Mathematical Modelling Course | IIT Roorkee | Prof. Ameeya Kumar Nayak

Course Details

Master the Art of Mathematical Modelling with IIT Roorkee's Expert-Led Course

In an increasingly data-driven world, the ability to translate real-world phenomena into precise mathematical frameworks is a superpower. Mathematical modelling sits at the heart of scientific discovery, engineering innovation, and technological advancement, particularly in the complex realm of biological and biomedical sciences. If you're an undergraduate or postgraduate student looking to build a formidable skill set in this domain, a new course offered by the prestigious Indian Institute of Technology (IIT) Roorkee presents an unparalleled opportunity.

Course Overview: Bridging Mathematics and Life Sciences

This intensive 4-week course, titled "Mathematical Modelling: Analysis and Applications," is meticulously designed to introduce the fundamental principles of creating and analyzing mathematical models for biological systems. It serves as a perfect bridge for students from diverse backgrounds—including applied mathematics, physics, chemistry, and biomedical sciences—to apply abstract mathematical concepts to tangible life science problems.

The curriculum is anchored in the study of dynamical systems, providing a robust framework for understanding how systems change over time. You will delve into both deterministic discrete-time and continuous-time models, gaining a comprehensive toolkit for tackling a wide array of scientific questions.

Learn from an Esteemed Expert: Prof. Ameeya Kumar Nayak

The course is led by Prof. Ameeya Kumar Nayak, a distinguished professor in the Department of Mathematics at IIT Roorkee. With over a decade of dedicated teaching and research experience, Prof. Nayak specializes in the numerical modeling of fluid flow problems. His research delves deep into species transport in macro and micro-scale environments, with significant applications in biomedical devices and micro-electromechanical systems (MEMS).

His academic excellence is evidenced by an impressive portfolio of over 70 peer-reviewed journal publications in top-tier outlets such as AIP, Proceedings of the Royal Society, Springer, ASME, American Chemical Society, and Elsevier. Learning from an instructor of this caliber ensures you are receiving knowledge at the very frontier of applied mathematical research.

Detailed Course Curriculum: A 4-Week Journey

The course is logically structured to build your expertise from the ground up over four weeks.

Week 1: Foundations of Discrete-Time Models

The journey begins with an overview of mathematical modelling, its types, and solution methods. You will then immerse yourself in the world of discrete-time linear models.

• Classic Models: Fibonacci's rabbit model, cell-growth models, and introductory prey-predator models.
• Analytical & Graphical Techniques: Learn stability analysis for systems of linear difference equations and interpret models using intuitive cobweb diagrams.
• Advanced Topics: Explore age-structured models (Leslie Model) and stability tests like Jury's criterion.
• Numerical Methods: Get hands-on with the Power and LR methods for finding eigenvalues.

Week 2: Navigating Non-Linear Discrete Systems

Building on week one, the focus shifts to the richer, more complex behavior of non-linear systems.

• Analyze different cell division and prey-predator models under non-linear assumptions.
• Understand stability for non-linear discrete-time models.
• Deep dive into the famous Logistic Difference Equation, exploring chaotic behavior and creating insightful bifurcation diagrams.

Week 3: Transition to Continuous-Time Models

This week marks the transition to continuous modelling, discussing the limitations of discrete models and the need for ODEs.

• ODE Fundamentals: Cover order, degree, solution techniques, and geometrical significance.
• Solution Methods: Master separation of variables, homogeneous equations, and the Bernoulli equation.
• Applied Modelling: Construct models for microbial growth and chemostats.
• System Analysis: Learn linearization methods and stability analysis for systems of ODEs.

Week 4: Advanced Continuous Models and Analysis

The final week consolidates your learning with advanced single and multi-species models.

• Study the Allee effect in single-species models.
• Employ phase diagram analysis for qualitative solutions of differential equations.
• Explore iconic ecological models: the Lotka-Volterra competition model and advanced prey-predator models.

Who Should Enroll?

Intended Audience: This course is ideal for UG and PG students from technical universities and colleges across disciplines like Mathematics, Physics, Chemistry, Engineering, and Biological Sciences.

Prerequisites: A foundational understanding of Basic Calculus and Introductory Numerical Methods is recommended to fully grasp the course content.

Essential Reference Materials

To supplement your learning, the course references seminal texts in the field:

• J.N. Kapur, Mathematical Models in Biology and Medicine
• Leah Edelstein-Keshet, Mathematical Models in Biology (SIAM)
• J.D. Murray, Mathematical Biology Vol. I & II (Springer)

Why Take This Course?

This course is more than just a series of lectures; it's a rigorous training program. You will emerge with the ability to:

• Formulate mathematical models for biological processes.
• Analyze the stability and long-term behavior of dynamical systems.
• Solve and interpret both linear and non-linear difference and differential equations.
• Use graphical tools like cobweb and phase diagrams for model analysis.
• Apply these skills to real-world problems in ecology, population biology, and biomedical engineering.

Under the guidance of a leading IIT Roorkee professor, you will gain not only theoretical knowledge but also the analytical mindset required to become a proficient mathematical modeller. Enroll today and take the first step towards mastering the language of nature through mathematics.

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