User Rating: 5 / 5

Course Code            : PMAT 32322

Title                          : Mathematical Methods

Pre-requisites          : PMAT 22293

 Learning Outcomes:

At the end of this course, the student should be able to

• classify partial differential equations using various techniques learned
• solve hyperbolic, parabolic and elliptic equations using fundamental principles
• apply range of techniques to find solutions of standard PDEs
• demonstrate accurate and efficient use of Fourier analysis techniques and their applications in the theory of PDE’s
• solve real world problems by identifying them appropriately from the perspective of partial derivative equations.

 Course Contents:

Partial Differential Equations (PDEs): Classification of PDE, First order partial differential equations: Lagrange’s method and Charpit’s method

Second order partial differential equations: Linear Partial Differential Equations with Constant Coefficients, Partial Differential Equations of Order two with Variables Coefficients, Classification of Partial Differential Equations Reduction to Canonical or Normal Form: Parabolic, elliptic and hyperbolic partial differential equations.

Fourier Series: Fourier Series, Even and Odd Functions, Half-Range Expansions, Fourier Integral, Fourier Cosine and Sine Transforms

Solutions of PDEs: Separations of variables, Fourier Transforms, Laplace Transforms, D’Alembert’s Solution of the Wave Equation

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. 2. Zill, D.G. (7th Ed., 2020). Advanced Engineering Mathematics, Jones & Bartlett Learning.
3. 3. Raisinghania, M.D. (19th Ed., 2018). Advanced Differential Equations, S.Chands, India.

User Rating: 5 / 5

Course Code            : PMAT 31312

Title                          : Abstract Algebra

Pre-requisites          : PMAT 21263

 Learning Outcomes:

At the end of this course, the student should be able to

• demonstrate factual knowledge including mathematical notation and terminology used in this course
• analyze and use basic definitions in Abstract algebra including binary operations, groups, subgroups, homomorphism, rings and ideals
• examine the fundamental principles including the laws and theorems arising from the concepts covered in this course
• develop and apply the fundamental properties of abstract algebraic structures, the substructures, their quotient structures and their mappings
• build experience and confidence in proving theorems about the structure size and nature of groups, subgroups, rings, subrings ideals and the associated mappings
• apply course materials along with techniques and procedures covered in this course to solve problems
• develop specific skills, competencies and thought processes sufficient to support further studies or work in this or related fields.

 Course Contents:

Binary Operations: Definition and properties

Groups: Definition and Examples, Basic properties, Subgroups, Cyclic Groups, Abelian Groups, Finite & Infinite Groups, order of a group, order of an element

Normal subgroups: Definition and examples, Quotient groups, Cosets, Lagrange theorem

Group isomorphism: Definition of Group Homomorphism, Kernel of a homomorphism, Image of a homomorphism, Group Isomorphism

Rings: Definition and examples, Basic properties, Subrings, Characteristic of a ring, Ideals, Integral domains; Fields.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Fraleigh, J.B. (8th Ed., 2020). First Course in Abstract Algebra, Pearson.

2. Dummit, D.S. & Foote, R.M. (3rd Ed., 2011). Abstract Algebra, Wiley.

3. Pinter, C.C. (2nd Ed., 2010). A Book of Abstract Algebra, Dover.

User Rating: 5 / 5

Course Code            : PMAT 31212

Title                          : Theory of Riemann Integration

Pre-requisites          : PMAT 12212

 Learning Outcomes:

At the end of the course the student should be able to

• demonstrate knowledge in basic mathematical concepts of calculus
• use basic mathematical concepts of calculus for further studies
• categorize ordinary differential equations
• solve linear ordinary differential equations using appropriate methods
• derive the ordinary differential equations for certain applications.

 Course Contents:

Limits and Derivatives: Limit of a Function, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity-Horizontal Asymptotes, Derivatives and Rates of Change, Derivative as a Function, Differential rules, Applications of Differentiation

Integrals: Definite Integral, Fundamental Theorem of Calculus, Substitution Rule

Applications of Integration: Areas between Curves, Volumes, Volumes by Cylindrical Shells, Arc Length, Area of a Surface of Revolution, Techniques of Integration

Ordinary Differential Equations (ODEs): Introduction to Differential Equations, ODEs, Order, Degree, classification of linear and non-linear ODEs, solution of a differential equation, Family of curves, first order ODEs: Separable ODEs, Exact ODEs, Integrating Factors, Linear ODEs, Higher Order Linear ODEs: Homogeneous Linear ODEs, Homogeneous Linear ODEs with Constant Coefficients, Differential Operators, Special types of ODEs: Bernoulli Equations, Euler–Cauchy Equations, Applications of ODEs.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
3. Stewart, J. (8th Ed., 2015). Calculus Early Transcendentals, Thomson Learning, Inc.

User Rating: 5 / 5

Course Code            : PMAT 31303

Title                          : Complex Variables

Pre-requisites          : PMAT 22293

Learning Outcomes:

At the end of this course, the student should be able to

• Perform basic mathematical operations with complex numbers in cartesian and polar form
• demonstrate knowledge of complex numbers and complex valued functions
• analyze and discuss limit, continuity and differentiability of complex valued functions
• analyze and interpret results of complex numbers and complex valued functions in applications
• evaluate real integrals using complex integrals.

Course Contents:

Complex Numbers: Basic Algebraic Properties, Exponential Form, De Movers’ Theorem, nth root of unity, Roots of Complex Numbers, Argand Diagram, Regions in the Complex Plane.

Analytic Functions: Complex Valued Functions, Limits, Theorems on Limits, Continuity and Uniform continuity, Derivatives, Differentiation Formulas, Cauchy–Riemann Equations, Sufficient Conditions for Differentiability, Analytic Functions, Harmonic Functions.

Elementary functions: Polynomial functions, rational functions, exponentials, trigonometric functions, hyperbolic functions, logarithmic functions, power series.

Integrals: Definite Integrals of Complex-Valued Function of a Complex Variable, Contour Integrals, Properties of integrals, Cauchy Theorem, Cauchy Integral Formula.

Series: Taylor Series, Laurent Series, Classification of Singular Points.

Residues and Poles: Residues, Residues at Poles, Cauchy’s Residue Theorem, Residue at Infinity.

Applications of Residues: Evaluation of Real Integrals.

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on tutorials, tests and end of course examination.

Recommended Reading:

1. Brown, J.W. & Churchill, R.V. (9th Ed., 2014). Complex variables and applications, McGraw-Hill
2. Spiegel, M., Lipschutz, S. & Schiller, J. (2nd Ed., 2009). Schaum’s Outline of Complex Variables, McGraw-Hill
3. Hann, L. & Epstein, B. (1st Ed., 1996). Classical Complex Analysis, Jones and Bartlett Publishers
4. Ponnusamy, S. (2nd Ed., 2005). Foundation of Complex Analysis, Alpha Science