenna wireless communication systems.
• Physical layer aspects of communication systems, terrestrial free-space optical communications, resource allocation, integrated sensing and communication, intelligent reflecting surfaces, communication for 5G and beyond, wireless communication, quantum communication, ultra-reliable Low Latency Communications (URLLC), and next-generation multiple access techniques.
• Control systems, machine learning control systems, control systems applications in robotics and power electronics, multiagent systems, networked control.
• Power electronics converter, electric machines and drives, photovoltaic power systems, integration of renewable sources with the grid, VLSI for smart power applications, WBG-material-based power electronics.

The department actively engages in interdisciplinary research in collaboration with groups across various schools at Shiv Nadar Institution of Eminence, as well as with other leading institutes and universities.

For more information, please visit the following link to explore the department and research interests: https://snu.edu.in/schools/school-of-engineering/departments/department-of-electrical-engineering/

Eligibility for Full-Time/Part-Time Candidate #:
Qualifying Degree & Domain	Minimum performance in qualifying degree
M.Tech./M.E. or equivalent degree in Electronics & Communication/Electrical & Electronics/relevant discipline	60 % or 6.0 CGPA from a recognized technical institute or university.
M.Sc. in Physics with specialization in electronics/solid state devices/Material Science/Nanotechnology/other-relevant-disciplines	65% or 6.5 CGPA from a recognized institute or university.
M.Sc. Electronics
M.Sc. in Mathematics with background in linear algebra and Stochastic processes and transform theory.
B.Tech./B.E. in Electronics & Communication/ Electrical & Electronics/ relevant discipline.	75% or 7.5 CGPA from a recognized technical institute or university.

^# Part-time Ph.D. candidates are not eligible for Institute Fellowships.

 

The eligibility criteria mentioned above are the minimum, and applications not meeting the same criteria will be summarily rejected.

 

Selection Process

·       There will be a written test followed by an interview. However: o   M.Tech./M.E./M.S. Research candidates who have-completed/are-pursuing their degree based on GATE Score will be exempted from the written test. o   Candidates with valid CSIR-JRF/UGC-NET/GATE certificates will be exempted from the written test. ·       Final selection of the candidates will be made by the Departmental Selection Committee after completing the process (written test and/or Interview).

Written Test Pattern and Syllabus: A candidate appearing in the Ph.D. Entrance written test has to answer questions from the following sections:

Section 1: General Aptitude Applicable to: All Candidates Syllabus: As per GATE (Refer annexure A)
Section 2: Engineering Mathematics Applicable to: All Candidates Syllabus: As per GATE (Refer annexure B)
Section 3: Any one of the below domain/core subjects #
Qualifying Degree & Domain	Test Code* (Domain/Core)	Syllabus
MTech./M.E. and BTech/B.E. candidates applying for Ph.D. in ECE	ECE	Based on GATE Syllabus of Electronics & Communication Engineering (Annexure C)
MTech./M.E. and BTech/B.E. candidates applying for Ph.D. in EE	EE	Based on GATE Syllabus of Electrical Engineering (Annexure C)
M.Sc. in Physics and M.Sc. in Electronics	PHY	Based on GATE Syllabus of Physics (Annexure C)
M.Sc. in Mathematics	MA	Based on GATE Syllabus of Mathematics (Annexure C)

^#Any Test Code can be chosen by an applicant, regardless of the applicant’s qualifying degree/domain. For instance, an applicant with MSc Electronics background can appear for the ECE Test Code if he/she so chooses at the time of the test.

Note: Preliminary Coursework Requirements if Admitted into the PhD Program:

• Tech./M.Sc. or equivalent degree holders are required to complete a minimum of 20 credits (about 6 courses) during the PhD program with a minimum required CGPA of 7.0.
• Tech. or equivalent degree holders are required to complete a minimum of 12 credits (about 3-4 courses) during the PhD program with a minimum required CGPA of 7.0.

Application Fee:

As prompted during the application process.

Application Instructions: All interested applicants should apply online by clicking on the “APPLY NOW” button or through the website. Please follow the instructions carefully and fill mandatory fields. Upload the following documents:

• Passport-sized color photograph
• Current CV
• All mark sheets/degree certificates (10^th grade onwards)
• National level examination certificate- CSIR, UGC, GATE (if applicable)
• A statement of purpose

Important Dates*

Milestone	Date
Application portal opens	24th September 2026
Last date for the receipt of completed application form & application fee	30th November 2026
Shortlisting of eligible candidates	2nd December 2026
Written test	5th December 2026
Interviews	10th and/or 11th December 2026
Result declaration & Release of offer letters	15th December 2026
Fee Payment Deadline	TO BE FILLED BY ADMISSIONS TEAM
Ph.D. Registration	8th January 2027
Spring 2027 semester, First Day (Academic Session Begins)	11th January 2027

* Tentative dates, subject to change. Details of the written test and interview will be communicated to shortlisted applicants via email.

Tuition Fee and Financial Assistance

All full-time Ph.D. students receive a doctoral award (teaching & research assistantship) that includes a tuition fee waiver of ₹60,000 and a monthly stipend of ₹45,000 (first 2 years) and ₹50,000 (next 3 years), subject to continuous satisfactory performance and compliance with university regulations (see fee structure and details here) .

In addition, a Research Grant of ₹1,50,000 is available during the 5-year program for presenting work at Scopus-indexed conferences (domestic/international). Details on hostel fees and other financial regulations are provided on the admission website.

Hostel Accommodation: As on-campus hostel accommodation is currently limited, the Institution may not be able to accommodate all Ph.D. scholars on campus. Admitted scholars may need to make their own off-campus accommodation arrangements until hostel rooms become available.

Contacts:

Dr. Upendra Kumar Pandey 

Associate Professor & Co-Ordinator, Ph.D. Program,

Department of Electrical Engineering, School of Engineering,

Shiv Nadar Institution of Eminence (Deemed to be University), Delhi NCR,

P.O. Shiv Nadar University, NH-91, Tehsil-Dadri

District Gautam Buddha Nagar, UP, 201314, India.

Email: [email protected] 

Telephone No.: 0120- 7170100, Ext. 690, Mob. +91 8105752985

OR

Ms. Priyanka Verma

Administrative Assistant, Department of Electrical Engineering

Email: [email protected] 

Telephone No.: 0120- 7170100, Ext. 428, Mob. +91 8077173721.

 

For further information visit our website www.snu.edu.in

More details about Doctoral admissions can be found at: https://snu.edu.in/admissions/graduate-programs

All interested applicants should apply online by clicking on “APPLY NOW” button:

  

Annexure A

 

Section 1: General Aptitude

 

Verbal Aptitude

Basic English grammar: tenses, articles, adjectives, prepositions, conjunctions, verb-noun agreement, and other parts of speech 

Basic vocabulary: words, idioms, and phrases in context.

Reading and comprehension, Narrative sequencing.

Quantitative Aptitude

Data interpretation: data graphs (bar graphs, pie charts, and other graphs representing data), 2- and 3-dimensional plots, maps, and tables

Numerical computation and estimation: ratios, percentages, powers, exponents and logarithms, permutations and combinations, and series Mensuration and geometry Elementary statistics and probability.

Analytical Aptitude

Logic: deduction and induction, Analogy, Numerical relations and reasoning.

Spatial Aptitude

Transformation of shapes: translation, rotation, scaling, mirroring, assembling, and grouping paper folding, cutting, and patterns in 2 and 3 dimensions.
 

Annexure B

 

Section 2: Engineering Mathematics

 

Linear Algebra

Matrix Algebra, Systems of Linear Equations, Eigenvalues and Eigenvectors, Linear Vector, Spaces: Basis, Orthogonality, and Completeness, Matrices: Similarity Transformations and Diagonalization

 

Calculus and Vector Calculus

Mean Value Theorems, Theorems of Integral Calculus, Evaluation of Definite and Improper Integrals, Partial Derivatives, Maxima and Minima, Multiple Integrals, Fourier Series, Fourie Analysis, Vector Identities, Directional Derivatives Line, Surface, and Volume Integrals, Stokes’s Theorem, Gauss’s (Divergence) Theorem, Green’s Theorem

 

Differential Equations

First-Order Equations (Linear and Nonlinear), Higher-Order Linear Differential Equations with Constant Coefficients, Method of Variation of Parameters, Cauchy’s and Euler’s Equations, Initial and Boundary Value Problems, Partial Differential Equations: Method of Separation of Variables, Second-Order Linear Differential Equations and Special Function Solutions

 

Complex Variables and Complex Analysis

Analytic Functions, Cauchy-Riemann Conditions, Cauchy’s Integral Theorem and Formula, Taylor and Laurent Series, Residue Theorem and Applications, Singularities and Solution Integrals

 

Transforms and Tensors

Laplace Transforms, Elementary Concepts of Tensors: Covariant and Contravariant Tensors

 

Probability and Statistics

Sampling Theorems, Conditional Probability, Descriptive Statistics: Mean, Median, Mode, Standard Deviation, Random Variables: Discrete and Continuous Distributions, Common Distributions: Poisson, Normal, and Binomial, Correlation and Regression Analysis

 

Annexure C

 

Electronics and Communication Engineering (Test Code: ECE)

 

Networks, Signals and Systems

Circuit Analysis: Node and mesh analysis, superposition, Thevenin's theorem, Norton’s theorem, reciprocity. Sinusoidal steady state analysis: phasors, complex power, maximum power transfer. Time and frequency domain analysis of linear circuits: RL, RC, and RLC circuits, solution of network equations using Laplace transform.

Linear 2-port network parameters, wye-delta transformation.

Continuous-time Signals: Fourier series and Fourier transform, sampling theorem and applications. 

Discrete-time Signals: DTFT, DFT, z-transform, discrete-time processing of continuous-time signals. LTI systems: definition and properties, causality, stability, impulse response, convolution, poles and zeroes, frequency response, group delay, phase delay.

 

Electronic Devices

Energy bands in intrinsic and extrinsic semiconductors, equilibrium carrier concentration, direct and indirect band-gap semiconductors.

Carrier Transport: Diffusion current, drift current, mobility and resistivity, generation, and recombination of carriers, Poisson and continuity equations. 

P-N junction, Zener diode, BJT, MOS capacitor, MOSFET, LED, photo diode, and solar cell.

 

Analog Circuits

Diode Circuits: Clipping, clamping, and rectifiers.

BJT and MOSFET Amplifiers: Biasing, AC coupling, small signal analysis, frequency response. Current mirrors and differential amplifiers.

Op-amp Circuits: Amplifiers, summers, differentiators, integrators, active filters, Schmitt triggers and oscillators.

 

Digital Circuits

Number Representations: Binary, integer and floating-point- numbers. Combinatorial circuits: Boolean algebra, minimization of functions using Boolean identities and Karnaugh map, logic gates and their static CMOS implementations, arithmetic circuits, code converters, multiplexers, decoders.

Sequential Circuits: Latches and flip-flops, counters, shift-registers, finite state machines, propagation delay, setup and hold time, critical path delay.

Data Converters: Sample and hold circuits, ADCs and DACs.

Semiconductor Memories: ROM, SRAM, DRAM.

Computer Organization: Machine instructions and addressing modes, ALU, data-path and control unit, instruction pipelining.

 

Control Systems

Basic control system components; Feedback principle; Transfer function; Block diagram representation; Signal flow graph; Transient and steady-state analysis of LTI systems; Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots; Lag, lead and lag-lead compensation; State variable model and solution of state equation of LTI systems. 

 

Communications

Random Processes: Auto correlation and power spectral density, properties of white noise, filtering of random signals through LTI systems.

Analog Communications: Amplitude modulation and demodulation, angle modulation and demodulation, spectra of AM and FM, super heterodyne receivers.

Information Theory: Entropy, mutual information and channel capacity theorem.

Digital Communications: PCM, DPCM, digital modulation schemes (ASK, PSK, FSK, QAM), bandwidth, inter-symbol interference, MAP, ML detection, matched filter receiver, SNR and BER. Fundamentals of error correction, Hamming codes, CRC.

 

Electromagnetics

Maxwell's Equations: Differential and integral forms and their interpretation, boundary conditions, wave equation, Poynting vector.

Plane Waves and Properties: Reflection and refraction, polarization, phase and group velocity, propagation through various media, skin depth.

Transmission Lines: Equations, characteristic impedance, impedance matching, impedance transformation, S-parameters, Smith chart. Rectangular and circular waveguides, light propagation in optical fibers, dipole and monopole antennas, linear antenna arrays.

 

Electrical Engineering (Test Code: EE)

 

Electric circuits

Network Elements: Ideal voltage and current sources, dependent sources, R, L, C, M elements; Network solution methods: KCL, KVL, Node and Mesh analysis; Network Theorems: Thevenin’s, Norton’s, Superposition and Maximum Power Transfer theorem; Transient response of DC and AC networks, sinusoidal steady-state analysis, resonance, two-port networks, balanced three-phase circuits, star-delta transformation, complex power and power factor in AC circuits. 

 

Electromagnetic Fields

Coulomb's Law, Electric Field Intensity, Electric Flux Density, Gauss's Law, Divergence, Electric field and potential due to point, line, plane and spherical charge distributions, Effect of dielectric medium, Capacitance of simple configurations, Biot‐Savart’s law, Ampere’s law, Curl, Faraday’s law, Lorentz force, Inductance, Magnetomotive force, Reluctance, Magnetic circuits, Self and Mutual inductance of simple configurations. 

 

Signals and Systems

Representation of continuous and discrete time signals, shifting and scaling properties, linear time-invariant and causal systems, Fourier series representation of continuous and discrete time periodic signals, sampling theorem, Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform. R.M.S. value, average value calculation for any general periodic waveform.

 

Electrical Machines

Single phase transformer: equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency; Three-phase transformers: connections, vector groups, parallel operation; Auto-transformer, Electromechanical energy conversion principles; DC machines: separately excited, series and shunt, motoring and generating mode of operation and their characteristics, speed control of dc motors; Three-phase induction machines: principle of operation, types, performance, torque-speed characteristics, no-load and blocked-rotor tests, equivalent circuit, starting and speed control; Operating principle of single-phase induction motors; Synchronous machines: cylindrical and salient pole machines, performance and characteristics, regulation and parallel operation of generators, starting of synchronous motors; Types of losses and efficiency calculations of electric machines.

 

Power Systems

Basic concepts of electrical power generation, AC and DC transmission concepts, Models and performance of transmission lines and cables, Economic Load Dispatch (with and without considering transmission losses), Series and shunt compensation, Electric field distribution and insulators, Distribution systems, Per‐unit quantities, Bus admittance matrix, Gauss- Seidel and Newton-Raphson load flow methods, Voltage and Frequency control, Power factor correction, Symmetrical components, Symmetrical and unsymmetrical fault analysis, Principles of over‐current, differential, directional and distance protection; Circuit breakers, System stability concepts, Equal area criterion.

 

Control Systems

Mathematical modelling and representation of systems, Feedback principle, transfer function, Block diagrams and Signal flow graphs, Transient and Steady‐state analysis of linear time invariant systems, Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci, Lag, Lead and Lead‐Lag compensators; P, PI and PID controllers; State space model, Solution of state equations of LTI systems.

 

Electrical and Electronic Measurements

Bridges and Potentiometers, Measurement of voltage, current, power, energy and power factor; Instrument transformers, Digital voltmeters and multi-meters, Phase, Time and Frequency measurement; Oscilloscopes, Error analysis. 

 

Analog and Digital Electronics

Simple diode circuits: clipping, clamping, rectifiers; Amplifiers: biasing, equivalent circuit and frequency response; oscillators and feedback amplifiers; operational amplifiers: characteristics and applications; single stage active filters, Active Filters: Sallen Key, Butterwoth, VCOs and timers, combinatorial and sequential logic circuits, multiplexers, demultiplexers, Schmitt triggers, sample and hold circuits, A/D and D/A converters. 

 

Physics (Test Code: PHY)

 

Classical Mechanics

Lagrangian Formulation: D'Alembert's principle, Euler-Lagrange equation, Hamilton's principle, calculus of variations; symmetry and conservation laws; central force motion: Kepler problem and Rutherford scattering; small oscillations: coupled oscillations and normal modes; rigid body dynamics: interia tensor, orthogonal transformations,  Euler angles, Torque free motion of a symmetric top; Hamiltonian and Hamilton's equations of motion; Liouville's theorem; canonical transformations: action-angle variables, Poisson brackets, Hamilton-Jacobi equation.

Special Theory of Relativity: Lorentz transformations, relativistic kinematics, mass-energy equivalence.

 

Electromagnetic Theory

Solutions of electrostatic and magnetostatic problems including boundary value problems; method of images; separation of variables; dielectrics and conductors; magnetic materials; multipole expansion; Maxwell's equations; scalar and vector potentials; Coulomb and Lorentz gauges; electromagnetic waves in free space, non-conducting and conducting media; reflection and transmission at normal and oblique incidences; polarization of electromagnetic waves; Poynting vector, Poynting theorem, energy and momentum of electromagnetic waves; radiation from a moving charge.

 

Quantum Mechanics

Postulates of quantum mechanics; uncertainty principle; Schrodinger equation; Dirac Bra-Ket notation, linear vectors and operators in Hilbert space; one dimensional potentials: step potential, finite rectangular well, tunneling from a potential barrier, particle in a box, harmonic oscillator; two and three dimensional systems: concept of degeneracy; hydrogen atom; angular momentum and spin; addition of angular momenta; variational method and WKB approximation, time independent perturbation theory; elementary scattering theory, Born approximation; symmetries in quantum mechanical systems.

 

Thermodynamics and Statistical Physics

Laws of thermodynamics; macrostates and microstates; phase space; ensembles; partition function, free energy, calculation of thermodynamic quantities; classical and quantum statistics; degenerate Fermi gas; black body radiation and Planck's distribution law; Bose-Einstein condensation; first and second order phase transitions, phase equilibria, critical point.

 

Atomic and Molecular Physics

Spectra of one-and many-electron atoms; spin-orbit interaction: LS and jj couplings; fine and hyperfine structures; Zeeman and Stark effects; electric dipole transitions and selection rules; rotational and vibrational spectra of diatomic molecules; electronic transitions in diatomic molecules, Franck-Condon principle; Raman effect; EPR, NMR, ESR, X-ray spectra; lasers: Einstein coefficients, population inversion, two and three level systems.

 

Solid State Physics

Elements of crystallography; diffraction methods for structure determination; bonding in solids; lattice vibrations and thermal properties of solids; free electron theory; band theory of solids: nearly free electron and tight binding models; metals, semiconductors, and insulators; conductivity, mobility and effective mass; Optical properties of solids; Kramer's-Kronig relation, intra- and inter-band transitions; dielectric properties of solid; dielectric function, polarizability, ferroelectricity; magnetic properties of solids; dia, para, ferro, antiferro and ferri-magnetism, domains and magnetic anisotropy; superconductivity: Type-I and Type II superconductors, Meissner effect, London equation, BCS Theory, flux quantization.

 

Electronics

Semiconductors in Equilibrium: Electron and hole statistics in intrinsic and extrinsic semiconductors; metal-semiconductor junctions; Ohmic and rectifying contacts; PN diodes, bipolar junction transistors, field effect transistors; negative and positive feedback circuits; oscillators, operational amplifiers, active filters; basics of digital logic circuits, combinational and sequential circuits, flip-flops, timers, counters, registers, A/D and D/A conversion.

 

Nuclear and Particle Physics

Nuclear radii and charge distributions, nuclear binding energy, electric and magnetic moments; semi-empirical mass formula; nuclear models; liquid drop model, nuclear shell model; nuclear force and two nucleon problem; alpha decay, beta-decay, electromagnetic transitions in nuclei; Rutherford scattering, nuclear reactions, conservation laws; fission and fusion; particle accelerators and detectors; elementary particles; photons, baryons, mesons and leptons; quark model; conservation laws, isospin symmetry, charge conjugation, parity and time-reversal invariance.

 

Mathematics (Test Code: MA)

 

Calculus

Functions of two or more variables, continuity, directional derivatives, partial derivatives, total derivative, maxima and minima, saddle point, method of Lagrange’s multipliers; Double and Triple integrals and their applications to area, volume, and surface area; Vector Calculus: gradient, divergence and curl, Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem.

Linear Algebra

Finite dimensional vector spaces over real or complex fields; Linear transformations and their matrix representations, rank and nullity; systems of linear equations, characteristic polynomial, eigen values and eigen vectors, diagonalization, minimal polynomial, Cayley-Hamilton Theorem, Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, symmetric, skew-symmetric, Hermitian, skew-Hermitian, normal, orthogonal and unitary matrices; diagonalization by a unitary matrix,  Jordan canonical form; bilinear and quadratic forms.

Real Analysis

Metric spaces, connectedness, compactness, completeness; Sequences and series of functions, uniform convergence, Ascoli-Arzela theorem; Weierstrass approximation theorem; contraction mapping principle, Power series; Differentiation of functions of several variables, Inverse and Implicit function theorems; Lebesgue measure on the real line, measurable functions; Lebesgue integral, Fatou’s lemma, monotone convergence theorem, dominated convergence theorem. 

Complex Analysis

Functions of a complex variable: continuity, differentiability, analytic functions, harmonic functions; Complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle, Morera’s theorem; zeros and singularities; Power series, radius of convergence, Taylor’s series and Laurent’s series; Residue theorem and applications for evaluating real integrals; Rouche’s theorem, Argument principle, Schwarz lemma; Conformal mappings, Mobius transformations.

Ordinary Differential Equations

First order ordinary differential equations, existence and uniqueness theorems for initial value problems, linear ordinary differential equations of higher order with constant coefficients; Second order linear ordinary differential equations with variable coefficients; Cauchy-Euler equation, method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method); Legendre and Bessel functions and their orthogonal properties; Systems of linear first order ordinary differential equations, Sturm's oscillation and separation theorems, Sturm-Liouville eigenvalue problems, Planar autonomous systems of ordinary differential equations: Stability of stationary points for linear systems with constant coefficients, Linearized stability, Lyapunov functions.

 

Algebra

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms; cyclic groups, permutation groups, Group action, Sylow’s theorems and their applications; Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisenstein’s irreducibility criterion; Fields, finite fields, field extensions, algebraic extensions, algebraically closed fields 

Functional Analysis

Normed linear spaces, Banach spaces, Hahn-Banach theorem, open mapping and closed graph theorems, principle of uniform boundedness; Inner-product spaces, Hilbert spaces, orthonormal bases, projection theorem, Riesz representation theorem, spectral theorem for compact self-adjoint operators.

Numerical Analysis

Systems of linear equations: Direct methods (Gaussian elimination, LU decomposition, Cholesky factorization), Iterative methods (Gauss-Seidel and Jacobi) and their convergence for diagonally dominant coefficient matrices; Numerical solutions of nonlinear equations: bisection method, secant method, Newton-Raphson method, fixed point iteration; Interpolation: Lagrange and Newton forms of interpolating polynomial, Error in polynomial interpolation of a function;  Numerical differentiation and error, Numerical integration: Trapezoidal and Simpson rules, Newton-Cotes integration formulas, composite rules, mathematical errors involved in numerical integration formulae; Numerical solution of initial value problems for ordinary differential equations: Methods of Euler, Runge-Kutta method of order 2.

Partial Differential Equations

Method of characteristics for first-order linear and quasilinear partial differential equations; Second order partial differential equations in two independent variables: classification and canonical forms, method of separation of variables for Laplace equation in Cartesian and polar coordinates, heat and wave equations in one space variable; Wave equation: Cauchy problem and d'Alembert formula, domains of dependence and influence, non-homogeneous wave equation; Heat equation: Cauchy problem; Laplace and Fourier transform methods.

Topology

Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, quotient topology, metric topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma. 

Linear Programming

Linear programming models, convex sets, extreme points; Basic feasible solution, graphical method, simplex method, two-phase methods, revised simplex method; Infeasible and unbounded linear programming models, alternate optima; Duality theory, weak duality, and strong duality; Balanced and unbalanced transportation problems, Initial basic feasible solution of balanced transportation problems (least cost method, north-west corner rule, Vogel’s approximation method); Optimal solution, modified distribution method; Solving assignment problems, Hungarian method.

 

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