TEW Differential Equations, Mathematical Modelling and Analysis (2018)

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Prof. I. K. Rana from IIT Bombay started his lecture with emphasis on the historical importance of Real Analysis and its evolution. In his series of lectures, he discussed development of numbers and importance of Pigeon Hole Principle. He also discussed irrationality of V2 and gave many different proofs of it. Also he discussed how to approximate v2. On other day he talked about i. Prof. I. K. Rana discussed a geometrical approach to approximate y using sequences. He also discussed intrinsic property of sequence and Cauchy sequence. In his concluding lecture, he discussed nested interval property, model of growth and decay, limit and continuity, properties of continuous functions, tangent to the curve, and differentiability.

Second lecture was on metric space and we are fortunate to have Dr. Tanvi Jain from ISI Delhi to enlighten us with the concepts of arithmetic mean, Frechet mean of m points. She took some informative examples of metric spaces like metric on Hyperspace, Hausdorff metric on CL(X), metric induced by operator norm, Bures-Wasserstein metric, orthogonal Procrustes problem, Wasserstein mean, Polar decomposition. In her subsequent lectures she discussed compactness, sequentially compact and limit point compactness. To make her talk more informative she discussed tutorial sheet comprising of variety of problems. In her last lecture, she discussed the proofs of some famous theorems like Arzela-Ascoli theorem, Implicit function theorem, Picard's existence theorem, Peano's theorem, Banach fixed point theorem and its applications, Fredholm's second integral equations and linear Volterra integral equations and Stone-Weierstrass theorem.

Prof. V. D. Sharma from IIT Bombay delivered a beautiful lecture series on partial differential equations and its geometrical interpretation. He also discussed quasilinear and non-linear partial differential equations with examples. Further he discussed Cauchy's problem and wave equation with their properties. On other day, he delivered a lecture on first order and second degree ODEs. He discussed linear and non-linear ODEs. Further, he explained various methods of solving first order and second degree ODES like variable separable, integrating factor method and he also talked about exact differential equations. In the series of his lectures, he defined different types of solutions of ODEs as general solution, singular solution and particular solution. Subsequently, he discussed envelope of family of solutions as singular solutions and other geometrical interpretation of solutions of ODEs. On his last lecture, he discussed Initial Value Problem (IVP) and Boundary Value Problem (BVP), asymptotic behaviour of IVP and BVP and their applications in real world.

Prof. Mani Mehra from IIT Delhi delivered the lecture on partial differential equations (PDEs). On her series of lectures, she discussed the classification of second-order partial differential equations in two independent variables. Further, she discussed several physical models (heat equations, wave equations and potential equations) to explain the partial differential equations. PDEs are classified into elliptic, parabolic and hyperbolic. This classification is defined by transforming the PDEs into canonical forms. Further, she talked about partial differential equations in n independent variables and discussed its classification using eigenvalue of the matrix associated with PDEs.

In her subsequent lecture, she discussed the existence, uniqueness, and stability of the partial differential equations. She defined well-posed and ill-posed problems. PDEs are called as well-posed if i. solution exists ii. solution should be unique, and iii. solution depends continuously on initial and boundary data of the problems This is called Hadamard principle of well-posedness of partial differential equations. She elaborated some ill-posed problems and explained its occurrence in nature as Inverse problems. She explains the difficulty in dealing with ill-posed problems. Well-posed are nice to handle and have nice stability features. Several numerical methods are available in the literature to deal with such problems. In her last talk, she discussed the analytical behaviour of heat equations. She explained the finite difference approximation of heat problems. She defined the consistency, stability to define the convergence of finite difference scheme. She uses Fourier's method to show the stability of this problem. At last, she explained the various norms to measure the error while discretizing the continuous problems. She defined maximum norm I. and l, norms and their importance.

Dr. Vivek from DTU delivered useful and informative lectures and lab sessions on "Maxima". He enlightened us with the basic commands of Maxima and their uses in solving ODE/PDE. Then he continued his lecture with the discussion of various famous problems and models like Logistic equation, Lake pollution problem, SIR model, SI model and SEIR model.

Dr. Mukund Madhav Mishra from Hansraj College of University of Delhi who discussed the existence and convergence of Fourier series involving! the notion of Weierstrass approximation theorem, Taylor series, Riesz product, Convolution and Fejer's theorem. In his second lecture, he discussed various problems of mathematics with the audience.

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