Masters Programs

MS Mathematics

Program Educational Objectives (PEOs)

  • The graduates would offer mathematical interpretation of the scientific data resulting from variety of scientific phenomena.
  • The graduates would capably employ the variety of analytical tools in support of commercial and professional research circles. 
  • The graduates would competently deliver existing knowledge in academic circles.

Program Learning Objectives (PLOs)





Mathematics and Knowledge

An ability to apply knowledge of Mathematics to address the industrial and everyday life problems.


Problem Analysis and Reasoning

An ability to survey existing literature, identification of gaps, permissible solution of problems to substantiate conclusions.


Investigation Tools

An ability to interact with methodological and computational advancements facilitating the permissible solutions.


Mathematics and Society

An ability to demonstrate the applicability of mathematical rigors in modeling of complex social and health phenomena.



An ability to communicate effectively the outcomes of Mathematical pathways.


Project Execution 

An ability to design and execute a research project as an independent researcher in a multidisciplinary environment.

Program Structure and Course Contents

MS Mathematics is spread over a minimum of 4 semesters and a maximum of 8 semesters. Each semester has 18 weeks, including one week for the mid-semester examination and one week for the final semester examination. MS Mathematics program has 32 credit hours in total, including 26 credit hours of course work and 6 credit hours for research thesis.


First Semester

CodeCourse TitleCredit Hours
 MA-5001 Riemannian Geometry  3
 MA-5002 Advanced Numerical Analysis  3
 MA-5003 Advanced Partial Differential Equations  3
 MA-5004 Fluid Mechanics  3
 Total 12

Second Semester

CodeCourse TitleCredit Hours
 MA-50XX Elective Course-I  3
 MA-50XX Elective Course-II  3
 MA-50XX Elective Course-III  3
 MA-50XX Elective Course-IV  3
 TEX-5078 Functional Textile  2
 Total 14

Third & Fourth Semester

Code Course Title Credit Hours
 MA-5090  Research Thesis  6(3+3)
 Total Credit Hours of the Programme 32


Sr. No.


Course Title

Credit Hours



Graph Theory

 3 (3, 0)



Integral Transform

3 (3,0)



Numerical Solutions of Partial Differential Equations

3 (3,0)



Compressible Fluid Flow

3 (3, 0)



Viscous Fluid Flow

3 (3, 0)




3 (3, 0)



Research Methodology


3 (3, 0)

Course Specifications

MA-5001: Riemannian Geometry

Manifolds, Differential Maps, Submanifolds, Tangents, Coordinate Vector Fields, Tangent Spaces, Dual Spaces, Multilinear Functions, Algebra of Tensors, Vector Fields, Tensor Fields, Integral Curves, Flows, Lie Derivatives, Brackets, Differential Forms, Integration Theory on Manifolds, Riemannian and Semi Riemannian Metrics, Flat Spaces, Affine Connection, Parallel Translations, Covariant Differentiation of Tensor Fields, Curvature Tensor, Torsion Tensor, Connection of a Semi-Riemannian Tensor, Killing Equation, Killing Vector Fields, Geodesics, Conformal Transformations, The Weyl Tensor.

Recommended Books:

  1. Chavel, Riemannian Geometry: A Modern Introduction, CUP, latest Edition.
  2. M. P. do-Carmo, Riemannian Geometry, Birkhauser, Boston, Latest Edition.
  3. V. Ferrari, L. Gualtieri and P. Pani, General Relativity and its Applications: Black Holes, Compact Stars, and Gravitational Waves, CRC Press, 2020.
  4. M. J. W. Hall, General Relativity: An Introduction to Black Holes, Gravitational Waves, and Cosmology,               Morgan & Claypool Publishers, 2018.
  5. J. H. Luscombe, Core Principles of Special and General Relativity, Taylor & Francis Group, 2021.

MA-5002: Advanced Numerical Analysis

Euler’s method, Improved and Modified Euler’s Method, Runge-Kutta Method, Milne’s Method, Hamming’s Methods, Initial Value Problem, Special Cases when First Derivative Missing, Boundary Value Problems, Simultaneous Algebraic Equations Method, Iterative Methods for Linear Equations, Gauss-Siedel Method, Relaxation Methods, Vector and Matrix Norms, Sequences and Series of Matrices, Graph Theory, Directed Graph of A Matrix, Strongly Connected and Irreducible Matrices, Grerschgoin Theorem, Symmetric and Positive Definite Matrices, Cyclic-Consistently Ordered Matrices, Choice of Optimum Value for Relaxation Parameter.

Recommended Books:

  1. Iserles, A first course in the Numerical Analysis of Differential Equations, Cambridge text in Applied Mathematics.
  2. J. H. Mathews and K.D. Fink, Numerical Methods using MATLAB, Prentice-Hall, Latest Edition.
  3. M. K. Jain, S. R. K. I. Yengar, R.K. Jain, Numerical methods for scientific and Engineering computations, New Age International (P), Ltd., latest Edition.
  4. W. Bohem and H. Prautzsch, Numerical Methods, A K Peters, Ltd., latest edition.

MA-5003: Advanced Partial Differential Equations

Cauchy’s Problems for Linear Second Order Equations in N-Independent Variables, Cauchy Kowalewski Theorem, Characteristics Surfaces, Adjoint Operations,  Bicharacteristics Spherical, and Cylindrical Waves, Heat  Equation, Wave Equation, Laplace Equation, Maximum- Minimum Principle, Integral Transforms.

Recommended Books:

  1. G .B.  Whitham, Linear and Nonlinear Waves, New York, NY: Wiley, latest edition.
  2. J. Kevorkian, Partial Differential Equations: Analytical Solution Techniques. Texts in Applied Mathematics, vol. 35. 2nd ed. New York, NY: Springer, latest edition.
  3. E. J. Hinch, Perturbation Methods. Cambridge Texts in Applied Mathematics. Cambridge, UK: Cambridge University Press, latest edition.
  4. R. B. Guenther, and W. John, Partial Differential Equations of Mathematical Physics and Integral Equations. New York, NY: Dover Publications, latest edition.

MA-5004: Fluid Mechanics

Navier-Stokes Equation and Exact Solutions, Dynamical Similarity and Reynold’s Number, Turbulent Flow, Boundary-Layer Concept and Governing Equations, Laminar Flat Plate, Boundary Layer, Exact Solution, Momentum, Integral Equation, Use of Momentum Integral Equation for Flow with Zero Pressure Gradient, Pressure Gradient in Boundary-Layer Flow, Reynold’s Equations of Turbulent Motion, Magnetohydrodynamics, MHD Equations, Fluid Drifts, Stability and Equilibrium Problems.

Recommended Books:

  1. F. M. White. Viscous fluid flow. McGraw-Hill latest edition.
  2. C. T. Crowe, D. F. Elger, B. C. Williams, and J. A. Roberson, Engineering fluid mechanics, latest edition, John Wiley & Sons.
  3. Y. Munson and H. J. Okiishi, Fundamentals of fluid mechanics, J. Wiley & Sons latest edition.
  4. G. K. Batchelor, An introduction to fluid dynamics. Cambridge University Press latest edition.
  5. J. Raudkivi, R. A. Callander, Advanced fluid mechanics: An Introduction, John Wiley & Sons.
  6. R. L. Daugherty, J. B. Franzini, and E. J. Finnemore, Fluid mechanics with engineering application. McGraw-Hill latest edition.

MA-5005: Advanced Graph Theory

Fundamentals of Graph Theory, Paths, Cycles, Trees, Hamilton Cycles, Euler Circuits, Planer Graphs, Flows, Connectivity, Matching Network Flows, Connectivity and Menger’s theorem, External Problems, Paths, and Complete Subgraphs, Hamilton Path and Cycles, Coloring, Vertex Coloring, Edge Coloring, Graphs on Surfaces.

Recommended Books:

  1. D. B. West, Introduction to Graph Theory, Prentice-Hall, latest edition.
  2. J. Kleinberg and E. Tardos, Algorithm Design, Addison-Wesley, latest edition.
  3. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, latest edition.
  4. R. Diestel, Graph Theory, Springer latest edition.
  5. F. Harary, Graph Theory, Narosa, latest edition.
  6. C. Berge, Graphs and Hypergraphs, North-Holland/Elsevier, latest edition.

MA-5006: Integral Transform

Laplace Transform, Applications to Integral Equations, Fourier Transforms, Fourier Sine and Cosine Transform, Inverse Transform, Applications to Differentiation, Convolutions Theorem, Applications to Partial Differential Equations, Hankel Transform and Its Applications, Applications to Integration, Mellin Transform and its Applications. 


Recommended Books:

  1. S. Howison, Practical Applied Mathematics latest edition.
  2. P. J. Collins, Differential and Integral Equations latest edition.
  3. W. E. Boyce & R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems.
  4. K. F. Riley & M. P. Hobson, Essential Mathematical Methods for the Physical Sciences, latest edition.
  5. H. A. Priestley, Introduction to Complex Analysis latest edition.
  6. L. Debnath & P. Mikusinski, Introduction to Hilbert Spaces with Applications, latest edition.

MA-5007: Numerical Solutions of Partial Differential Equations

Boundary and Initial Conditions, Polynomial Approximations in Higher Dimensions, Finite Element Method, Galerkin Method in One and More Dimensions, Error Bound on Galerkin Method, The Method of Collocation, Error Bounds on The Collocation Method.

Recommended Books:

  1. L. Burden and J. D.  Faires,  Numerical Analysis, 7th ed. Brooks/Cole, latest edition.
  2. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations, SIAM, latest edition.
  3. L. F. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with Matlab, Cambridge, latest edition.
  4. J .D.  Lambert, Numerical Methods for Ordinary Differential Systems , Wiley, latest edition.
  5. R. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, latest edition.
  6. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, latest edition.
  7. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge, latest edition.

MA-5008: Compressible Fluid Flow

Introduction to inviscid compressible flow, Concepts of thermodynamics, Types of processes, Second law of thermodynamics, Energy equation, Stream function for steady compressible flow, Velocity of sound, Mach number, Types of compressible flows, Distinction between Subsonic and Supersonic flows, Isentropic and non-isentropic inviscid compressible flow, Flow-through varying-area ducts, Normal shock waves, Prandtl relation, Fanno flow, Rayleigh flow, the Hodograph method, Introduction to viscous compressible flow, Navier-Stokes equations for a viscous compressible flow, Energy equation for a viscous compressible flow, Basic equations for three-dimensional viscous compressible flow, Exact solutions of Navier-Stokes equations for a viscous compressible flow, Boundary layer equation for two-dimensional viscous compressible flow, Momental Integral equation.

Recommended Books: 

  1.   J. D. Anderson, Modern Compressible Flow, 3rd edition, McGraw Hill, latest edition
  2. J. D. Anderson, Modern Compressible Flow with Historical Perspective, McGraw-Hill, Inc: New York,       latest edition.
  3. F. M. White, Viscous fluid flow, McGraw-Hill latest edition.
  4. C. T. Crowe, D. F. Elger, B. C. Williams, and J. A. Roberson, Engineering fluid mechanics, latest edition, John Wiley & Sons.
  5.  Y. Munson and H. J. Okiishi, Fundamentals of fluid mechanics, J. Wiley & Sons latest edition.

MA-5009: Viscous Fluid Flow

Some examples of viscous flow phenomena, properties of fluids, boundary conditions, equation of continuity, the Navier-Stokes’ equations, the energy equation; boundary conditions, orthogonal coordinate system, dimensionless parameters, velocity considerations, two-dimensional considerations, and the stream functions, Couette flows, Poissillee flow, unsteady duct flows, similarity solutions, some exact analytic solution from the paper, introduction to laminar boundary layers equations, similarity solutions, two-dimensional solutions, thermal boundary layer, some exposure will also be given from the recent literature appearing in the journals.

Recommended Books:

  1. F. M. White, Viscous fluid flow, McGraw-Hill, latest edition.
  2. P. K. Kundu, Fluid mechanics. Academic Press latest edition.
  3. H. Ockendon, Viscous flow. Cambridge University Press latest edition.
  4. G. K. Batchelor, An introduction to fluid dynamics Cambridge University Press, latest edition.
  5. Y. Munson and H. J. Okiishi, Fundamentals of fluid mechanics, J. Wiley & Sons latest edition.

MA-5010: Cosmology

Principles of Relativity: Overview of Special Relativity - spacetime interval and Lorentz metricfour vectors - Introduction to general relativity (GR) - equivalence principle - notions of curvature - gravitation as a manifestation of the curvature of spacetime - gravitational redshift and clock corrections - orbits in strong gravity, light bending and gravitational lensing - concept of horizon and ergosphere, hydrostatic equilibrium in GR - gravitational radiation. Cosmological Models: Universe at large scales – Homogeneity and isotropy – distance ladder – Newtonian cosmology - expansion and redshift - Cosmological Principle - Hubble’s law - Robertson-Walker metric - Observable quantities – luminosity and angular diameter distances - Horizon distance- Dynamics of Friedman- Robertson-Walker models: Friedmann equations for sources with p=wu and w =−1, 0, 1/3, discussion of closed, open and flat Universes.

Recommended Books:

  1. J.  V.  Narlikar,  An Introduction to Relativity, Cambridge University Press, latest edition.
  2. T. Padmanabhan, Theoretical Astrophysics: Galaxies and Cosmology, vol. 3:, Cambridge University Press, latest edition.
  3. L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, vol. 2, Oxford : Pergamon Press, latest edition.
  4. J. V. Narlikar,  Introduction to Cosmology, Cambridge University Press, latest edition.
  5. B. F. Schutz, First course in general relativity, Cambridge university press, latest edition.

RM-5011: Research Methodology

Scientific Statements, Hypothesis, Model, Theory and Law, Types of Research, Problem Definition, Objectives of Research, Research Design, Data Collection, Data Analysis, Interpretation of Results, Validation of Results, Literature Search, Formal Research Proposal, Budgeting and Funding, Sampling, Systematic Sampling, Stratified Sampling, Cluster Sampling, Convenience Sampling, Judgment Sampling, Quota Sampling, Snow  Ball Sampling, Identifying Variables of Interest and their Interactions, Operating Characteristic Curves, Power Curves, Surveys and Field Trials, Submission of a Paper, Role of Editor, Peer-Review Process, Importance of Citations, Impact Factor, Plagiarism, Protection of Research Work from Misuse.

Recommended Books:

  1. Dr. R. Kumar Research Methodology: A Step-by-Step Guide for Beginners, 2nd Edition, Sage Publica        tions latest edition.
  2. M. J. Anderson, Doe Simplified 2E: Practical Tools for Effective Experimentation, 2nd Edition, Produc   tivity Press, latest edition.
  3. M. J Anderson and P. J. Whitcomb, RSM Simplified: Optimizing Processes Using Response Surface Methods for Design of Experiments, Productivity Press, latest edition.

TEX -5078: Functional Textile

Basics of textiles and raw materials, Preparatory processes of Spinning, Types of yarns and spinning, Mathematical Modeling regarding fiber and yarn properties, Woven Fabric Production, Knitted Fabric Production, Mathematical Modeling regarding fiber, yarn, and woven fabric properties, Mathematical Modeling regarding fiber, yarn, and knitted fabric properties, Nonwoven fabric formation, and operations, Introduction to textile processing, Pretreatment and dyeing of textiles, Printing, and finishing of textiles, Application of mathematical modeling in textile processing, Clothing Product design, and development, Clothing preparatory processes, Clothing manufacturing processes, Applications of mathematical modeling in clothing.

Recommended Books:

  1. Y. Nawab, Textile Engineering an Introduction, 2016.
  2. T. Gries, D. Veit and Burkhard, Textile Technology, latest edition.
  3. B Neckar, Theory of structure and mechanics of fiber assemblies, latest edition.
  4. O. Kyosev, Topology-Based Modeling of Textile Structures and Their Joint Assemblies: Principles, Algorithms, and Limitations, 2018.

MS Thesis Evaluation

The MS thesis will only be reviewed for evaluation when the research paper is “Under Review” or “Under Consideration” by a journal.  The reviewer will be a PhD examiner of the relevant field from an external university/institute to evaluate the thesis in addition to the departmental evaluation committee. The Plagiarism test must be conducted on the dissertation before its submission to the external expert as per HEC criteria.

Carrier Opportunities for Students after Completion of the Program

Mathematics enhances the analytical skills that help in almost all disciplines of life. In addition, it helps in problem-solving, logical thinking, and decision-making skills. Thus, a mathematician can avail several opportunities in data sciences, artificial intelligence, and areas related to research and development in engineering and science.  Jobs directly related to your degree include:

  1. Research scientist in strategic organizations
  2. Teaching and research
  3. Data scientist
  4. Data analyst
  5. Numerical Analyst
  6. Technical Programmer
  7. Investment analyst
  8. Astronomer
  9. Computational Fluid Dynamics

  1. MSc/BS in Pure Mathematics/Applied Mathematics/Computational Mathematics (minimum 16-year education) or equivalent degree with minimum CGPA 2.00/4.00 in semester system or 60% in annual system/term system from an HEC recognized institute/university.
  2. The applicant must pass NTS/NTU-GAT (General) test with minimum 50/100 marks prior to apply (please see Test Banner for more information on main page of this website).
  3. The applicant must not be already registered as a student in any other academic program in Pakistan or abroad.
  4. Result waiting applicants may apply for admission, however their merit will be finalized only on submission of final BS/M.Sc or equivalent official transcript or degree.
  5. Relevant Admission Committee will determine relevancy of terminal degree and decide deficiency course/s (if any) at the time of admission interview, the detail of which will be provided to the student in his/her admission letter/email.  
  6. Deficiency course/s will be treated as non-credit and qualifying course/s for which student will also pay extra dues as per fee policy. Those course/s will neither be mentioned in student’s final transcript nor will be included for calculation of CGPA. However, the student may obtain his/her a separate transcript for completion of deficiency course/s.

NoteThe student will submit his/her publication from his/her thesis research work and submit to his/her supervisor. Final defense will be held after the submitted publication of student will be notified as “Under Review” or “Under Consideration” by a journal. It will be compulsory for graduate student to include his/her Supervisor’s name in his/her publication.

Merit Criteria

Admission merit will be prepared according to the following criteria:

 BS/MSc or Equivalent  60% weightage
 NTS-GAT (General) Test  30% weightage
 Interview  10% weightage


Fee Head1st 2nd3rd4th
Admission Fee (Once) 25000 - - -
Certificate Verification Fee (Once) 2000 - - -
University Security (Refundable) 5000 - - -
Red Crescent  Donation (Once) 100 - - -
University Card Fee (Once) 300 - - -
Degree Fee (Once) - - - 5000
Tuition Fee (Per Semester) 30,000 30,000 21,000 21,000
Library Fee (Per Semester) 3000 3000 3000 3000
Examination Fee (Per Semester) 3000 3000 3000 3000
Medical Fee (Per Semester) 2000 2000 2000 2000
Student Activity Fund (Per Semester) 2000 2000 2000 2000
Endowment Fund (Per Semester) 1000 1000 1000 1000
TOTAL 73,400 41,000 32,000 37,000